Conjectures and results on modular representations of \mathrm{GL}_n(K) for a p-adic field K
Christophe Breuil, Florian Herzig, Yongquan Hu, Stefano Morra, Benjamin Schraen
aa r X i v : . [ m a t h . N T ] F e b Conjectures and results on modular representationsof GL n ( K ) for a p -adic field K Christophe Breuil ∗ Florian Herzig † Yongquan Hu ‡ Stefano Morra § Benjamin Schraen ¶ Abstract
Let p be a prime number and K a finite extension of Q p . We state conjec-tures on the smooth representations of GL n ( K ) that occur in spaces of mod p automorphic forms (for compact unitary groups). In particular, when K isunramified, we conjecture that they are of finite length and predict their inter-nal structure (extensions, form of subquotients) from the structure of a certainalgebraic representation of GL n . When n = 2 and K is unramified, we proveseveral cases of our conjectures, including new finite length results. Contents ∗ CNRS, Bâtiment 307, Faculté d’Orsay, Université Paris-Saclay, 91405 Orsay Cedex, France † Dept. of Math., Univ. of Toronto, 40 St. George St., BA6290, Toronto, ON M5S 2E4, Canada ‡ Morningside Center of Math., No. 55, Zhongguancun East Road, Beijing, 100190, China § Lab. d’Analyse, Géométrie, Algèbre, 99 Av. Jean Baptiste Clément, 93430 Villetaneuse, France ¶ Bâtiment 307, Faculté d’Orsay, Université Paris-Saclay, 91405 Orsay Cedex, France Local-global compatibility conjectures 19 D ∨ ξ H and V H . . . . . . . . . . . . . . . . . . . . 202.1.2 Global setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.1.3 Weak local-global compatibility conjecture . . . . . . . . . . . 292.1.4 A reformulation using C -groups . . . . . . . . . . . . . . . . . 322.2 Good subquotients of L ⊗ . . . . . . . . . . . . . . . . . . . . . . . . . 392.2.1 Definition and first properties . . . . . . . . . . . . . . . . . . 392.2.2 The parabolic group associated to an isotypic component . . . 422.2.3 The structure of isotypic components of L ⊗ . . . . . . . . . . 472.2.4 From one isotypic component to another . . . . . . . . . . . . 542.3 Good conjugates of ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.3.1 Some preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 582.3.2 Good conjugates of a generic ρ . . . . . . . . . . . . . . . . . 612.4 The definition of compatibility . . . . . . . . . . . . . . . . . . . . . . 652.4.1 Compatibility with e P . . . . . . . . . . . . . . . . . . . . . . . 652.4.2 Compatibility with ρ . . . . . . . . . . . . . . . . . . . . . . . 752.4.3 Explicit examples . . . . . . . . . . . . . . . . . . . . . . . . . 842.5 Strong local-global compatibility conjecture . . . . . . . . . . . . . . 94 GL ( Q p f ) ϕ, O × K )-modules and ( ϕ, Γ)-modules . . . . . . . . . . . . . . . . . . 993.1.1 The ring A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.1.2 Multivariable ( ψ, O × K )-modules . . . . . . . . . . . . . . . . . 1053.1.3 Multivariable ( ϕ, O × K )-modules . . . . . . . . . . . . . . . . . . 1143.1.4 An upper bound for the ranks of D A ( π ) ´et and D ∨ ξ ( π ) . . . . . 1193.2 Tensor induction for GL ( Q p f ) . . . . . . . . . . . . . . . . . . . . . . 1212.2.1 Lower bound for V GL ( π ): statement . . . . . . . . . . . . . . 1223.2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 1243.2.3 A computation for the operator F . . . . . . . . . . . . . . . . 1303.2.4 Lower bound for V GL ( π ): proof . . . . . . . . . . . . . . . . . 1393.3 On the structure of some representations of GL ( K ) . . . . . . . . . . 1473.3.1 Combinatorial results . . . . . . . . . . . . . . . . . . . . . . . 1483.3.2 On the structure of gr( π ∨ ) . . . . . . . . . . . . . . . . . . . . 1533.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1563.3.4 Characteristic cycles . . . . . . . . . . . . . . . . . . . . . . . 1603.3.5 On the length of π in the semisimple case . . . . . . . . . . . 1653.4 Local-global compatibility results for GL ( Q p f ) . . . . . . . . . . . . 1703.4.1 Global setting and results . . . . . . . . . . . . . . . . . . . . 1713.4.2 Review of patching functors . . . . . . . . . . . . . . . . . . . 1733.4.3 Direct sums of diagrams . . . . . . . . . . . . . . . . . . . . . 1753.4.4 Local-global compatibility results . . . . . . . . . . . . . . . . 180 References 183 Introduction
Let p be a prime number and K a local field of residue characteristic p . In theearly nineties, Barthel and Livné had the fancy idea to start classifying irreducible(admissible) smooth representations of GL ( K ) over an algebraically closed fieldof characteristic p ([BL94], [BL95]). They found four nonempty distinct classes ofsuch representations: 1-dimensional ones, irreducible principal series, special series,and those which are not an irreducible constituent of a principal series that theycalled supersingular. In 2001, one of us classified supersingular representations ofGL ( Q p ) with a central character ([Bre03a]) and showed that they are in “natural”bijection with 2-dimensional irreducible representations of Gal( Q p / Q p ) in character-istic p . This was one of the starting points of the mod p and p -adic Langlandsprogrammes for GL ( Q p ), which was developed essentially during the decade 2000-2010 (see for instance [Bre03b], [Bre10], [Eme10b], [Kis10], [Col10], [Ber10], [Paš13],[Emea], [CDP14], [CEG + p local Langlands correspondencefor GL ( Q p ) (compared to previous Langlands correspondences). The first one is thatit involves reducible representations of GL ( Q p ). More precisely, the representationof GL ( Q p ) is irreducible (resp. semisimple, resp. indecomposable) if and only if itscorresponding 2-dimensional representation of Gal( Q p / Q p ) is, and, in the reduciblecase, is given (at least generically) by an extension between two specific principalseries. The second one, found by Colmez in [Col10], is that the correspondence can bemade functorial by an exact functor from finite length representations of GL ( Q p ) toétale ( ϕ, Γ)-modules, i.e. to finite length representations of Gal( Q p / Q p ) by Fontaine’sequivalence. Thanks to this exact functor, one can extend the correspondence firstto extensions of representations, and then to deformations on both sides.When K is not Q p , trouble comes from supersingular representations. Contraryto the case K = Q p , they can be more numerous than 2-dimensional irreducible rep-resentations of Gal( K/K ) ([BP12]) and they cannot be described as quotients of acompact induction by a finite number of equations ([Hu12, Cor.5.5], [Sch15, Thm.0.1],[Wu, Thm.1.1]), justifying a posteriori the terminology “very strange” that was usedto describe them in the introduction of [BL95]. As a consequence, no classification ofsupersingular representations of GL ( K ) is known so far, which has hitherto made im-possible to find a definition of a hypothetical local mod p correspondence for GL ( K )by purely local (either representation theoretic or geometric) means.Fortunately, the global theory comes to the rescue. If a local correspondenceexists, there is a place where it should be realized: the mod p cohomology of Shimuravarieties. Let us assume now that K is a finite unramified extension of Q p with residue4eld F p f and let K def = 1 + pM ( O K ) ⊆ GL ( O K ). Following the pioneering work of[BDJ10] on Serre weight conjectures, a series of articles ([BP12], [EGS15], [HW18],[LMS], [Le19]) led to a complete description of the K -invariants of the GL ( K )-representations carried by Hecke isotypic subspaces in such mod p cohomology groups.Although these invariants are only a tiny piece of the representations of GL ( K ),combined with weight cycling this turned out to give a strong hint on the form ofthese representations, as well as being a useful technical result. Indeed, very recently,building on this description and on results of [BHH + ], Hu and Wang could provethat, at least when K is quadratic unramified and the representation of Gal( Q p /K )is a nonsplit extension between two (sufficiently generic) characters, these GL ( K )-representations are indecomposable of length 3 (in particular are of finite length),with the similar principal series as in the case K = Q p in socle and cosocle, and asupersingular representation “in the middle” ([HW, Thm.10.37]).These recent results maintain the hope of a local Langlands correspondence forGL ( K ). They also prompted us to make public some conjectures we had in mind formany years on the form of the GL n ( K )-representations carried by Hecke isotypic sub-spaces, and on a functorial link to representations of Gal( Q p / Q p ) via ( ϕ, Γ)-modules.We state such conjectures in the present work (Conjecture 2.1.3.1, Conjecture 2.1.4.5,Conjecture 2.5.1) and we prove some special cases in the case n = 2 and K unramified,including some new finite length results (Theorem 3.4.4.3, Theorem 3.4.4.6, Corollary3.4.4.7). Moreover, when n = 2 and K is unramified, we also define (and use in theproofs!) an abelian category C of smooth admissible representations of GL ( K ) over F (containing the representations coming from the global theory) together with anexact functor from C to a new category of multivariable ( ϕ, Γ)-modules.
Let us first describe our conjectures with some details. As usual, we mostly work inthe setting of compact unitary groups (except in §2.1.4), so that we do not (yet) mixdelicate representation theoretic issues with difficult geometric problems (ultimately,we think that the representations of GL n ( K ) should not change from one globalsetting to another). We fix F a CM-field, i.e. a totally imaginary quadratic extensionof a totally real number field F + , and we assume for simplicity in this introduction that p is inert in F + . We also assume (not for simplicity) that the unique p -adic place v of F + splits in F . We fix a continuous absolutely irreducible representation r : Gal( F /F ) −→ GL n ( F ) , where F is a (sufficiently large) extension of F p and we assume that r is automorphicfor a unitary group H over F + that is compact at all infinite places and becomes GL n over F . Equivalently there exists a compact open subgroup U v ⊆ H ( A ∞ ,vF + ) such that S ( U v , F )[ m ] def = { f : H ( F + ) \ H ( A ∞ F + ) /U v → F locally constant } [ m ] = 0 , m ] means the Hecke-isotypic subspace associated to r (one has to choose afinite set of bad places Σ in the definition of m , but we forget this issue here, see§2.1.3 below).Let ˜ v | v in F , K def = F ˜ v the corresponding completion and r ˜ v the restriction of r to a decomposition subgroup at ˜ v . Then S ( U v , F )[ m ] is an admissible smoothrepresentation of GL n ( K ) over F by the usual right translation action on functions.Our main conjecture gives the form of this GL n ( K )-representation (assuming it isof finite length) as well as a functorial link to r ˜ v . But to state it we need a fewpreliminaries on certain algebraic representations of GL n over F .Let us first assume for simplicity that K = Q p . We let Std be the standard n -dimensional algebraic representation of GL n over F and define the following algebraicrepresentation of GL n over F : L ⊗ def = n − O i =1 ^ i F Std . We fix P ⊆ GL n a parabolic subgroup containing the Borel B of upper-triangularmatrices, and let M P be its Levi subgroup containing the torus T of diagonal matrices.We fix e P ⊆ P a Zariski closed algebraic subgroup containing M P and we consider thealgebraic representation L ⊗ | e P of e P over F . Definition 1.2.1 (Definition 2.2.1.3) . A subquotient of L ⊗ | e P is a good subquotientif its restriction to the center Z M P of M P is a (direct) sum of isotypic components of L ⊗ | Z MP .Note that an isotypic component of L ⊗ | Z MP carries an action of M P (Lemma2.2.1.2). Hence, viewing an isotypic component of L ⊗ | Z MP as a representation of e P via the surjection e P ։ M P , one can see L ⊗ | e P as a successive extension of suchisotypic components (Lemma 2.2.1.5). On the GL n ( Q p )-side, the isotypic compo-nents of L ⊗ | Z MP will play the role of irreducible constituents. Note that the isotypiccomponents of L ⊗ | Z MP are by definition all distinct.To an isotypic component C of L ⊗ | Z MP , we associate a parabolic subgroup P ( C )of GL n containing B as follows. Let λ ∈ X ( T ) = Hom Gr ( T, G m ) be any weight suchthat C is the isotypic component of λ | Z MP and define (see (36)) λ ′ def = 1 | W ( P ) | X w ′ ∈ W ( P ) w ′ ( λ ) ∈ X ( T ) ⊗ Z Q , where W ( P ) is the Weyl group of M P . Let θ be the highest weight of L ⊗ | T and w in the Weyl group of GL n such that w ( λ ′ ) is dominant with respect to B . Then one6an check that (see Proposition 2.2.2.6) θ − w ( λ ′ ) = X α ∈ S n α α, where S is the set of simple roots of GL n (with respect to B ) and the n α are in Q ≥ . Then P ( C ) is by definition the parabolic subgroup of GL n corresponding to thesubset { α ∈ S, n α = 0 } of S . We denote by P ( C ) − its opposite parabolic subgroup.We now go back to the above global setting. Assuming a weak genericity conditionon r ˜ v , one can replace r ˜ v by a suitable conjugate so that the image of r ˜ v is containedin the F -points of a Zariski closed algebraic subgroup e P r ˜ v of a parabolic P r ˜ v as abovewhich is “as small as possible” (see Definition 2.3.2.2 and Theorem 2.3.2.5). Thefollowing conjecture is part of Conjecture 2.5.1 (see Definition 2.4.2.7 and Definition2.4.1.5). Conjecture 1.2.2.
Assume that r ˜ v has distinct irreducible constituents and that theratio of any two -dimensional constituents is not in { ω, ω − } , where ω is the mod p cyclotomic character. Then we have a GL n ( Q p ) -equivariant isomorphism for someinteger d ≥ : S ( U v , F )[ m ] ∼ = (cid:16) Π ˜ v ⊗ ( ω n − ◦ det) (cid:17) ⊕ d , where Π ˜ v is an admissible smooth representation of GL n ( Q p ) over F of finite lengthwith distinct irreducible constituents such that there exists a bijection Φ between the ( finite ) set of subquotients of Π ˜ v and the ( finite ) set of good subquotients of L ⊗ | e P r ˜ v satisfying the following properties: (i) Φ respects inclusions, and thus extends to a bijection between the sets of allsubquotients on both sides; (ii) Φ − sends an isotypic component C of L ⊗ | Z MPr ˜ v to an irreducible constituent of Π ˜ v of the form Ind GL n ( Q p ) P ( C ) − ( Q p ) π ( C ) , where π ( C ) is a supersingular representationof M P ( C ) ( Q p ) over F . When K is not necessarily Q p , the conjecture is completely analogous, defining L ⊗ by L ⊗ def = O Gal( K/ Q p ) (cid:18) n − O i =1 ^ i F Std (cid:19) , replacing e P by e P Gal( K/ Q p ) def = e P × · · · × e P | {z } Gal( K/ Q p ) and taking isotypic components of L ⊗ | Z MP for the diagonal embedding Z M P ֒ → Z Gal( K/ Q p ) M P in the definition of good subquotientsof L ⊗ | e P Gal( K/ Q p ) . 7 xample 1.2.3. (i) If r ˜ v is irreducible, then e P r ˜ v = GL n = M P r ˜ v and there is only oneisotypic component C in L ⊗ | Z GL n . It is such that P ( C ) = GL n : the representationΠ ˜ v in Conjecture 1.2.2 is irreducible and supersingular.(ii) If r ˜ v is semisimple, then e P r ˜ v = M P r ˜ v , and since the direct sum decompositionof L ⊗ | Z MPr ˜ v into isotypic components for the (diagonal) Z M Pr ˜ v -action is a direct sumdecomposition as a e P r ˜ v = M P r ˜ v -representation, we see that the representation Π ˜ v inConjecture 1.2.2 is also semisimple.(iii) If K = Q p and n = 2, we have L ⊗ = Std. When r ˜ v is irreducible, by (i) the rep-resentation Π ˜ v of GL ( Q p ) in Conjecture 1.2.2 is supersingular. When r ˜ v is reduciblesplit, then e P r ˜ v = T = M P r ˜ v , and L ⊗ | T = F λ ⊕ F λ , where λ i : diag( x , x ) x i , i ∈ { , } . There are two isotypic components C = F λ or C = F λ , both with P ( C ) = B : the representation Π ˜ v in Conjecture 1.2.2 is a direct sum of two irre-ducible principal series. Finally, when r ˜ v is reducible nonsplit, then e P r ˜ v = B , L ⊗ | B isa nonsplit extension of F λ by F λ and Π ˜ v is a nonsplit extension between two irre-ducible principal series. Note that Conjecture 1.2.2 is known in that case ([CS17b],[CS17a] for r ˜ v irreducible, [BD20, Cor.7.40] for arbitrary r ˜ v , all generalizing methodsof [Emea]).(iv) For K arbitrary (unramified) and n = 2, see Example 2.2.2.9 and Example 1 of§2.4.3.Conjecture 1.2.2 only gives part of the picture. For instance there should bereducible subquotients of Π ˜ v which are also parabolic inductions Ind GL n ( Q p ) P ( C ) − ( Q p ) π ( C )with π ( C ) of the form π ( C ) ∼ = π ( C ) ⊗ · · · ⊗ π d ( C ), where the (reducible) π i ( C )have themselves the same form as Π ˜ v but for the smaller GL n i ( K ) appearing inthe Levi M P ( C ) ( K ) (which gives a “fractal” flavour to the whole picture!). In fact,it is possible that, in the end, this “fractal” picture will automatically follow fromproperty (ii) in Conjecture 1.2.2 (i.e. from the statement for irreducible subquotientsonly), as one can already see in many of the examples of §2.4.3 using the work ofHauseux ([Hau18], [Hau19]), see Remark 2.4.1.6(iv). Also some parabolic (possiblyreducible) inductions as above should be deduced from others by a permutation onthe factors π i ( C ). Tracking down all these internal symmetries (with the varioustwists by characters that occur) and all the implications between them is not reallydifficult but a bit tedious, as the reader will see from the technical lemmas in §2.4.1(see e.g. Proposition 2.4.1.8). The interested reader should maybe first have a lookat the various examples in §2.4.3 before going into the full combinatorics.Finally, the full picture has to take into account the Galois action. There is asimple way to extend Colmez’s functor from representations of GL ( Q p ) to represen-tations of GL n ( K ) that we recall now (see [Bre15] or §2.1.1). Let ξ : G m → T be thecocharacter x diag( x n − , x n − , . . . ,
1) and N def = Ker( N ℓ −→ O K trace −→ Z p ), where N is the unipotent radical of B ( O K ) and the map ℓ is the sum of the entries on8he first diagonal (following the notation of [SV11]). Let π be a smooth representa-tion of GL n ( K ) over F and endow the algebraic dual ( π N ) ∨ of π N with the residual F J N /N K ∼ = F J X K -module structure (where X def = h(cid:16) (cid:17)i − Z × p andan endomorphism ψ which commutes with the Z × p -action by ( xf )( v ) def = f ( ξ ( x − ) v ) , x ∈ Z × p , f ∈ ( π N ) ∨ , v ∈ π N ψ ( f )( v ) def = f (cid:16) P N /ξ ( p ) N ξ ( p ) − n ξ ( p ) v (cid:17) , f ∈ ( π N ) ∨ , v ∈ π N . Then one defines a covariant left exact functor V from the category of smooth repre-sentations of GL n ( K ) over F to the category of (filtered) direct limits of continuousfinite-dimensional representations of Gal( Q p / Q p ) over F by V ( π ) def = (cid:16) lim −→ D V ∨ ( D ) (cid:17) ⊗ δ, (1)where the inductive limit is taken over the continuous morphisms of F J X K -modules h : ( π N ) ∨ → D , where D is an étale ( ϕ, Γ)-module of finite rank over F (( X )) and h intertwines the actions of Z × p (recall Γ ∼ = Z × p ), commutes with ψ and is surjectivewhen tensored by F (( X )). (Here V ∨ is Fontaine’s contravariant functor associating arepresentation of Gal( Q p / Q p ) to D and recall that any étale ( ϕ, Γ)-module is endowedwith an endomorphism ψ which is left inverse to the Frobenius ϕ .) In (1), δ is acertain power of ω which is here for normalization issues (see Example 2.1.1.3, seealso the end of §2.1.4). In general, one doesn’t know when V ( π ) is nonzero or if it isfinite-dimensional.Using (1), one can strengthen Conjecture 1.2.2 (when K = Q p ) so that it takesinto account the action of Gal( Q p / Q p ) as follows. Conjecture 1.2.4 (see Definition 2.4.1.5 and Conjecture 2.5.1) . There is a bijection Φ as in Conjecture 1.2.2 which moreover commutes with the action of Gal( Q p / Q p ) inthe following sense: for each subquotient Π ′ ˜ v of Π ˜ v one has V (Π ′ ˜ v ) = Φ(Π ′ ˜ v ) ◦ r ˜ v ( recallthat Φ(Π ′ ˜ v ) is an algebraic representation of e P r ˜ v over F and that r ˜ v takes values in e P r ˜ v ( F )) . If K is not necessarily Q p , then by definition Φ(Π ′ ˜ v ) is an algebraic representationof e P Gal( K/ Q p ) r ˜ v and there is a completely analogous conjecture replacing Φ(Π ′ ˜ v ) ◦ r ˜ v byΦ(Π ′ ˜ v ) ◦ ( r σ ˜ v ) σ ∈ Gal( K/ Q p ) , which is again a representation of Gal( Q p / Q p ).In particular the functor V , when applied to Π ˜ v and its subquotients Π ′ ˜ v , shouldbehave like an exact functor. Note that Conjecture 1.2.4 is known when K = Q p and n = 2 by the same references as in Example 1.2.3(iii). In the special case Π ′ ˜ v = Π ˜ v ,Conjecture 1.2.4 implies in particular Conjecture 1.2.5 (Conjecture 2.1.3.1) . The functor V induces an isomorphism V (cid:16) S ( U v , F )[ m ] ⊗ ( ω − ( n − ◦ det) (cid:17) ∼ = (cid:18) ind ⊗ Q p K (cid:16) n − O i =1 ^ i F r ˜ v (cid:17)(cid:19) ⊕ d here ind ⊗ Q p K is the tensor induction from Gal( Q p /K ) to Gal( Q p / Q p ) . The statement in Conjecture 1.2.5 makes sense even if K is ramified, and weconjecture it for an arbitrary finite extension K of Q p and an arbitrary representation r ˜ v (see Conjecture 2.1.3.1). In fact, using C -parameters ([BG14]), it can even beformulated in a more intrinsic way and in a more general global setting, see Conjecture2.1.4.5. Remark 1.2.6.
Assuming K = Q p , the first appearance of the Gal( Q p / Q p )-represen-tation on the right-hand side of the isomorphism in Conjecture 1.2.5 is in [BH15],where its “ordinary part” was related to the “ordinary part” of S ( U v , F )[ m ] (seeTheorem 2.5.9 for an improvement). Note that the algebraic representation L ⊗ ofGL n is not irreducible for n >
2. One could have thought about using the irreduciblealgebraic representation of GL n of highest weight θ instead of the reducible L ⊗ tomake predictions (at least for p big enough the latter strictly contains the former asa direct factor). However, we chose the representation L ⊗ . One reason is that itcan also be seen as a representation of GL n × · · · × GL n ( n − L ⊠ def = ⊠ n − i =1 V i F Std – and one can hope tostate a stronger variant of Conjecture 1.2.4 replacing L ⊗ by L ⊠ and Φ(Π ′ ˜ v ) ◦ r ˜ v byΦ(Π ′ ˜ v ) ◦ ( r ˜ v , r ˜ v , . . . , r ˜ v ) (see [Záb18b], [Záb18a] where such a possibility is mentioned).However one has to be careful with defining a “multivariable” functor V in thatcontext (there is a tentative definition in [Záb18b] when K = Q p generalizing (1), butsee Remark 3.1.2.12 when n = 2 and K = Q p ).If a representation Π ˜ v as in Conjecture 1.2.4 exists, we do hope that it will realizea mod p local Langlands correspondence for GL n ( K ). Let us now describe our main results when n = 2 and K = Q p f is unramified. Fora finite place ˜ w of F we denote by R (cid:3) r ˜ w the (unrestricted) framed deformation ringof r ˜ w def = r | Gal( F ˜ w /F ˜ w ) over W ( F ). We let I K ⊆ Gal( Q p /K ) be the inertia subgroupand ω f ′ for f ′ ∈ { f, f } be Serre’s fundamental character of level f ′ . We make thefollowing extra assumptions on F , H , r and U v = Q w = v U w (recall we assumed p inertin F + for simplicity):(i) F/F + is unramified at all finite places of F + ;(ii) H is quasi-split at all finite places of F + ;(iii) r | Gal(
F /F ( p √ is adequate ([Tho17, Def.2.20]);10iv) r ˜ w is unramified if ˜ w | F + is inert in F ;(v) R (cid:3) r ˜ w is formally smooth over W ( F ) if r ˜ w is ramified and ˜ w | F + = v ;(vi) r ˜ v | I K is, up to twist, of one of the following forms: r ˜ v | I K ∼ = ω ( r +1)+ ··· + p f − ( r f − +1) f
00 1 ! ,r ˜ v | I K ∼ = ω ( r +1)+ ··· + p f − ( r f − +1)2 f ω p f (same)2 f , where the r i satisfy the following bounds: ( max { , f − } ≤ r j ≤ p − max { , f +2 } if j > r ˜ v is reducible,max { , f } ≤ r ≤ p − max { , f +1 } if r ˜ v is irreducible; (2)(vii) U w is maximal hyperspecial in H ( F + w ) if w is inert in F .(We also need to fix a place v which splits in F , where nothing ramifies and U v iscontained in the Iwahori subgroup at v , we forget that here along with the set Σ ofbad places and the definition of the ideal m .) Theorem 1.3.1 (Theorem 3.4.4.3) . Assume (i) to (vii), then Conjecture 1.2.5 holdsfor n = 2 , K unramified and ρ semisimple. We sketch the proof of Theorem 1.3.1. We denote by I the pro- p Iwahori subgroupin GL ( O K ) and set ρ def = r ˜ v (1) Π def = S ( U v , F )[ m ] . Note that the central character of Π is det( ρ ) ω − (Lemma 2.1.3.3). There are twomain steps in the proof which involve quite different arguments:(i) one proves a Gal( Q p / Q p )-equivariant injection (ind ⊗ Q p K ρ ) ⊕ d ֒ → V (Π);(ii) one proves dim F V (Π) ≤ f d (= dim F (ind ⊗ Q p K ρ ) ⊕ d ).We first sketch the proof of (i). Arguing as in the proof of [BHH + , Thm.1.2(i)],there is an integer d ≥ ( O K ) K × -equivariant isomorphism Π K ∼ = D ( ρ ) ⊕ d ,where D ( ρ ) is defined as in [BP12, §13] (see Corollary 3.4.2.2). Taking into accountthe action of (cid:16) p (cid:17) on Π I ⊆ Π K , one can promote this isomorphism to an isomor-phism of diagrams : 11 heorem 1.3.2 ([DL, Thm.1.3] when d = 1, Theorem 3.4.1.1 when d > . Thereis a diagram D ( ρ ) = ( D ( ρ ) ֒ → D ( ρ )) only depending on ρ such that one has anisomorphism of diagrams: D ( ρ ) ⊕ d ∼ = (Π I ֒ → Π K ) . Theorem 1.3.2 can actually be made stronger, i.e. one can show that certainconstants ν i ∈ F × associated to the weight cycling on D ( ρ ) ∼ = D ( ρ ) I as in [Bre11,§6] (up to suitable normalization) are as predicted in [Bre11, Thm.6.4]. When d = 1,Theorem 1.3.2 (and its strengthening) is entirely due to Dotto and Le ([DL, Thm.1.3]).When d >
1, we check from their proof that the action of (cid:16) p (cid:17) on Π I ∼ = ( D ( ρ ) I ) ⊕ d “respects” each copy of D ( ρ ) I . Note that Theorem 1.3.2 holds under much weakerbounds on the r i than the bounds (2), see §3.4.1.Then item (i) above follows from the following purely local result. Theorem 1.3.3 (Theorem 3.2.1.1) . Let π be an ( admissible ) smooth representation of GL ( K ) over F such that one has an isomorphism of diagrams D ( ρ ) ⊕ d ∼ = ( π I ֒ → π K ) .Then one has a Gal( Q p / Q p ) -equivariant injection (ind ⊗ Q p K ρ ) ⊕ d ֒ → V ( π ) . The proof of Theorem 1.3.3 is a long and technical computation of ( ϕ, Γ)-modulesthat is given in §3.2. It uses the previous computations in [Bre11] and the bounds(2) (though one can slightly weaken them, see (127)).We now sketch the (longer) proof of (ii). We let Z be the center of I (orof K ) and m I /Z the maximal ideal of the Iwasawa algebra F J I /Z K . The mainidea is to focus on the structure of the (algebraic) dual π ∨ as F J I /Z K -moduleand to use the results of [BHH + ]. Recall that the graded ring gr( F J I /Z K ) forthe m I /Z -adic filtration (we use the normalization of [LvO96, §I.2.3]) is not com-mutative, but contains a regular sequence of central elements ( h , . . . , h f − ) suchthat R def = gr( F J I /Z K ) / ( h , . . . , h f − ) is a commutative polynomial algebra in 2 f variables F [ y i , z i , ≤ i ≤ f −
1] (see [BHH + , §5.3] and (100), (118)). We let J def = ( y i z i , h i , ≤ i ≤ f −
1) (an ideal of gr( F J I /Z K )) and define R def = gr( F J I /Z K ) /J ∼ = F [ y i , z i , ≤ i ≤ f − / ( y i z i , ≤ i ≤ f − . (3)Then p def = ( z i , ≤ i ≤ f −
1) is one of the 2 f minimal prime ideals of R . If N is anyfinite type gr( F J I /Z K )-module killed by a power of J , one can define its multiplicity m p ( N ) ∈ Z ≥ at p , see (124).For π a smooth representation of GL ( K ) over F with a central character,we endow π ∨ with the m I /Z -adic filtration and we let gr( π ∨ ) be the associated gradedgr( F J I /Z K )-module. 12 heorem 1.3.4 (Theorem 3.3.2.3) . Let π be an ( admissible ) smooth representationof GL ( K ) over F satisfying the following two properties: (i) there is a GL ( O K ) K × -equivariant isomorphism D ( ρ ) ⊕ d ∼ = π K ; (ii) for any character χ : I → F × appearing in π [ m I /Z ] there is an equality ofmultiplicities [ π [ m I /Z ] : χ ] = [ π [ m I /Z ] : χ ] . Then gr( π ∨ ) is killed by J and one has m p (gr( π ∨ )) ≤ f d . By the proof of [BHH + , Cor.5.3.5], property (ii) in Theorem 1.3.4 implies thatgr( π ∨ ) is killed by J . By an explicit computation (using both properties (i) and (ii)),one proves in Theorem 3.3.2.1 that there is a surjection of R -modules( ⊕ λ ∈ P R/ a ( λ )) ⊕ d ։ gr( π ∨ ) , where P is a combinatorial finite set associated to ρ (in bijection with the set of χ appearing in π [ m I /Z ], see §3.3.1) and the a ( λ ) are explicit ideals of R containingthe image of J (see Definition 3.3.1.1). Then Theorem 1.3.4 follows from the equality m p ( ⊕ λ ∈ P R/ a ( λ )) = 2 f which is an easy computation.Arguing as in [BHH + ], the representation Π satisfies all assumptions of Theorem1.3.4, see Corollary 3.4.2.2 and Theorem 3.4.4.1. Hence the upper bound in item (ii)below Theorem 1.3.1 follows from Theorem 1.3.4 combined with the next result: Theorem 1.3.5 (Corollary 3.1.4.5) . Let π be an admissible smooth representation of GL ( K ) over F with a central character such that gr( π ∨ ) is killed by some power of J . Then one has dim F V ( π ) ≤ m p (gr( π ∨ )) . We prove Theorem 1.3.5 by first associating to π an “étale ( ϕ, O × K )-module over A ” (Definition 3.1.3.1). This is the “multivariable ( ϕ, Γ)-module” mentioned at theend of §1.1. Though one could probably give a more direct proof without explicitlyintroducing them, these étale ( ϕ, O × K )-modules are important for our finite lengthresults below and are likely to play a role later, so we describe them now.We start with the ring A . Let F J N K ∼ = F J O K K be the Iwasawa algebra of theunipotent radical N of B ( O K ). Then F J N K ∼ = F J Y , . . . , Y f − K , where the Y i areeigenvectors for the action of the finite torus on F J N K (see (100)). Let S be the multiplicative system in F J N K generated by the Y i . The filtration on F J N K by powersof its maximal ideal m N naturally extends to a filtration on the localized ring F J N K S and we define A to be the completion of F J N K S for this filtration ([LvO96, §I.3.4]).The ring A is not local, but it is a regular noetherian domain (Corollary 3.1.1.2) anda complete filtered ring in the sense of [LvO96, §I.3.3] with associated graded ring13r( A ) ∼ = gr( F J N K S ) (see Remark 3.1.1.3(iii) for a concrete description of A ). Mostimportantly, the natural action of O × K on F J N K ∼ = F J O K K by multiplication on O K extends by continuity to A (Lemma 3.1.1.4) and any ideal of A preserved by O × K iseither 0 or A (Corollary 3.1.1.7).Let π be an admissible smooth representation of GL ( K ) over F with a centralcharacter and recall that π ∨ is endowed with the m I /Z -adic filtration (which, ingeneral, strictly contains the m N -adic filtration). We endow ( π ∨ ) S def = F J N K S ⊗ F J N K π ∨ with the tensor product filtration and define D A ( π ) as the completion of ( π ∨ ) S . Then D A ( π ) is a complete filtered A -module such that gr( D A ( π )) ∼ = gr(( π ∨ ) S ) (Lemma3.1.1.1). The action of O × K on π ∨ extends by continuity to D A ( π ), as well as the map ψ : π ∨ −→ π ∨ , f ψ ( f ) def = (cid:18) v ∈ π f ( ξ ( p ) v ) = f (cid:16)(cid:16) p
00 1 (cid:17) v (cid:17)(cid:19) (Lemma 3.1.2.5). The latter can be linearized into an A -linear morphism β : D A ( π ) −→ A ⊗ φ,A D A ( π ) , where φ is the usual Frobenius on the characteristic p ring A (see (117)).We let C be the abelian category of admissible smooth representations π with acentral character such that gr(( π ∨ ) S ) is a finite type gr( F J N K S )-module. It followsfrom (3) that (cid:16) gr( F J I /Z K ) /J (cid:17) [( y · · · y f − ) − ] ∼ = F [ y , . . . , y f − ][( y · · · y f − ) − ] ∼ = gr( F J N K S )which easily implies that, if gr( π ∨ ) is killed by a power of J , then π is in C (Proposition3.1.2.11). In particular the representation Π is in C . Note that any finite lengthadmissible smooth representation π of GL ( Q p ) over F with a central character issuch that gr( π ∨ ) is killed by a power of J (Corollary 3.3.3.5), hence is in C .For π in C , by general results of [Lyu97], there exists a largest quotient D A ( π ) ´et of D A ( π ) such that the map β induces an isomorphism β ´et : D A ( π ) ´et ∼ → A ⊗ φ,A D A ( π ) ´et (see the beginning of §3.1.2). We let ϕ : D A ( π ) ´et → D A ( π ) ´et such that Id ⊗ ϕ =( β ´et ) − . Then D A ( π ) ´et equipped with ϕ and the induced action of O × K is our étale ( ϕ, O × K ) -module over A associated to π in C . Theorem 1.3.6 (Corollary 3.1.2.9, Theorem 3.1.3.3 and Corollary 3.1.4.5) . (i) If π is in C , then D A ( π ) ´et is a finite projective A -module and rk A ( D A ( π ) ´et ) ≤ m p (gr( π ∨ )) . (ii) The ( contravariant ) functor π → D A ( π ) ´et is exact on the abelian category C . A -module endowed with an A -semilinear O × K -action is nonzero, then this annihilator is A (since there are no proper nonzero idealsof A which are preserved by O × K , see above) and hence the A -module must be 0.For a smooth representation π of GL ( K ) over F such that dim F V ( π ) < + ∞ ,we denote by D ∨ ξ ( π ) the unique étale ( ϕ, Γ)-module over F (( X )) such that V ( π ) = V ∨ ( D ∨ ξ ( π )) ⊗ δ (see (1)). We denote by tr : A → F (( X )) the ring morphism inducedby the trace tr : F J N K → F J Z p K ∼ = F J X K . Theorem 1.3.7 (Theorem 3.1.3.6) . If π is in C , then we have an isomorphism ofétale ( ϕ, Γ) -modules over F (( X )) : D A ( π ) ´et ⊗ A F (( X )) ∼ −→ D ∨ ξ ( π ) . In particular dim F V ( π ) = rk A ( D A ( π ) ´et ) < + ∞ and the functor π V ( π ) in (1) isexact on the category C . The proof essentially follows by a careful unravelling of all the definitions andconstructions involved. The last statement follows from the first and from Theorem1.3.6.Theorem 1.3.7 and Theorem 1.3.6(i) imply in particular the bound on V ( π ) inTheorem 1.3.5, which finally proves Theorem 1.3.1.We see that the multivariable ( ϕ, O × K )-module D A ( π ) ´et plays an important role inthe proof of Theorem 1.3.5. One natural question therefore is to understand morethe internal structure of D A (Π) ´et (at least conjecturally): is D A (Π) ´et purely local?Can we recover D A (Π) ´et from the Lubin–Tate ( ϕ q , Γ)-modules associated to ρ ? Weplan to come back to these questions, as well as generalizations in higher dimension,in future work.The modules D A (Π) ´et and D ∨ ξ (Π) are also crucial tools in the proof of our finitelength results on the representation Π which provide evidence to Conjecture 1.2.2 andConjecture 1.2.4 and that we describe now. Theorem 1.3.8 (Theorem 3.4.4.5) . Assume moreover d = 1 , i.e. Π K ∼ = D ( ρ )( the so-called minimal case ) . Then the GL ( K ) -representation Π is generated by its GL ( O K ) -socle, in particular is of finite type. Note that the last finiteness assertion in Theorem 1.3.8 was known for ρ non -semisimple (and sufficiently generic) by [HW, Thm.1.6], but the proof there doesn’textend to the semisimple case.We sketch the proof of Theorem 1.3.8. Let Π ′ ⊆ Π be a nonzero subrepresen-tation and Π ′′ def = Π / Π ′ . As gr(Π ∨ ) and hence its quotient gr(Π ′∨ ) are killed by J ,15he representations Π, Π ′ , Π ′′ are all in C , thus Theorem 1.3.6(i) and Theorem 1.3.7imply dim F V (Π ′ ) ≤ m p (gr(Π ′∨ )) and dim F V (Π ′′ ) ≤ m p (gr(Π ′′∨ )). Since V (Π ′′ ) ∼ = V (Π) /V (Π ′ ) by the last statement in Theorem 1.3.7, and since m p is an additive func-tion by Lemma 3.3.4.4 (and Definition 3.3.4.1), we deduce dim F V (Π ′ ) = m p (gr(Π ′∨ ))and dim F V (Π ′′ ) = m p (gr(Π ′′∨ )) as we have seen that dim F V (Π) = m p (gr(Π ∨ )) (=2 f ). On the other hand, by computations analogous to the ones used in the proofs ofTheorem 1.3.3 and Theorem 1.3.4, we also have inequalities m p (gr(Π ′∨ )) ≤ lg(soc GL ( O K ) (Π ′ )) ≤ dim F V (Π ′ )and thus we deduce m p (gr(Π ′∨ )) = lg(soc GL ( O K ) (Π ′ )) = dim F V (Π ′ ) = 0 . (4)Now take Π ′ to be the nonzero subrepresentation generated over GL ( K ) by theGL ( O K )-socle of Π. We wish to prove Π ′′ = 0. Aslg(soc GL ( O K ) (Π ′ )) = lg(soc GL ( O K ) (Π)) = 2 f = dim F V (Π)we already have by (4) and the exactness of V that m p (gr(Π ′′∨ )) = dim F V (Π ′′ ) = 0 . (5)To deduce Π ′′ = 0 from (5), we need the following key new ingredient: Π is essentiallyself-dual of grade (or codimension) f , i.e. Ext j F J I /Z K (cid:16) Π ∨ , F J I /Z K (cid:17) = 0 if j < f and there is a GL ( K )-equivariant isomorphismExt f F J I /Z K (cid:16) Π ∨ , F J I /Z K (cid:17) ∼ = Π ∨ ⊗ (det( ρ ) ω − ) , (6)where Ext f F J I /Z K (Π ∨ , F J I /Z K ) is endowed with the action of GL ( K ) defined byKohlhaase in [Koh17, Prop.3.2]. This follows by the same argument as in [HW,Thm.8.2] (using Remark 3.4.4.2). We then define e Π as the admissible smooth repre-sentation of GL ( K ) over F such that e Π ∨ ⊗ (det( ρ ) ω − ) ∼ = Im (cid:18) Ext f F J I /Z K (cid:16) Π ∨ , F J I /Z K (cid:17) → Ext f F J I /Z K (cid:16) Π ′′∨ , F J I /Z K (cid:17)(cid:19) , and by (6) e Π is a subrepresentation of Π. By (6) and general results on Ext j Λ ( − , Λ)for Auslander regular rings Λ, Π ′′∨ ⊆ Π ∨ is also of grade 2 f if it is nonzero, and henceExt f F J I /Z K (Π ′′∨ , F J I /Z K ) is nonzero if and only if Π ′′ = 0. From the short exactsequence0 → e Π ∨ ⊗ (det( ρ ) ω − ) → Ext f F J I /Z K (cid:16) Π ′′∨ , F J I /Z K (cid:17) → Ext f +1 F J I /Z K (cid:16) Π ′∨ , F J I /Z K (cid:17) (7)and the fact that the last Ext f +1 has grade ≥ f + 1, we finally obtain: e Π is nonzero if and only if Π ′′ is nonzero. (8)We now use the following general theorem.16 heorem 1.3.9 (Theorem 3.3.4.5) . Let π be an admissible smooth representationof GL ( K ) over F with a central character such that gr( π ∨ ) is killed by a power of J . Then the gr( F J I /Z K ) -module ( for the m I /Z -adic filtration on Ext f F J I /Z K ( π ∨ , F J I /Z K )) : gr (cid:18) Ext f F J I /Z K (cid:16) π ∨ , F J I /Z K (cid:17)(cid:19) is also finitely generated and annihilated by a power of J , and we have m p (gr( π ∨ )) = m p gr (cid:18) Ext f F J I /Z K (cid:16) π ∨ , F J I /Z K (cid:17)(cid:19)! . From the injection in (7) and from Theorem 1.3.9 applied to π = Π ′′ we have m p (gr( e Π ∨ )) ≤ m p (gr(Π ′′∨ )), hence we obtain m p (gr( e Π ∨ )) = m p (gr(Π ′′∨ )) ( ) = 0 . This implies e Π = 0 by (4) (applied to the subrepresentation Π ′ = e Π) and thus Π ′′ = 0by (8), finishing the proof of Theorem 1.3.8.The following corollary immediately follows from Theorem 1.3.8 and from [BP12,Thm.19.10(i)]. Corollary 1.3.10 (Theorem 3.4.4.5) . Assume moreover d = 1 and ρ irreducible.Then the GL ( K ) -representation Π is irreducible and is a supersingular representa-tion. When ρ is reducible (split), we can prove the following result. Theorem 1.3.11 (Theorem 3.4.4.6) . Assume moreover d = 1 and ρ reducible, i.e. ρ = χ χ ! . Then one has Π = Ind GL ( K ) B ( K ) ( χ ⊗ χ ω − ) ⊕ Π ′ ⊕ Ind GL ( K ) B ( K ) ( χ ⊗ χ ω − ) , where Π ′ is generated by its GL ( O K ) -socle and Π ′∨ is essentially self-dual of grade f , i.e. satisfies (6) . Moreover, when f = 2 , Π ′ is irreducible and supersingular ( andhence Π is semisimple ) . The fact that the two principal series in Theorem 1.3.11 occur as subobjects of Πwas already known (and is not difficult). To prove that they also occur as quotients(and that the obvious composition is the identity), we again crucially use the essentialself-duality (6). The rest of the statement follows from Theorem 1.3.8 and [BP12,Thm.19.10(ii)].The following last corollary sums up the above results.17 orollary 1.3.12 (Theorem 3.4.4.7) . Assume (i) to (vii) as at the beginning of §1.3and assume d = 1 as in Theorem 1.3.8. Then Conjecture 1.2.4 holds for n = 2 and ρ irreducible, or for n = 2 , K quadratic and ρ semisimple. Note finally that when f = 2, ρ is non -semisimple (sufficiently generic) and d = 1,Conjecture 1.2.2 at least is known and follows from [HW, Thm.10.37]. We finish this introduction with some very general notation (many more will bedefined in the text).Throughout the text, we fix Q p an algebraic closure of Q p and K an arbitraryfinite extension of Q p in Q p with residue field F q , q = p f ( f ∈ Z ≥ ). The field K isunramified from §2.2 on. We also fix a finite extension E of Q p , with ring of integers O E , uniformizer ̟ E and residue field F , and we assume that F contains F q . The finitefield F is the main coefficient field in this work. We denote by ε the p -adic cyclotomiccharacter of Gal( Q p / Q p ) and by ω its reduction mod p . We normalize local class fieldtheory so that uniformizers correspond to geometric Frobeniuses.If H is any split connected reductive algebraic group, we denote by Z H the centerof H and by T H a split maximal torus. If P H is a parabolic subgroup of H containing T H , we denote by M P H its Levi subgroup containing T H , N P H its unipotent radicaland P − H its opposite parabolic subgroup with respect to T H (so P H ∩ P − H = M H ).We let n ≥ G the algebraic group GL n over Z . Theinteger n is arbitrary in §2 and is 2 in §3.Irreducible for a representation always means absolutely irreducible.Finally, though we mainly work with the group GL n , several proofs in §2 can beextended more or less verbatim to a split connected reductive algebraic group over Z with connected center, and §2.1.4 deals with possibly nonsplit reductive groups. Acknowledgements : B. S. would like to thank Gabriel Dospinescu and VytasPašk¯unas for stimulating discussions around C -groups. Y. H. is partially supportedby National Key R&D Program of China 2020YFA0712600, National Natural ScienceFoundation of China Grants 11688101 and 11971028; National Center for Mathemat-ics and Interdisciplinary Sciences and Hua Loo-Keng Key Laboratory of Mathematics,Chinese Academy of Sciences. F. H. is partially supported by an NSERC grant. C. B.,S. M. and B. S. are members of the A.N.R. project CLap-CLap ANR-18-CE40-0026.18 Local-global compatibility conjectures
We state local-global compatibility conjectures (Conjecture 2.1.3.1, Conjecture 2.1.4.5and Conjecture 2.5.1) which “functorially” relate Hecke-isotypic components withtheir action of GL n ( K ) in spaces of mod p automorphic forms to representationsof Gal( Q p / Q p ). Conjecture 2.5.1 assumes K is unramified but is much stronger andmore precise than Conjecture 2.1.3.1 and Conjecture 2.1.4.5 as it predicts the number,position and form of the irreducible constituents of these Hecke-isotypic components,as well as their contribution on the Galois side.Throughout this section, we let T ⊆ G = GL n the diagonal torus over Z and X ( T ) the Z -module Hom Gr ( T, G m ). As usual, we identify X ( T ) with ⊕ ni =1 Z e i via e i (cid:16) diag( x , . . . , x n ) x i (cid:17) and define h , i : X ( T ) × X ( T ) → Z , h e i , e j i def = δ i,j ,which we extend by Q -bilinearity to X ( T ) ⊗ Z Q . This provides an isomorphism of Z -modules X ( T ) ∼ → Hom Z ( X ( T ) , Z ) ∼ = Hom Gr ( G m , T ) given by e i e ∗ i def = (cid:16) x diag(1 , . . . , | {z } i − , x, , . . . , (cid:17) , i ∈ { , . . . , n } . (9)We denote by R = { e i − e j , ≤ i = j ≤ n } ⊆ X ( T ) the roots of ( G, T ), by B ⊆ G the Borel subgroup (over Z ) of upper-triangular matrices and by N the unipotentradical of B , so that the positive roots are R + = { e i − e j , ≤ i < j ≤ n } ⊆ R andthe simple roots are S = { e i − e i +1 , ≤ i ≤ n − } ⊆ R + . An element of X ( T ) ⊗ Z Q is dominant if h λ, e i − e i +1 i ≥ i ∈ { , . . . , n − } . If λ, µ ∈ X ( T ) ⊗ Z Q , wewrite λ ≤ µ if µ − λ ∈ P n − i =1 Q ≥ ( e i − e i +1 ). If λ = P n − i =1 n i ( e i − e i +1 ) for some n i ∈ Q ,its support is by definition the set of simple roots e i − e i +1 such that n i = 0. Finally,we denote by W ∼ = S n the Weyl group of ( G, T ), which acts on the left on X ( T ) by w ( λ )( t ) def = λ ( w − tw ) for λ ∈ X ( T ) and t ∈ T .If P is a standard parabolic subgroup of G (that is, containing B ), we denote by S ( P ) ⊆ S the subset of simple roots of M P , R ( P ) + ⊆ R + the positive roots of M P (generated by S ( P )) and W ( P ) ⊆ W its Weyl group. We state our first local-global compatibility conjecture (see Conjecture 2.1.3.1 andits generalization Conjecture 2.1.4.5) which relate Hecke-isotypic components withtheir action of GL n ( K ) to representations of Gal( Q p / Q p ) without taking care of theirirreducible constituents. 19 .1.1 The functors D ∨ ξ H and V H We review the simple generalization of Colmez’s functor defined in [Bre15].Throughout this section, we fix a connected reductive algebraic group H whichis split over K with a connected center, B H ⊆ H a Borel subgroup and T H ⊆ B H asplit maximal torus in B H . We let (cid:16) X ( T H ) , R H , X ∨ ( T H ) , R ∨ H (cid:17) be the associated rootdatum, R + H ⊆ X ( T H ) the (positive) roots of B H , S H ⊆ R + H the simple roots and S ∨ H the associated simple coroots.We need to recall some notation of [Bre15] (to which we refer the reader for anyfurther details). For α ∈ R + H , we let N α ⊆ N H be the associated (commutative) rootsubgroup, where N H def = N B H is the unipotent radical of B H . For α ∈ S H , we fix anisomorphism ι α : N α ∼ → G a of algebraic groups over K such that ι α ( tn α t − ) = α ( t ) ι α ( n α ) ∀ t ∈ T H , ∀ n α ∈ N α . (10)We fix an open compact subgroup N ⊆ N H ( K ) such that Q α ∈ R + H N α ∼ → N H inducesa bijection Q α ∈ R + H N α ( K ) ∩ N ∼ → N for any order on the α ∈ R + H and such that ι α induces isomorphisms for α ∈ S H : N α ( K ) ∩ N ∼ −→ O K ⊆ K = G a ( K ) . We denote by ℓ the composite N H ։ Q α ∈ S H N α P α ∈ SH ι α −→ G a (a morphism of algebraicgroups over K ). The morphism ℓ thus induces a group morphism still denoted ℓ : N → O K and we define N def = Ker (cid:16) N ℓ → O K Tr K/ Q p −→ Q p (cid:17) (11)which is a normal open compact subgroup of N . We fix an isomorphism of Z p -modules ψ : Tr K/ Q p ( O K ) ∼ → Z p . When N H = 0, i.e. when H = T H , this fixes anisomorphism N /N K/ Q p ◦ ℓ ∼ −→ Tr K/ Q p ( O K ) ψ ∼ −→ Z p . (12)We fix fundamental coweights ( λ α ∨ ) α ∈ S H (which exist since H has a connected center)and set ξ H def = X α ∨ ∈ S ∨ H λ α ∨ ∈ Hom Gr ( G m , T H ) = X ∨ ( T H ) . (13)Note that ξ H ( x ) N ξ H ( x − ) ⊆ N for any x ∈ Z p \{ } . Let F J X K [ F ] be the noncom-mutative polynomial ring in F over the ring of formal power series F J X K such that F S ( X ) = S ( X p ) F .For π a smooth representation of B H ( K ) over F , we endow the invariant subspace π N ⊆ π with a structure of an F J X K [ F ]-module as follows:20i) F J X K ∼ = F J Z p K acts via F J N /N K ( ) ∼ = F J Z p K (here X def = [1] − F acts via the “Hecke” action F ( v ) def = P n ∈ N /ξ H ( p ) N ξ H ( p − ) n ξ H ( p ) v ∈ π N for v ∈ π N .Note that π N is a torsion F J X K -module (but not a torsion F [ F ]-module in general).We also endow π N with an action of Z × p by making x ∈ Z × p act by ξ H ( x ). This actioncommutes with F and satisfies ξ H ( x ) ◦ (1 + X ) = (1 + X ) x ◦ ξ H ( x ).As in [Bre15], we denote by ΦΓ ´et F the category of finite-dimensional étale ( ϕ, Γ)-modules over F J X K [ X − ] = F (( X )) and by d ΦΓ ´et F the corresponding category of (pseudo-compact) pro-objects, see [Bre15, §2] for more details. Both ΦΓ ´et F and d ΦΓ ´et F are abeliancategories. Let M ⊆ π N be a finite type F J X K [ F ]-submodule which is Z × p -stable andassume that M is admissible as an F J X K -module, that is, M [ X ] def = { m ∈ M, Xm = 0 } is finite-dimensional over F . Let M ∨ def = Hom F ( M, F ) (algebraic F -linear dual) whichis also an F J X K -module (but not a torsion F J X K -module in general). Then by akey result of Colmez M ∨ [ X − ] can be endowed with the structure of an object ofΦΓ ´et F ([Col10], see also [Bre15, Lemma 2.6]). More precisely X acts on f ∈ M ∨ by( Xf )( m ) def = f ( Xm ) ( m ∈ M ), x ∈ Z × p acts by ( xf )( m ) def = f ( x − m ), and the operator ϕ is defined as follows. Take the F -linear dual of Id ⊗ F : F J X K ⊗ ϕ, F J X K M −→ M ,compose with ( F J X K ⊗ ϕ, F J X K M ) ∨ ∼ −→ F J X K ⊗ ϕ, F J X K M ∨ f p − X i =0 (1 + X ) i ⊗ f (cid:16) X ) i ⊗ · (cid:17) (14)and invert X : the resulting morphism M ∨ [ X − ] → F J X K ⊗ ϕ, F J X K M ∨ [ X − ] turns outto be an F (( X ))-linear isomorphism whose inverse is by definition Id ⊗ ϕ .When H = T H we then define D ∨ ξ H ( π ) def = lim ←− M M ∨ [ X − ] , (15)where the projective limit is taken over the finite type F J X K [ F ]-submodules M of π N (for the preorder defined by inclusion) which are admissible as F J X K -modulesand invariant under the action of Z × p . When H = T H , one has to replace M ∨ [ X − ]by F (( X )) ⊗ F M ∨ , we refer the reader to [Bre15, §3]. The functor D ∨ ξ H is right exactcontravariant from the category of smooth representations of B H ( K ) over F to thecategory d ΦΓ ´et F and, up to isomorphism, only depends on the choice of the cocharacter The formula for this isomorphism given in the proof of [Bre15, Lemma 2.6] is actually wrong, thepresent formula is the correct one. Note that it is also the same as f P p − i =0 1(1+ X ) i ⊗ f ((1+ X ) i ⊗· ). H . Moreover, if D ∨ ξ H ( π ) turns out to be in ΦΓ ´et F (and not just d ΦΓ ´et F ), then D ∨ ξ H ( π ) isexactly the maximal étale ( ϕ, Γ)-module which occurs as a quotient of ( π N ) ∨ [ X − ],see [Bre15, Rem.5.6(iii)]. Remark 2.1.1.1. If H = G m = T H , then by definition ξ H = 1. It follows, fordim F π = 1, that D ∨ ξ H ( π ) is always the trivial (rank one) ( ϕ, Γ)-module (even if π is anontrivial character).Let us now assume that the dual group c H of H also has a connected center, andlet us fix θ H ∈ X ( T H ) such that θ H ◦ α ∨ = Id G m for all α ∈ S H ([BH15, Prop.2.1.1],such an element is called a twisting element ). In §2.1.4 below, it is possible to avoidthis assumption using C -parameters, but since our main aim is G = GL n in the restof the paper, there is no harm making this assumption.Consider the smooth character K × −→ F × , x ω (cid:16) θ H ( ξ H ( x )) (cid:17) and denote by δ H the restriction of this character to Q × p ⊆ K × . Seeing ω ◦ θ H ◦ ξ H as a character of Gal( Q p /K ) via local class field theory for K (as normalized in §1),and remembering that the restriction from K × to Q × p corresponds via local class fieldtheory to the composition with the transfer Gal( Q p / Q p ) ab → Gal( Q p /K ) ab , we seethat δ H ∼ = ind ⊗ Q p K ( ω ◦ θ H ◦ ξ H ) , where ind ⊗ Q p K is the tensor induction from Gal( Q p /K ) to Gal( Q p / Q p ) (see the end of§2.1.2 below).Denote by Rep F the abelian category of continuous linear representations ofGal( Q p / Q p ) on finite-dimensional F -vector spaces (equipped with the discrete topol-ogy) and IndRep F the corresponding category of ind-objects, i.e. the category offiltered direct limits of objects of Rep F . Recall that there is a covariant equivalence ofcategories V : ΦΓ ´et F ∼ → Rep F (see [Fon90, Thm.A.3.4.3] where this functor is denoted V E ) compatible with tensor products and duals on both sides. We denote by V ∨ thedual of V (i.e. the dual Galois representation). When H = T H , we then define thecovariant functor V H from the category of smooth representations of B H ( K ) over F to the category IndRep F by V H ( π ) def = lim −→ M (cid:16) V ∨ ( M ∨ [ X − ]) (cid:17) ⊗ δ H , (16)where the inductive limit is taken over the finite type F J X K [ F ]-submodules of π N which are admissible as F J X K -modules and preserved by Z × p . Likewise, when H = T H ,with F (( X )) ⊗ F M ∨ instead of M ∨ [ X − ] (note that δ H is then 1). Lemma 2.1.1.2.
The functor V H is left exact. roof. We give the proof for H = T H , leaving the case H = T H to the reader. Let0 → π ′ → π s → π ′′ → B H ( K )-representations over F , which gives a short exact sequence 0 → π ′ N → π N s → π ′′ N . If M is a finite type F J X K [ F ]-submodule of π N which is admissible as F J X K -module and stable under theaction of Z × p , then so are M ∩ π ′ N and s ( M ) (see e.g. [Bre15, Lemma 2.1(i)]). Thefunctor M → V ∨ ( M ∨ [ X − ]) being covariant exact (since both M M ∨ [ X − ] and V ∨ are contravariant exact), each such M ⊆ π N gives rise to a short exact sequencein Rep F :0 → V ∨ (cid:16) ( M ∩ π ′ N ) ∨ [ X − ] (cid:17) → V ∨ (cid:16) M ∨ [ X − ] (cid:17) → V ∨ (cid:16) s ( M ) ∨ [ X − ] (cid:17) → . Twisting by δ H and taking the inductive limit over such M , we obtain a short exactsequence 0 → V H ( π ′ ) → V H ( π ) → lim −→ M V ∨ (cid:16) s ( M ) ∨ [ X − ] (cid:17) ⊗ δ H → F . Butwe have an injection lim −→ M V ∨ (cid:16) s ( M ) ∨ [ X − ] (cid:17) ⊗ δ H ֒ → V H ( π ′′ )in IndRep F since all transitions maps in the inductive limits are injective, thereforewe end up with an exact sequence 0 → V H ( π ′ ) → V H ( π ) → V H ( π ′′ ). Example 2.1.1.3.
For H = G × Z K = GL n/K (so H ∼ = c H ), we take in the sequel(writing just G as a subscript instead of G × Z K ) ξ G ( x ) def = diag( x n − , . . . , x,
1) and θ G (cid:16) diag( x , . . . , x n ) (cid:17) = x n − x n − · · · x n − , so that δ G = ind ⊗ Q p K ( ω ( n − +( n − + ··· +4+1 ). Remark 2.1.1.4. (i) The covariant functor V H depends on the choices of ξ H and δ H (though we don’t include it in the notation). The reader may wonder why we needto assume the existence of θ H and normalize V H using the strange twist δ H above.This comes from the local-global compatibility: it turns out that this normalizationis essentially what is going on in spaces of mod p automorphic forms (see [BH15, §4],[Bre15, Cor.9.8], Example 2.1.1.6 and §§2.1.3, 2.5 below). This normalization is alsonatural if one uses C -parameters, see §2.1.4.(ii) For H as in Example 2.1.1.3, π a smooth representation of B ( K ) over F and χ : K × → F × a smooth character, one checks that V G ( π ⊗ ( χ ◦ det)) ∼ = V G ( π ) ⊗ δ , where δ is the continuous character of Gal( Q p / Q p ) associated via local class field theory to x χ (cid:16) det( ξ G ( x )) (cid:17) for x ∈ Q × p . An explicit computation gives δ = ( χ | Q × p ) n ( n − ∼ =ind ⊗ Q p K ( χ n ( n − ).When restricted to the abelian category of finite length admissible smooth repre-sentations of H ( K ) over F with all irreducible constituents isomorphic to irreducible23onstituents of principal series, it is proven in [Bre15, §9] that the functors D ∨ ξ H and V H are exact. It seems reasonable to us, and also consistent with the conjecturalformalism developed in the sequel (see e.g. Remark 2.4.2.8(iii)), to hope that thereexists a suitable abelian category of admissible smooth representations of H ( K ) over F containing the previous abelian category and the representations “coming from theglobal theory” on which the functors D ∨ ξ H and V H are still exact. See for instance thecategory C in §3.1.2 when H = GL /K and K is unramified.We now recall the behaviour of the functor V H with respect to parabolic induction.We assume for simplicity H = G × Z K = GL n/K and let ξ G , θ G as in Example2.1.1.3. We let P be a standard parabolic subgroup of G × Z K and write M P = Q di =1 M i with M i ∼ = GL n i /K . We define V M P as in (16) using ξ M P def = ξ G and θ M P def = θ G (todefine D ∨ ξ MP and δ M P ). We write ξ M P = ⊕ di =1 ξ M P ,i in X ∨ ( T ) = ⊕ di =1 X ∨ ( T i ) and θ M P = ⊕ di =1 θ M P ,i in X ( T ) = ⊕ di =1 X ( T i ), where T i is the diagonal torus in M i , and let V M P ,i def = V GL ni but defined with ξ M P ,i and θ M P ,i . Finally we define V M i def = V GL ni with ξ M i and θ M i as in Example 2.1.1.3 replacing n by n i , and we recall that ξ M i , θ M i and δ M i are trivial characters if n i = 1.If π P is a smooth representation of M P ( K ) over F , that we see as a representationof P − ( K ) via P − ( K ) ։ M P ( K ), we define the usual smooth parabolic inductionInd G ( K ) P − ( K ) π P def = { f : G ( K ) → π P loc . const ., f ( px ) = p ( f ( x )) , p ∈ P − ( K ) , x ∈ π P } , with G ( K ) acting (smoothly) on the left by ( gf )( g ′ ) def = f ( g ′ g ). Lemma 2.1.1.5.
Let π P be a smooth representation of M P ( K ) over F of the form π P = π ⊗ · · · ⊗ π d , where the π i are smooth representations of M i ( K ) over F . As-sume that the π i have central characters Z ( π i ) : K × → F × and that V M P ( π P ) ∼ = N di =1 V M P ,i ( π i ) . Then we have an isomorphism in IndRep F ( using implicitly localclass field theory for Gal( Q p / Q p )) : V G (cid:18) Ind G ( K ) P − ( K ) π P (cid:19) ⊗ δ − G ∼ = d O i =1 (cid:18) V M i ( π i ) ⊗ (cid:16) Z ( π i ) n − P ij =1 n j (cid:17) | Q × p δ − M i (cid:19) . Proof.
By [Bre15, Thm.6.1] we have V G (cid:16) Ind G ( K ) P − ( K ) π P (cid:17) ∼ = V M P ( π P ) so that from theassumption (all isomorphisms are in IndRep F ): V G (cid:16) Ind G ( K ) P − ( K ) π P (cid:17) ∼ = d O i =1 V M P ,i ( π i ) . (17)An easy computation yields in M i ( K ) for x ∈ K × : ξ M P ,i ( x ) = diag( x n − P ij =1 n j , . . . , x n − P ij =1 n j | {z } n i ) ξ M i ( x )24hich implies by [Bre15, Rem.4.3] that V M P ,i ( π i ) ⊗ δ − M P ,i ∼ = V M i ( π i ) ⊗ (cid:16) Z ( π i ) n − P ij =1 n j (cid:17) | Q × p δ − M i , (18)where δ M P ,i def = ind ⊗ Q p K ( ω ◦ θ M P ,i ◦ ξ M P ,i ) (and recall V M i ( π i ) = 1 if n i = dim F π i = 1,see Remark 2.1.1.1). Since δ G = Q di =1 δ M P ,i , twisting (17) by δ − G gives the result by(18). Example 2.1.1.6.
An enlightening and important example is the case of principalseries Ind G ( K ) B − ( K ) ( χ ⊗ · · · ⊗ χ n ), where the χ i : K × → F × are smooth characters. Theassumptions of Lemma 2.1.1.5 are then trivially satisfied and thus we have V G (cid:16) Ind G ( K ) B − ( K ) ( χ ⊗ · · · ⊗ χ n ) (cid:17) ⊗ δ − G ∼ = ( χ n − χ n − · · · χ n − ) | Q × p . In particular we deduce (using Example 2.1.1.3 for δ G ) that V G (cid:16) Ind G ( K ) B − ( K ) ( χ ω − ( n − ⊗ χ ω − ( n − ⊗ · · · ⊗ χ n ) (cid:17) ∼ = ( χ n − χ n − · · · χ n − ) | Q × p ∼ = ind ⊗ Q p K ( χ n − χ n − · · · χ n − ) , where χ n − χ n − · · · χ n − on the last line is seen as a character of Gal( Q p /K ) via localclass field theory for K . Remark 2.1.1.7.
Using [Bre15, Prop.5.5] the assumptions of Lemma 2.1.1.5 are sat-isfied when all finite type F J X K [ F ]-submodules of π N i for i ∈ { , . . . , d } are automati-cally admissible as F J X K -modules. This happens for instance if the π i are principal se-ries or (when K = Q p ) are finite length representations of GL ( Q p ) with a central char-acter, but is not known otherwise. Contrary to what is stated in [Bre15, Rem.5.6(ii)],we currently do not have a proof of an isomorphism V M P ( π P ) ∼ = N di =1 V M P ,i ( π i ) for anysmooth representations π i , though we expect that it will indeed be satisfied for repre-sentations “coming from” the global theory. Note that, in [Záb18b, Prop.3.2], Zábrádidoes prove a compatibility of his functor with the tensor product which looks closeto the isomorphism above. However, loc.cit. deals with an external tensor product,whereas we have an internal tensor product. In particular he has two operators F ,one for each factor in the external tensor product (whereas we consider the resultingdiagonal operator), and his argument doesn’t extend. We recall our global setting (see e.g. [EGH13, §7.1] or [Tho12, §6] or [BH15, §4.1]or many other references) and define the Gal( Q p / Q p )-representation L ⊗ ( ρ ) for ρ :Gal( Q p /K ) −→ G ( F ).We let F + be a totally real finite extension of Q with ring of integers O F + , F/F + a totally imaginary quadratic extension with ring of integers O F (do not confuse F F of §2.1.1!) and c the nontrivial element of Gal( F/F + ). If v (resp.˜ v ) is a finite place of F + (resp. F ), we let F + v (resp. F ˜ v ) be the completion of F + (resp. F ) at v (resp. ˜ v ) and O F + v (resp. O F ˜ v ) the ring of integers of F + v (resp. F ˜ v ). If v splits in F and ˜ v, ˜ v c are the two places of F above v , we have O F + v = O F ˜ v c ∼ = O F ˜ vc ,where the last isomorphism is induced by c . We let A ∞ F + (resp. A ∞ ,vF + ) denote the finiteadèles of F + (resp. the finite adèles of F + outside v ). Finally we always assume thatall places of F + above p split in F .We let n ∈ Z > , N a positive integer prime to p and H a connected reductivealgebraic group over O F + [1 /N ] satisfying the following conditions:(i) there is an isomorphism ι : H × O F + [1 /N ] O F [1 /N ] ∼ −→ G × Z O F [1 /N ];(ii) H × O F + [1 /N ] F + is an outer form of G × Z F + = GL n/F + ;(iii) H × O F + [1 /N ] F + is isomorphic to U n ( R ) at all infinite places of F + .One can prove that such groups exist (cf. e.g. [EGH13, §7.1.1]). Condition (i) impliesthat if v is any finite place of F + that splits in F and if ˜ v | v in F the isomorphism ι induces ι ˜ v : H ( F + v ) ∼ → GL n ( F ˜ v ) = G ( F ˜ v ) which restricts to an isomorphism stilldenoted by ι ˜ v : H ( O F + v ) ∼ → GL n ( O F ˜ v ) if v doesn’t divide N . Condition (ii) impliesthat c ◦ ι ˜ v : H ( F + v ) ∼ → GL n ( F ˜ v c ) (resp. c ◦ ι ˜ v : H ( O F + v ) ∼ → GL n ( O F ˜ vc ) if v doesn’tdivide N ) is conjugate in GL n ( F ˜ v c ) (resp. in GL n ( O F ˜ vc )) to τ − ◦ ι ˜ v c , where τ is thetranspose in GL n ( F ˜ v c ) (resp. in GL n ( O F ˜ vc )).If U is any compact open subgroup of H ( A ∞ F + ) then S ( U, F ) def = { f : H ( F + ) \ H ( A ∞ F + ) /U → F } is a finite-dimensional F -vector space since H ( F + ) \ H ( A ∞ F + ) /U is a finite set. Fix v | p in F + and a compact open subgroup U v of H ( A ∞ ,vF + ), we define S ( U v , F ) def = lim −→ U v S ( U v U v , F ) , where U v runs among compact open subgroups of H ( O F + v ). We endow S ( U v , F ) witha linear left action of H ( F + v ) by ( h v f )( h ) def = f ( hh v ) ( h v ∈ H ( F + v ), h ∈ H ( A ∞ F + )).Thus, for ˜ v dividing v in F , the isomorphism ι ˜ v gives an admissible smooth action of G ( F + v ) = GL n ( F ˜ v ) on S ( U v , F ). By what is above, the action of G ( F + v ) induced by ι ˜ v is the inverse transpose of the one induced by ι ˜ v c .If U is a compact open subgroup of H ( A ∞ F + ), following [EGH13, §7.1.2] we saythat U is unramified at a finite place v of F + which splits in F and doesn’t divide N ifwe have U = U v × H ( O F + v ), where U v is a compact open subgroup of H ( A ∞ ,vF + ). Notethat a compact open subgroup of H ( A ∞ F + ) is unramified at all but a finite number26f finite places of F + which split in F . If U is a compact open subgroup of H ( A ∞ F + )and Σ a finite set of finite places of F + containing the set of places of F + that splitin F and divide pN and the set of places of F + that split in F at which U is not unramified, we denote by T Σ def = O E [ T ( j )˜ w ] the commutative polynomial O E -algebragenerated by formal variables T ( j )˜ w for j ∈ { , . . . , n } and ˜ w a place of F lying abovea finite place w of F + that splits in F and doesn’t belong to Σ. The algebra T Σ actson S ( U, F ) by making T ( j )˜ w act by the double coset ι − w " GL n ( O F ˜ w ) n − j ̟ ˜ w j ! GL n ( O F ˜ w ) , where ̟ ˜ w is a uniformizer in O F ˜ w . Explicitly, if we writeGL n ( O F ˜ w ) (cid:16) n − j ̟ ˜ w j (cid:17) GL n ( O F ˜ w ) = a i g i (cid:16) n − j ̟ ˜ w j (cid:17) GL n ( O F ˜ w ) , we have for f ∈ S ( U, F ) and g ∈ H ( A ∞ F + ):( T ( j )˜ w f )( g ) def = X i f gι − w (cid:18) g i (cid:16) n − j ̟ ˜ w j (cid:17) (cid:19)! . One checks that T ( j )˜ w c = ( T ( n )˜ w ) − T ( n − j )˜ w on S ( U, F ). We let T Σ ( U, F ) be the image of T Σ in End O E ( S ( U, F )) (if U ′ ⊆ U , we thus have S ( U, F ) ⊆ S ( U ′ , F ) and T Σ ( U ′ , F ) ։ T Σ ( U, F )). If S is any T Σ -module and I any ideal of T Σ , we set in the sequel S [ I ] def = { x ∈ S : Ix = 0 } .We now fix v | p and a compact open subgroup U v of H ( A ∞ ,vF + ). If Σ a finite set offinite places of F + containing the set of places of F + that split in F and divide pN and the set of places of F + prime to p that split in F and at which U v U v (for any U v )is not unramified, the algebra T Σ acts on S ( U v U v , F ) (via its quotient T Σ ( U v U v , F ))for any U v and thus also on S ( U v , F ). This action commutes with that of H ( F + v ). If m Σ is a maximal ideal of T Σ with residue field F , we can define the localized subspaces S ( U v U v , F ) m Σ and their inductive limitlim −→ U v S ( U v U v , F ) m Σ = S ( U v , F ) m Σ , which inherits an induced (admissible smooth) action of H ( F + v ) together with a com-muting action of lim ←− U v T Σ ( U v U v , F ) m Σ . We have S ( U v U v , F )[ m Σ ] ⊆ S ( U v U v , F ) m Σ ⊆ S ( U v U v , F )and thus inclusions of admissible smooth H ( F + v )-representations over F : S ( U v , F )[ m Σ ] ⊆ S ( U v , F ) m Σ ⊆ S ( U v , F ) . H ( F + v ), S ( U v , F ) m Σ is a direct summand of S ( U v , F )(= the maximal vector subspace on which the elements of m Σ act nilpotently).We now go back to the notation of §2.1.1. For λ ∈ X ( T ) a dominant weight withrespect to B , we consider the following algebraic representation of G × Z F over F : L ( λ ) def = (cid:16) ind GB − λ (cid:17) / Z ⊗ Z F = (cid:16) ind G × Z F B − × Z F λ (cid:17) / F , (19)where ind means the algebraic induction functor of [Jan03, §I.3.3] and the last equalityfollows from [Jan03, II.8.8(1)]. For α = e i − e i +1 ∈ S , we set λ α def = e + · · · + e i ∈ X ( T ) , so that the λ α for α ∈ S are fundamental weights of G (see e.g. [BH15, §2.1]).Let ρ : Gal( Q p /K ) −→ G ( F ) be a continuous homomorphism, viewing L ( λ α ) as acontinuous homomorphism G ( F ) −→ Aut (cid:16) L ( λ α )( F ) (cid:17) (where L ( λ α )( F ) is the underlying F -vector space of the algebraic representation L ( λ α )), we define the Galois representations for α ∈ S : L ( λ α )( ρ ) : Gal( Q p /K ) ρ −→ G ( F ) L ( λ α ) −→ Aut (cid:16) L ( λ α )( F ) (cid:17) . Recall that L ( λ α )( ρ ) = V i F ρ if α = e i − e i +1 ([BH15, Ex.2.1.3]). We let O α ∈ S (cid:16) L ( λ α )( ρ ) (cid:17) ∼ = n − O i =1 ^ i F ρ be the tensor product of the representations L ( λ α )( ρ ) (over F ) and define the followingfinite-dimensional continuous representation of Gal( Q p / Q p ) over F : L ⊗ ( ρ ) def = ind ⊗ Q p K (cid:18) O α ∈ S (cid:16) L ( λ α )( ρ ) (cid:17)(cid:19) , (20)where ind ⊗ Q p K means the tensor induction from Gal( Q p /K ) to Gal( Q p / Q p ) ([Col89],[CR81, §13], see also the end of the proof of Lemma 2.4.2.3). Note that there areGal( Q p / Q p )-equivariant isomorphisms L ⊗ ( ρ ∨ ) ∼ = L ⊗ ( ρ ) ∨ ∼ = L ⊗ ( ρ ) ⊗ ind ⊗ Q p K (cid:16) det( ρ ) − ( n − (cid:17) (21)(recall ind ⊗ Q p K (cid:16) det( ρ ) − ( n − (cid:17) is still one dimensional). Example 2.1.2.1.
For n = 2, we thus just have L ⊗ ( ρ ) = ind ⊗ Q p K ( ρ ).28 .1.3 Weak local-global compatibility conjecture We state our weak local-global compatibility conjecture (Conjecture 2.1.3.1).Let r : Gal( F /F ) → GL n ( F ) be a continuous representation and r ∨ its dual. Weassume:(i) r c ∼ = r ∨ ⊗ ω − n (where r c ( g ) def = r ( cgc ) for g ∈ Gal(
F /F ));(ii) r is an absolutely irreducible representation of Gal( F /F ).Fix v | p in F + , V v ⊆ U v ⊆ H ( A ∞ ,vF + ) compact open subgroups and Σ a finite set offinite places of F + containing(a) the set of places of F + that split in F and divide pN ;(b) the set of places of F + that split in F at which V v is not unramified;(c) the set of places of F + that split in F at which r is ramified.We associate to r and Σ the maximal ideal m Σ in T Σ with residue field F generatedby ̟ E and all elements (cid:18) ( − j Norm( ˜ w ) j ( j − / T ( j )˜ w − a ( j )˜ w (cid:19) j, ˜ w , where j ∈ { , . . . , n } , ˜ w is a place of F lying above a finite place w of F + that splitsin F and doesn’t belong to Σ, X n + a (1)˜ w X n − + · · · + a ( n − w X + a ( n )˜ w is the characteristicpolynomial of r (Frob ˜ w ) (an element of F [ X ], Frob ˜ w is a geometric Frobenius at ˜ w )and where a ( j )˜ w is any element in O E lifting a ( j )˜ w . Note that S ( V v , F )[ m Σ ] = 0 in factimplies assumption (i) above on r (though strictly speaking we need (i) to define m Σ in T Σ ). Note also that if U is any subgroup of H ( A ∞ F + ) containing V v as a normalsubgroup, then U naturally acts on S ( V v , F ) and S ( V v , F )[ m Σ ].For ˜ v | v in F , we denote by V G, ˜ v the functor defined in (16) applied to smoothrepresentations of H ( F + v ) over F , where we identify H ( F + v ) with GL n ( F ˜ v ) = G ( F ˜ v ) via ι ˜ v . For any finite place ˜ w of F , we denote by r ˜ w the restriction of r to a decompositionsubgroup at ˜ w . Conjecture 2.1.3.1.
Let r : Gal( F /F ) → GL n ( F ) be a continuous representationthat satisfies conditions (i) and (ii) above and fix a place v of F + which divides p .Assume that there exist compact open subgroups V v ⊆ U v ⊆ H ( A ∞ ,vF + ) with V v normalin U v , a finite-dimensional representation σ v of U v /V v over F and a finite set Σ offinite places of F + as above such that Hom U v ( σ v , S ( V v , F )[ m Σ ]) = 0 . Let ˜ v | v in F . hen there is an integer d ∈ Z > depending only on v , U v , V v , σ v and r such thatthere is an isomorphism of representations of Gal( Q p / Q p ) on F : V G, ˜ v (cid:16) Hom U v ( σ v , S ( V v , F )[ m Σ ]) ⊗ ( ω − ( n − ◦ det) (cid:17) ∼ = L ⊗ ( r ˜ v ) ⊕ d . (22) Remark 2.1.3.2. (i) In the special case σ v = 1, Conjecture 2.1.3.1 boils down to V G, ˜ v ( S ( U v , F )[ m Σ ] ⊗ ( ω − ( n − ◦ det)) ∼ = L ⊗ ( r ˜ v ) ⊕ d .(ii) Conjecture 2.1.3.1 implies that the G ( F ˜ v )-representation Hom U v ( σ v , S ( V v , F )[ m Σ ])determines the Gal( Q p / Q p )-representation L ⊗ ( r ˜ v ). Note that this doesn’t imply ingeneral that Hom U v ( σ v , S ( V v , F )[ m Σ ]) determines the Gal( F ˜ v /F ˜ v )-representation r ˜ v itself (though this is also expected, see [PQ] and the references therein).(iii) See §§3.2, 3.4 below for nontrivial evidence on Conjecture 2.1.3.1 when K isunramified and n = 2.We now check that, at least when p is odd, F/F + is unramified at finite placesand H × O F + [1 /N ] F + is quasi-split at finite places, Conjecture 2.1.3.1 holds for ˜ v if andonly if it holds for ˜ v c (these extra assumptions come from the use of [Tho12, §6] inthe next lemma). Lemma 2.1.3.3.
Assume p > , F/F + unramified at finite places and H × O F + [1 /N ] F + quasi-split at finite places of F + . Let ˜ v | v in F . Then the action of the center ( F + v ) × ⊆ GL n ( F + v ) on S ( V v , F )[ m Σ ] via ι ˜ v is given by det( r ˜ v ) ω n ( n − ( via local class field theoryfor F + v ) .Proof. We can assume S ( V v , F )[ m Σ ] = 0. The map S ( V v U v , O E ) −→ S ( V v U v , F ) be-ing surjective for U v small enough (see e.g. [BH15, Lemma 4.4.1] or [EGH13, §7.1.2]),we have a surjection of smooth H ( F + v )-representations: S ( V v , O E ) m Σ ։ S ( V v , F ) m Σ (23)(where S ( V v U v , O E ), S ( V v , O E ) m Σ are defined as S ( V v U v , F ), S ( V v , F ) m Σ replacing F by O E ). By classical local-global compatibility applied to (cid:16) lim −→ U S ( U, O E ) (cid:17) ⊗ O E E ,see e.g. [EGH13, Thm.7.2.1], we easily deduce with (23) that if ( F + v ) × acts via ι ˜ v onthe whole S ( V v , F )[ m Σ ] (inside S ( V v , F ) m Σ ) by a single character, then this charactermust be det( r ˜ v ) ω n ( n − .Let us prove that ( F + v ) × indeed acts by a character. The functor associating toany local artinian O E -algebra A with residue field F the set of isomorphism classesof deformations r A of r to A such that r cA ∼ = r ∨ A ⊗ ε − n is pro-representable by alocal complete noetherian O E -algebra R r, Σ of residue field F . When p > F/F + isunramified at finite places and H × O F + [1 /N ] F + is quasi-split at finite places of F + , itfollows from [Tho12, Prop.6.7] that there is a natural such deformation with valuesin T Σ ( V v U v , O E ) m Σ for any U v (where T Σ ( V v U v , O E ) m Σ is defined as T Σ ( V v U v , F ) m Σ
30n §2.1.2 replacing F by O E ), and hence by universality a continuous morphism oflocal O E -algebras: R r, Σ −→ T Σ ( V v U v , O E ) m Σ . (24)Likewise, the functor associating to any A as above the set of isomorphism classes ofGal( F ˜ v /F ˜ v ) ab -deformations of det( r ˜ v ) over A is pro-representable by the Iwasawaalgebra O E J Gal( F ˜ v /F ˜ v ) ab K , and considering det A ( r A | Gal( F ˜ v /F ˜ v ) ) for A = R r, Σ providesby the universal property again a continuous morphism of local O E -algebras: O E J Gal( F ˜ v /F ˜ v ) ab K −→ R r, Σ . (25)Since T Σ ( V v U v , O E ) m Σ acts by a character on S ( V v U v , F )[ m Σ ] for any U v , so is thecase of R r, Σ on S ( V v , F )[ m Σ ] by (24). Using (23), we see that it is enough to provethat the induced morphism O E J Gal( F ˜ v /F ˜ v ) ab K ( ) −→ R r, Σ ( ) −→ lim ←− U v T Σ ( V v U v , O E ) m Σ gives an action of Gal( F ˜ v /F ˜ v ) ab on S ( V v , O E ) m Σ which, when restricted to F × ˜ v ֒ → Gal( F ˜ v /F ˜ v ) ab (via the local reciprocity map), coincides with the action of F × ˜ v on S ( V v , O E ) m Σ as center of H ( F + v ) ι ˜ v ∼ = G ( F ˜ v ). We can work in S ( V v , O E ) m Σ ⊗ O E E , inwhich case this follows from local-global compatibility (as in [EGH13, Thm.7.2.1])and from the fact that, by construction of the map (24) (see [Tho12, §6]) and by(25), Gal( F ˜ v /F ˜ v ) ab acts on π V v ⊆ S ( V v , O E ) m Σ ⊗ O E E by multiplication by the char-acter det( r π ) | Gal( F ˜ v /F ˜ v ) , where π is an irreducible H ( A ∞ F + )-subrepresentation of (cid:16) lim −→ U S ( U, O E ) (cid:17) ⊗ O E E such that π V v occurs in S ( V v , O E ) m Σ ⊗ O E E and where r π isits associated (irreducible) p -adic representation of Gal( F /F ) ([EGH13, Thm.7.2.1]again).Let π be a smooth representation of G ( K ) = GL n ( K ) over F with central character Z ( π ) and denote by π ⋆ the smooth representation of G ( K ) with the same underlyingvector space as π but where g ∈ G ( K ) acts by τ ( g ) − . Lemma 2.1.3.4.
There is a
Gal( Q p / Q p ) -equivariant isomorphism V G ( π ⋆ ) ∼ = V G ( π ) ⊗ Z ( π ) − ( n − | Q × p , where Z ( π ) | Q × p is seen as a character of Gal( Q p / Q p ) via local class field theory.Proof. We use the notation of §2.1.1. Let w ∈ W be the element of maximallength, the isomorphism π N ∼ → π w N w , v w v shows that one can compute V G ( π ) using w N w instead of N and conjugating everything by w (e.g. x ∈ Z × p acts by w ξ G ( x ) w , etc.). Now, it is easy to check that the F -linear isomorphism31 π ⋆ ) N ∼ → π w N w , v w v is compatible with the F J X K [ F ]-module structure onboth sides but where we twist the F J X K [ F ]-action as follows on the right-hand side: X acts by (1 + X ) − − F acts by p − ( n − F , p − ( n − being here in the center of G ( K ). Likewise, it is compatible with the action of Z × p but where x ∈ Z × p acts by x − ( n − ξ G ( x ) on the right-hand side (with x − ( n − in the center of G ( K )). All thiseasily implies the lemma. Lemma 2.1.3.5.
Assume p > , F/F + unramified at finite places and H × O F + [1 /N ] F + quasi-split at finite places of F + . We have a Gal( Q p / Q p ) -equivariant isomorphism V G, ˜ v c (cid:16) Hom U v ( σ v , S ( V v , F )[ m Σ ]) (cid:17) ∼ = V G, ˜ v (cid:16) Hom U v ( σ v , S ( V v , F )[ m Σ ]) (cid:17) ⊗ ind ⊗ Q p F ˜ v (cid:16) det( r ˜ v ) − ( n − ω − n ( n − (cid:17) . Proof.
This follows from Lemma 2.1.3.4 applied to π = Hom U v ( σ v , S ( V v , F )[ m Σ ]) to-gether with Lemma 2.1.3.3, recalling that Z ( π ) | Q × p , seen as a character of Gal( Q p / Q p )via local class field theory, is ind ⊗ Q p F ˜ v ( Z ( π )) (where Z ( π ) is here seen as a character ofGal( F ˜ v /F ˜ v )). Proposition 2.1.3.6.
Assume p > , F/F + unramified at finite places and H × O F + [1 /N ] F + quasi-split at finite places of F + . Conjecture 2.1.3.1 holds for ˜ v ifand only if it holds for ˜ v c .Proof. This follows from Lemma 2.1.3.5 together with r ˜ v c ∼ = r ∨ ˜ v ⊗ ω − n , (21) and aneasy computation. C -groups We show that one can give a more general and more natural formulation of Conjecture2.1.3.1 (in the special case of Remark 2.1.3.2(i)) using C -parameters (Conjecture2.1.4.5).We start by some reminders about L -groups and C -groups.Let k be a field and k sep a separable closure of k . We note Γ k def = Gal( k sep /k ).Let H be a connected reductive group defined over k , let c H be its dual group, L H its L -group and C H its C -group. We refer to [Bor79, §2], [BG14, §§2,5], [GHS18,§9] and [Zhu, §1.1] for more details concerning these L -groups and C -groups. Notethat these two groups can be defined over Z . Their construction depends on thechoice of a pinning ( B H , T H , { x α } α ∈ S H ) of H k sep . The dual group c H has a naturalpinned structure ( B b H , T b H , { x b α } α ∈ S H ) with B b H a Borel subgroup of c H , T b H ⊆ B b H a maximal split torus and { x b α } α ∈ S H a pinning of ( B b H , T b H ) (see [Con14, §§5,6] for32he fact that everything can be defined over Z ) on which the group Γ k is acting.Let 1 → G m → f H → H → G m -extension of H (over k ) whoseexistence is proved in [BG14, Prop.5.3.1(a)]. The inverse images T e H and B e H of T H and B H in f H k sep are respectively a maximal torus and a Borel subgroup of f H k sep .Moreover, since the above extension is central, there is a unique pinning { e x α } α ∈ S H of ( B e H , T e H ) inducing { x α } α ∈ S H on ( B, T ) via the map f H k sep → H k sep . This gives riseto a pinned dual data ( cf H, B be B , T be H , { e x b α } α ∈ S H ) with an action of Γ k (trivial on someopen subgroup) and a Γ k -equivariant injection ( c H, B b H , T b H ) ֒ → ( cf H, B be H , T be H ) such that { x b α } α ∈ S H induces { e x b α } α ∈ S H .The L -groups and C -groups are then defined as the group schemes L H def = c H ⋊ Γ k C H def = cf H ⋊ Γ k . (26)We have the following simple description of cf H given in [Zhu, §1.1]. Let c H ad and T ad b H be the quotients of c H and T b H by the center of c H and let δ ad be the cocharacter of T ad b H ⊆ c H ad defined as the half sum of positive roots of c H with respect to ( B b H , T b H ).The group c H ad acts on c H by the adjoint action and, after precomposition with δ ad ,this defines an action, in the category of Z -group schemes, of G m on c H . There is anisomorphism of Z -group schemes cf H ∼ = c H ⋊ G m identifying B be H with B b H ⋊ G m and T be H with T b H ⋊ G m = T b H × G m . We note that, since δ ad is fixed by the Galois action, thisisomorphism is Galois equivariant. Using this isomorphism, we identify X ( T be H ) with X ( T b H ) × Z ∼ = X ∨ ( T H ) × Z . This shows that we have an exact sequence of Z -groupschemes: 1 −→ L H −→ C H d −−→ G m −→ . Let A be a topological Z p -algebra and assume from now on that k is either anumber field or a finite extension of Q p , so that we have an A -valued p -adic cyclotomiccharacter. We recall that a morphism ρ : Γ k → L H ( A ) is called admissible if itscomposition with the second projection L H ( A ) → Γ k is the identity (see [Bor79, §3]). Definition 2.1.4.1. An L -parameter (resp. C -parameter ) of H over A is an admis-sible continuous morphism ρ : Γ k −→ L H ( A ) (resp. ρ : Γ k −→ C H ( A ) such that d ◦ ρ is the p -adic cyclotomic character). When A is moreover an algebraically closed field,we say that two L -parameters (resp. C -parameters) of H over A are equivalent if theyare conjugate by an element of c H ( A ) (resp. cf H ( A )). Remark 2.1.4.2.
Assume A is an algebraically closed field. Each element of cf H ( A ) isthe product of an element of c H ( A ) and an element of the center of cf H ( A ). This can bededuced from [BG14, Prop.5.3.3] or [Zhu, (1.2)]. This implies that two C -parametersof H over A are equivalent if and only if they are conjugate by an element of c H ( A ).33or simplicity, we assume from now on that A is moreover an algebraically closedfield. We also assume (not for simplicity) that H has a connected center. We gener-alize now the representation L ⊗ ( ρ ) ⊗ F F p (see (20) for L ⊗ ( ρ )).Let ( λ α ∨ ) α ∈ S H be a family of fundamental coweights of H such that ξ H def = X α ∈ S H λ α ∨ ∈ X ( T b H ) ∼ = X ∨ ( T H ) (27)is fixed under the action of Γ k (compare with (13) and note that the cocharacters λ α ∨ exist since H has a connected center but each of them doesn’t have to be fixed by Γ k ).Let ( r λ α ∨ , V λ α ∨ ) be the irreducible algebraic representation of c H of highest weight λ α ∨ over A and let ( r ⊗ ξ H , V ⊗ ξ H ) be the irreducible algebraic representation of c H S H over A ofhighest weight ( λ α ∨ ) α ∈ S H = the character of T S H b H defined by ( x α ) α ∈ S H P α λ α ∨ ( x α ).Note that we have an isomorphism of algebraic representations of c H S H :( r ⊗ ξ H , V ⊗ ξ H ) ∼ = O α ∈ S H ( r λ α ∨ , V λ α ∨ ) . (28)Let γ ∈ Γ k and χ α,γ be the character of c H corresponding to the cocharacter γ ( λ α ∨ ) − λ γα ∨ ∈ X ∨ ( Z H ) ⊆ X ∨ ( T H ). Comparing the highest weights, for γ ∈ Γ k there is anisomorphism of algebraic irreducible representations of c H S H : (cid:16) r ⊗ ξ H ( γ − · ) , V ⊗ ξ H (cid:17) ∼ = (cid:16) ⊗ α ∈ S H ( r λ α ∨ ⊗ χ γ − α,γ ) ◦ c γ , V ⊗ ξ H (cid:17) , where c γ is the automorphism of c H S H defined by ( x α ) α ∈ S H ( x γ − α ) α ∈ S H . Thereforethere exists an A -linear automorphism M γ of V ⊗ ξ H , well defined up to a nonzero scalar,such that, for ( x α ) α ∈ S H ∈ c H ( A ) S H : M γ (cid:16) r ⊗ ξ H (( γ − x α ) α ∈ S H ) (cid:17) M − γ = (cid:16) ⊗ α ∈ S H r λ α ∨ ( x γ − α ) (cid:17) Y α ∈ S H χ α,γ ( x α ) . (29)Moreover the subspaces of highest weight of these two representations over V ⊗ ξ H beingthe same, we can choose M γ such that it induces the identity on this line. With thischoice, the map γ M γ is a representation of Γ k over V ⊗ ξ H . Since ξ H ∈ X ∨ ( T H ) Γ k ,we have Q α ∈ S H χ α,γ = 1 for all γ ∈ Γ k so that, for x ∈ c H ( A ), we have from (29) and(28) (replacing γ − x α by x for all α ∈ S H ): M γ (cid:16) ⊗ α ∈ S H r λ α ∨ ( x ) (cid:17) M − γ = (cid:16) ⊗ α ∈ S H r λ α ∨ ( γx ) (cid:17) . All this proves that there is an algebraic representation ( L ⊗ ξ H , V ⊗ ξ H ) of L H on V ⊗ ξ H defined by L ⊗ ξ H ( x, γ ) def = (cid:16) ⊗ α ∈ S H r λ α ∨ ( x ) (cid:17) M γ x ∈ c H ( A ) and γ ∈ Γ k . The isomorphism class of this representation does notdepend on the choice of the λ α ∨ such that ξ H = P λ α ∨ . Namely any other choice willtwist each r λ α ∨ by a character whose product over all α is trivial.If ρ is an L -parameter of H over A we define the Γ k -representation L ⊗ ξ H ( ρ ) as thecomposition L ⊗ ξ H ◦ ρ . Moreover if two L -parameters ρ and ρ are equivalent, therepresentations L ⊗ ξ H ( ρ ) and L ⊗ ξ H ( ρ ) are clearly isomorphic. If ρ is a C -parameter of H over A , ρ is in particular an L -parameter of f H over A by (26), and we define theΓ k -representation L ⊗ ,Cξ H ( ρ ) def = L ⊗ ξ e H ( ρ ), where ξ e H def = ( ξ H , ∈ X ( T be H ) ∼ = X ( T b H ) × Z . (30)We now compare L ⊗ ξ H ( ρ ), L ⊗ ,Cξ H ( ρ ) between k and finite extensions k ′ of k .We fix k ′ ⊆ k sep a finite extension of k , H ′ a connected reductive group over k ′ and we let H def = Res k ′ /k ( H ′ ). We let Σ k ′ be the set of embeddings k ′ ֒ → k sep inducingthe identity on k and τ ∈ Σ k ′ the inclusion k ′ ⊆ k sep . For τ ∈ Σ k ′ we choose g τ ∈ Γ k such that τ = g τ ◦ τ , and we have Γ k = ` τ ∈ Σ k ′ g τ Γ k ′ . The dual group c H of H isisomorphic to ind Γ k Γ k ′ c H ′ , i.e. the group scheme of functions f : Γ k → c H ′ such that f ( gh ) = h − f ( g ) for all g ∈ Γ k and h ∈ Γ k ′ (see [Bor79, §5.1(4)]). More explicitly,the map f ( f ( g τ )) τ ∈ Σ k ′ induces an isomorphism ind Γ k Γ k ′ c H ′ ∼ = c H ′ Σ k ′ and the actionof Γ k on c H ′ Σ k ′ is given by g · ( x τ ) τ ∈ Σ k ′ = (cid:16) ( g − τ gg g − ◦ τ ) x g − ◦ τ (cid:17) τ ∈ Σ k ′ . The map ( x τ ) τ ∈ Σ k ′ x τ is a Γ k ′ -equivariant map c H → c H ′ . It extends to a morphismof group schemes c H ⋊ Γ k ′ → L H ′ (resp. cf H ⋊ Γ k ′ → C H ′ ) inducing the identity onthe Γ k ′ factor (resp. the G m and Γ k ′ factors). If ρ is an L -parameter (resp. a C -parameter) of H over A , we can define an L -parameter (resp. a C -parameter) ρ ′ of H ′ by restriction of ρ to Γ k ′ and composition with the above morphism. Lemma 2.1.4.3.
The map ρ ρ ′ induces a bijection between equivalence classesof L -parameters ( resp. of C -parameters ) of H over A and equivalence classes of L -parameters ( resp. C -parameters ) of H ′ over A .Proof. A map ρ from Γ k to L H ( A ) of the form ( c ρ , Id) is admissible if and only if c ρ is a 1-cocycle of Γ k in c H ( A ) and is continuous if and only if c ρ is continuous.Moreover two admissible ρ are equivalent if and only if they are conjugate by anelement of c H ( A ). Therefore the map associating to ρ the class [ c ρ ] of c ρ inducesa bijection between the set of equivalence classes of L -parameters and the set ofclasses [ c ] ∈ H (Γ k , c H ( A )). The fact that the above map ρ ρ ′ induces an35somorphism H (Γ k , c H ( A )) ∼ → H (Γ k ′ , c H ′ ( A )) is a consequence of a nonabelianversion of Shapiro’s Lemma (see for example [Sti10, Prop.8] noting that everythingcan be made continuous there or [GHS18, Lemma 9.4.1] in a more restricted context).Therefore the map associated to a C -parameter ρ the class [ c ρ ] of c ρ induces abijection between the set of equivalence classes of C -parameters and the set of classes c ∈ H (Γ k , cf H ( A )) such that d ( c ) ∈ H (Γ k , A × ) ∼ = Hom contgp (Γ k , A × ) coincides withthe p -adic cyclotomic character. Let f H def = Res k ′ /k f H ′ , so that f H can be identifiedto a quotient of f H . It follows from Remark 2.1.4.2 that H (Γ k , dg H ( A )) is the set ofclasses of 1-cocycles of Γ k with values in dg H ( A ) up to c H ( A )-conjugation. It followsagain from Remark 2.1.4.2 that the set of equivalence classes of C -parameters of H over A is in bijection with the subset of H (Γ k , df H ( A )) of classes whose image in H (Γ k , ( A × ) [ k ′ : k ] ) ∼ = Hom contgp (Γ k , ( A × ) [ k ′ : k ] ) is the image of the p -adic cyclotomiccharacter via the diagonal embedding A × ֒ → ( A × ) [ k ′ : k ] . The conclusion follows fromthe commutativity of the following diagram H (cid:16) Γ k , cf H ( A ) (cid:17) H (cid:16) Γ k , ( \ Res k ′ /k G m )( A ) (cid:17) H (cid:16) Γ k ′ , cf H ′ ( A ) (cid:17) H (cid:16) Γ k ′ , d G m ( A ) (cid:17) ≀ ≀ and from the fact that the classes corresponding to the cyclotomic characters corre-spond under the right vertical arrow. Lemma 2.1.4.4.
Let ρ be an L -parameter, resp. a C -parameter, of H over A and ρ ′ the L -parameter, resp. C -parameter, of H ′ over A corresponding to ρ by Lemma2.1.4.3. Let ξ H ′ ∈ X ( T c H ′ ) be as in (27) ( with H ′ instead of H ) and let ξ H ∈ X ( T b H ) ∼ = X ( T c H ′ ) Σ k ′ be the character ( ξ H ′ ) τ ∈ Σ k ′ ( which is fixed by Γ k ) . Then we have an iso-morphism of representations of Γ k over A : L ⊗ ξ H ( ρ ) ∼ = ind ⊗ kk ′ (cid:16) L ⊗ ξ H ′ ( ρ ′ ) (cid:17) resp. L ⊗ ,Cξ H ( ρ ) ∼ = ind ⊗ kk ′ (cid:16) L ⊗ ,Cξ H ′ ( ρ ′ ) (cid:17) . Proof.
Let ρ ′ be an L -parameter of H ′ over A . If g ∈ Γ k and τ ∈ Σ k ′ , let gg τ = g g ◦ τ h ( g, τ ) with h ( g, τ ) ∈ Γ k ′ . For g ∈ Γ k , we can check that the above automorphism M g of V ⊗ ξ H = ( V ⊗ ξ H ′ ) ⊗ [ k ′ : k ] is defined by M g ( ⊗ τ ∈ Σ k ′ v τ ) = ⊗ τ ∈ Σ k ′ ( M h ( g,g − ◦ τ ) v g − ◦ τ ).Moreover, setting for g ∈ Γ k : ρ ( g ) def = (cid:16) ( ρ ′ ( h ( g, g − ◦ τ )) τ ∈ Σ k ′ , g (cid:17) ∈ c H ′ ( A ) Σ k ′ ⋊ Γ k it is easy to check that ρ is an admissible morphism and that the equivalence class of ρ corresponds to ρ ′ via Lemma 2.1.4.3. The result follows from an explicit computationtogether with the definition of the tensor induction ([CR81, §13], see also the end of36he proof of Lemma 2.4.2.3 below). The case of C -parameters can be deduced fromthe case of L -parameters as in the proof of Lemma 2.1.4.3.We will later need to “untwist” a C -parameter into an L -parameter. This canbe done when the group H has a twisting element (as we assumed in §2.1.1), i.e. acharacter θ H ∈ X ( T H ) Γ k ∼ = X ∨ ( T b H ) Γ k such that h θ H , α ∨ i = 1 for all α ∈ S H . By [Zhu,(1.3)], there exists a Galois equivariant isomorphism cf H ∼ = c H × G m given explicitly by t θ H : c H ⋊ G m ∼ = c H × G m ( h, t ) ( hθ H ( t ) , t ) . This induces an isomorphism of group schemes C H ∼ = L H × G m . The choice of θ H gives a bijection between the equivalence classes of C -parameters and of L -parametersof H over A given by ρ C ρ , so that t θ H ◦ ρ C ∼ = ( ρ, ε ), where ε is (the image in A × )of the p -adic cyclotomic character.Let ξ H ∈ X ∨ ( T H ) Γ k ∼ = X ( T b H ) Γ k be a dominant character of c H fixed by Γ k as above.The algebraic representation r ξ e H ◦ t − θ H of c H × G m (see (30) for ξ e H ) is the representationof highest weight ( ξ H , −h ξ H , θ H i ) and similarly L ⊗ ξ e H ◦ t − θ H = L ⊗ ξ H ⊗ x −h ξ H ,θ H i (where wenote x h the character x x h of G m ). This proves that we have L ⊗ ,Cξ H ( ρ C ) ∼ = L ⊗ ξ H ( ρ ) ⊗ ε −h ξ H ,θ H i . (31)On order to state the reformulation/generalization Conjecture 2.1.3.1 (more pre-cisely of its variant in Remark 2.1.3.2(i) and extending scalars from F to F p ), webroaden the global setting of §2.1.2 following [DPS].We now let H be a connected reductive group defined over Q . We fix some compactopen subgroup U p ⊆ H ( A ∞ ,p Q ) satisfying the hypotheses of [DPS, §9.2] ( U p there isdenoted K pf ). For i ≥ f H i ( F p ) be the completed cohomology of thetower of locally symmetric spaces associated to H of tame level U p defined in [Eme06](see [DPS, §9.2]). Let Σ be a set of finite places of Q containing p and the places of Q where H is not unramified or U p is not hyperspecial. Let T Σ be the abstract Heckealgebra defined as the tensor product of the spherical Z [ p − ]-Hecke algebras H ℓ of H ( Q ℓ ) with respect to U pℓ . We recall that a maximal open ideal m ⊆ T Σ is weaklynon-Eisenstein [DPS, Def.9.13] if the equivalent assumptions of [DPS, Lemma 9.10]are satisfied. In this case there is a unique q ≥ f H q ( F p ) m = 0. Then the H ( Q p )-representation f H q ( F p )[ m ] is smooth and admissible and the residue field of m is finite. We choose an embedding T / m ֒ → F p .Considering [DPS, Conj.9.3.1], the following construction is natural. Let r C :Gal( Q / Q ) → C H ( F p ) be a C -parameter unramified outside a finite number of primesand choose Σ big enough to contain all the primes of ramification of r C . For each37 / ∈ Σ, let x ℓ : H ℓ → F p be the character such that the semisimplification of r C (Frob ℓ )is contained in the c H ( F p )-conjugacy class CC ( x ℓ ) ζ ( ℓ − ) of C H ( F p ) defined by theversion of Satake isomorphism for C -groups in [Zhu] and ζ is the cocharacter t (2 δ ad ( t − ) , t ) of cf H (recall δ ad is defined at the beginning of this section). We define m Σ as the maximal ideal of T Σ generated by the kernels of all the x ℓ with ℓ / ∈ Σ. Notethat this gives us a natural embedding T Σ / m Σ ֒ → F p .Assume that H Q p def = H × Q Q p is isomorphic to Res K/ Q p ( H ′ ) for a finite extension K of Q p and some split connected reductive group H ′ over K (in particular H Q p is quasi-split) and that H ′ has a connected center. Then we can fix a cocharacter ξ H ′ of H ′ such that h ξ H ′ , α i = 1 for all α ∈ S H ′ and define ξ H Q p def = Res K/ Q p ( ξ H ′ ) | G m (restrictionto the diagonal embedding G m ֒ → Res K/ Q p ( G m ) = G [ K : Q p ] m ), which is a cocharacter of H Q p satisfying h ξ H Q p , α i = 1 for all α ∈ S H Q p . We can finally conjecture: Conjecture 2.1.4.5.
Assume that the H ( Q p ) -representation π def = f H q ( F )[ m Σ ] isnonzero. Then D ∨ ξ H ′ ( π ) ( defined similarly to ( )) is finite-dimensional over F p (( X )) and there is an integer d ∈ Z > such that we have an isomorphism of representationsof Gal( Q p / Q p ) over F p : V ∨ (cid:16) D ∨ ξ H ′ ( π ) (cid:17) ⊗ T Σ / m Σ F p ∼ = (cid:16) L ⊗ ,Cξ H Q p (cid:16) r C | Gal( Q p / Q p ) (cid:17)(cid:17) ⊕ d . We now check that, when H is the restriction of scalars of a compact unitarygroup as in §2.1.2, Conjecture 2.1.4.5 is equivalent to the special case of Conjecture2.1.3.1 in Remark 2.1.3.2(i) where the coefficient field is F p instead of F .We go back to the notation of §§2.1.2, 2.1.3 and we fix an embedding F ֒ → F p . Forsimplification we assume that there is a unique place v of F + over p and we fix ˜ v in F above v , so that we have an isomorphism (Res F + / Q H ) × Q Q p ∼ = Res F ˜ v / Q p GL n . Thebase field k at the beginning is now F + , the connected reductive group H over k isthe compact unitary group H of §2.1.2 (so that c H ∼ = G = GL n ), ξ H is the cocharacter ξ G of Example 2.1.1.3, the twisting element θ H is the character θ G of Example 2.1.1.3and the algebraically closed field A is F p .Let r be a continuous irreducible representation Gal( Q /F ) → GL n ( F p ) as in§2.1.3 (composed with our embedding F ֒ → F p ). Let r ′ : Gal( Q /F + ) → G n ( F p ) bethe continuous morphism associated to r using [CHT08, Lemma 2.1.4] and denote by( r ′ ) C : Gal( Q /F + ) → C H ( F p ) the C -parameter of H over F p obtained by the con-struction of [BG14, §8.3]. A simple computation shows that ( r ′ ) C (or more preciselyits composition with cf H ⋊ Gal( Q /F + ) ։ cf H ⋊ Gal(
F/F + )) is the composition of ( r ′ , ω )with G n × G m −→ c H ⋊ ( G m × Gal(
F/F + ))( g, µ, γ, λ ) ( gθ ′ H ( λ ) − , λ, γ ) (32)38here g ∈ GL n ( F p ), µ, λ ∈ F × p , γ ∈ Gal(
F/F + ) and θ ′ H ∈ X ( T ) is the character θ ′ H (diag( x , . . . , x n )) = x − x − · · · x − nn . Finally we define r C as the C -parameter ofRes F + / Q ( H ) over F p obtained from the application of Lemma 2.1.4.3 to ( r ′ ) C . Wecan check that the maximal ideal m Σ of T Σ defined by r C coincides with the ideal m Σ defined in §2.1.3. This can be checked using the formulas relating the Satakeisomorphism for C -groups with the usual Satake isomorphism ([Zhu, §1.4]) and theexplicit formulas [Gro98, (3.13)], [Gro98, (3.14)].Note that, seeing now θ H and θ ′ H as cocharacters of T (recall d GL n ∼ = GL n ), wehave θ H ◦ ω = ( θ ′ H ◦ ω ) ω n − , so that we have, using (32):( r ′ ) C = t − θ H ◦ (( r ⊗ ω n − ) , ω ) . Let ξ v def = ξ H × F + F + v and ξ p def = Res F + v / Q p ( ξ v ) | G m . Then (31) and Lemma 2.1.4.4 imply(note that ξ v is fixed by Gal( Q p /F + v ) since H × F + F + v is split): L ⊗ ,Cξ p ( r C | Gal( Q p / Q p ) ) ∼ = ind ⊗ Q p F + v (cid:16) r ⊗ ξ v ( r ˜ v ⊗ ω n − ) ω −h ξ H ,θ H i (cid:17) = L ⊗ ( r ˜ v ) ⊗ δ − G . This shows that Conjecture 2.1.4.5 is equivalent to the special case of Conjecture2.1.3.1 in Remark 2.1.3.2(i) (with F p instead of F ). L ⊗ From now on we assume that K is unramified (i.e. K = Q p f ). We define the algebraicrepresentation L ⊗ of Q σ ∈ Gal( K/ Q p ) G together with “good subquotients” of L ⊗ , andprove various properties of these good subquotients. This section is entirely on the“Galois side” (though no Galois representation appears yet). All the results, exceptRemark 2.2.3.12, in fact hold for any split reductive connected algebraic group G/ Z with connected center. We define good subquotients of L ⊗ .If H is an algebraic group over Z , we now write H instead of H × Z F (in ordernot to burden the notation) and H Gal( K/ Q p ) for the group product Q σ ∈ Gal( K/ Q p ) H (itis not a subgroup of H !).We define the following algebraic representation of G Gal( K/ Q p ) over F : L ⊗ def = O Gal( K/ Q p ) (cid:18) O α ∈ S L ( λ α ) (cid:19) (33)39recall L ( λ α ) is defined in (19)). Note that L ⊗ is also the tensor product of allfundamental representations of the product group G Gal( K/ Q p ) . In particular the center Z Gal( K/ Q p ) G acts on L ⊗ by the character θ G | Z G ⊗ · · · ⊗ θ G | Z G | {z } Gal( K/ Q p ) , where θ G is as in Example2.1.1.3, i.e. θ G = X α ∈ S λ α ∈ X ( T ) . (34) Remark 2.2.1.1. (i) With the notation of §2.1.4, the representation L ⊗ is the re-striction to c H of the representation ( L ⊗ ξ H , V ⊗ ξ H ) of L H , where k = F , H = Res K/ Q p ( G )and ξ H = ( ξ G , . . . , ξ G ) ∈ X ( T b H ) ( ξ G as in Example 2.1.1.3).(ii) Since λ α ∈ ⊕ ni =1 Z ≥ e i , all the weights of X ( T ) appearing in each L ( λ α ) | T are alsoin ⊕ ni =1 Z ≥ e i , and thus the same holds for the weights of L ⊗ | T (where T is diagonallyembedded into G Gal( K/ Q p ) ). This follows from the classical fact that the weights ap-pearing in L ( λ ) | T for any dominant λ ∈ X ( T ) are the points in ⊕ ni =1 Z e i ∼ = X ( T ) ofthe convex hull in ⊕ ni =1 R e i of the weights w ( λ ), w ∈ W .Fix P a standard parabolic subgroup of G , if R is a finite-dimensional algebraicrepresentation of P Gal( K/ Q p ) over F , we write R | Z MP for the restriction of R to Z M P acting via the diagonal embedding Z M P ֒ → Z Gal( K/ Q p ) M P ⊆ G Gal( K/ Q p ) . (35)Since Z M P is a torus, it follows from [Jan03, §I.2.11] that R | Z MP is the direct sum of its isotypic components. For instance, if P = G and R = L ⊗ , there is only oneisotypic component as Z M G = Z G acts on L ⊗ via the character f θ G | Z G . Lemma 2.2.1.2.
Any isotypic component of R | Z MP carries an action of M Gal( K/ Q p ) P when viewed inside R | M Gal( K/ Q p ) P .Proof. This just comes from the fact that the action of Z M P commutes with that of M Gal( K/ Q p ) P . Definition 2.2.1.3.
Let e P ⊆ P be a Zariski closed algebraic subgroup containing M P and R an algebraic representation of P Gal( K/ Q p ) over F , a subquotient (resp. subrepre-sentation, resp. quotient) of R | e P Gal( K/ Q p ) is a good subquotient (resp. subrepresentation,resp. quotient) if its restriction to Z M P is a (direct) sum of isotypic components of R | Z MP . Remark 2.2.1.4.
A Zariski closed subgroup e P as in Definition 2.2.1.3 actually de-termines the standard parabolic subgroup P that contains it. Indeed, assume thereis another standard parabolic subgroup P ′ such that M P ′ ⊆ e P ⊆ P ′ . Then we have M P ′ ⊆ P which implies P ′ ⊆ P . Symmetrically, we also have P ⊆ P ′ , hence P = P ′ .40ince isotypic components of R | Z MP tautologically occur with multiplicity 1, wesee in particular that there is only a finite number of good subquotients of R | e P Gal( K/ Q p ) .For instance the entire L ⊗ is the only good subquotient of L ⊗ | G Gal( K/ Q p ) . If ee P ⊆ e P is another Zariski closed algebraic subgroup as in Definition 2.2.1.3, any good sub-quotient (resp. subrepresentation, resp. quotient) of R | e P Gal( K/ Q p ) is a good subquotient(resp. subrepresentation, resp. quotient) of R | ee P Gal( K/ Q p ) (but the converse is wrong). Lemma 2.2.1.5.
There exists a filtration on L ⊗ | e P Gal( K/ Q p ) by good subrepresentationssuch that the graded pieces exhaust the isotypic components of L ⊗ | Z MP seen as rep-resentations of e P Gal( K/ Q p ) via the surjection e P Gal( K/ Q p ) ։ M Gal( K/ Q p ) P and Lemma2.2.1.2.Proof. It is enough to prove the lemma for e P = P . We prove the following statement(which implies the lemma): let H be a split connected reductive algebraic group over Z with connected center, T H ⊆ H a split maximal torus in H , B H ⊆ H a Borelsubgroup containing T H with set of (positive) roots R + H , V a finite-dimensional H -module over F , Q H ⊆ H a parabolic subgroup containing B H with Levi decomposition M Q H N Q H and center Z M QH ⊆ T H , Z ′ M QH a subtorus of Z M QH and λ ′ Q H ∈ X ( Z ′ M QH ) def =Hom Gr ( Z ′ M QH , G m ). Then the Z ′ M QH -isotypic component V λ ′ QH of V is a quotient oftwo subrepresentations in V | Q H which are both direct sums of isotypic componentsof V | Z ′ MQH (one applies this result to H = G Gal( K/ Q p ) , V = L ⊗ , Q H = P Gal( K/ Q p ) and Z ′ M QH = Z M P ). Note that as above V = ⊕ λ ′ QH V λ ′ QH and that V λ ′ QH carries from V | M QH an action of M Q H by the same proof as for Lemma 2.2.1.2. Let R ( Q H ) + ⊆ R + H be the positive roots of M Q H , if α ∈ R + H \ R ( Q H ) + , denote by α its image via thequotient map X ( T H ) ։ X ( Z ′ M QH ) and N α ⊆ N Q H the root subgroup. If n α ∈ N α and λ ′ Q H ∈ X ( Z ′ M QH ), then we have n α ( V λ ′ QH ) ⊆ P + ∞ i =0 V λ ′ QH + iα by [Jan03, §II.1.19](the sum being finite inside V ). Fix λ ′ Q H ∈ X ( Z ′ M QH ) that occurs in V | Z ′ MQH and let W ( λ ′ Q H ) be the set of λ ′′ Q H ∈ X ( Z ′ M QH ) of the form λ ′ Q H + (cid:16) P α ∈ R + H \ R ( Q H ) + Z ≥ α (cid:17) thatoccur in V | Z ′ MQH , we deduce that both subspaces X λ ′′ QH ∈W ( λ ′ QH ) \{ λ ′ QH } V λ ′′ QH ( X λ ′′ QH ∈W ( λ ′ QH ) V λ ′′ QH are preserved by N Q H , hence by Q H , inside V . Since their cokernel is exactly V λ ′ QH ,this proves the statement.We will use the following lemma extensively. Lemma 2.2.1.6. If Q is a ( standard ) parabolic subgroup of G containing P , any iso-typic component of R | Z MQ is a good subquotient of R | P Gal( K/ Q p ) ( hence of R | e P Gal( K/ Q p ) ) . roof. By Lemma 2.2.1.5 (applied in the case e P = P and with P there being Q ), suchan isotypic component is a good subquotient of R | Q Gal( K/ Q p ) , and thus is a subquotientof R | P Gal( K/ Q p ) since P ⊆ Q . It is also obviously a direct sum of isotypic componentsof R | Z MP since Z M Q ⊆ Z M P . This proves the lemma. Remark 2.2.1.7.
Let e P , P and R as in Definition 2.2.1.3 and define a good sub-quotient of R | e P (for the diagonal embedding e P ֒ → e P Gal( K/ Q p ) similar to (35)) as asubquotient of R | e P such that its restriction to Z M P is a sum of isotypic components of R | Z MP . Then, using the same kind of argument as for the proof of Lemma 2.2.1.5, onecan prove that a good subquotient of R | e P is also a good subquotient of R | e P Gal( K/ Q p ) ,so that good subquotients of R | e P and of R | e P Gal( K/ Q p ) are actually the same. Fix P ⊆ G a standard parabolic subgroup and C P an isotypic component of L ⊗ | Z MP ,we associate to C P a subset of the set of simple roots S (see (37)), as well as thestandard parabolic subgroup of G , denoted by P ( C P ), corresponding to this subset.We will use the following two lemmas, the first being well-known. Lemma 2.2.2.1.
Let λ ∈ X ( T ) ⊗ Z Q be dominant. Then the Weyl group of theroot subsystem of R generated by the simple roots α ∈ S such that s α fixes λ is thesubgroup of W of elements fixing λ . Lemma 2.2.2.2.
Let α ∈ S . Then P w ∈ W ( P ) w ( α ) ≥ , and we have P w ∈ W ( P ) w ( α ) =0 if and only if α ∈ S ( P ) . Moreover, if α ∈ S \ S ( P ) , then α is in the support of P w ∈ W ( P ) w ( α ) .Proof. If α ∈ S ( P ), it is clear that P w ∈ W ( P ) w ( α ) = 0 since, for each w ∈ W ( P ),we also have ws α ∈ W ( P ). If α ∈ S \ S ( P ), then − α is dominant for M P , thatis, −h α, β i ≥ β ∈ S ( P ). This implies that w ( − α ) ≤ − α for w ∈ W ( P ).Summing over W ( P ) gives − P w ∈ W ( P ) w ( α ) ≤ −| W ( P ) | α or equivalently | W ( P ) | α ≤ P w ∈ W ( P ) w ( α ). This proves the lemma.If w ∈ W satisfies w ( S ( P )) ⊆ S , we denote by w P the standard parabolic subgroupof G whose associated set of simple roots is w ( S ( P )). It has Levi subgroup M w P = wM P w − (so w P = ( wM P w − ) N ) and Weyl group W ( w P ) = wW ( P ) w − (caution: w P is not wP w − if w = 1!). If λ ∈ X ( T ), we define λ ′ def = 1 | W ( P ) | X w ′ ∈ W ( P ) w ′ ( λ ) ∈ ( X ( T ) ⊗ Z Q ) W ( P ) . (36)42 emark 2.2.2.3. (i) Note that λ ′ only depends on λ | Z MP since two distinct λ with thesame restriction to Z M P differ by an element in P α ∈ S ( P ) Z α and since P w ′ ∈ W ( P ) w ( α ) =0 for α ∈ S ( P ) by Lemma 2.2.2.2.(ii) It easily follows from the definitions and Lemma 2.2.2.2 that if w ∈ W satisfies w ( S ( P )) ⊆ S and λ ∈ X ( T ) is any weight, then w ( λ ′ ) = ( w ( λ )) ′ , where ( w ( λ )) ′ isgiven by the same formula as in (36) applied to the parabolic w P and the character w ( λ ). Lemma 2.2.2.4.
Let P be a standard parabolic subgroup of G . (i) Let λ ∈ X ( T ) , there exists w ∈ W such that w ( S ( P )) ⊆ S and w ( λ ) | Z MwP coincides with the restriction to Z M wP of a dominant weight of X ( T ) ⊗ Z Q . (ii) Let λ ∈ X ( T ) such that λ | Z MP occurs in L ⊗ | Z MP and let w as in (i). Thenwe have f θ G − w ( λ ) = P α ∈ S n α α for some n α ∈ Z ≥ ( see (34) for θ G ) and thesubset w ( S ( P )) ∪ { α ∈ S, n α = 0 } ⊆ S (37) only depends on λ | Z MP .Proof. (i) We first claim that it is equivalent to find w such that w ( S ( P )) ⊆ S and w ( λ ′ ) is dominant with λ ′ as in (36). Assume we have such a w , since w ′ ( λ ) | Z MP = λ | Z MP for all w ′ ∈ W ( P ), we have λ ′ | Z MP = λ | Z MP and thus w ( λ ) | Z MwP = w ( λ ′ ) | Z MwP .Conversely, assume that there is w such that w ( S ( P )) ⊆ S and w ( λ ) | Z MwP = µ | Z MwP for some dominant µ in X ( T ) ⊗ Z Q , and set µ ′ def = | W ( P ) | P w ′ ∈ W ( w P ) w ′ ( µ ) ∈ ( X ( T ) ⊗ Z Q ) W ( w P ) . Then we have µ ′ = w ( λ ′ ) by Remark 2.2.2.3(ii) and µ ≥ µ ′ (as µ ≥ w ′ ( µ ) for any w ′ ∈ W since µ is dominant). Thus µ − w ( λ ′ ) = µ − µ ′ = P α ∈ S ( w P ) n α α for some n α ∈ Q ≥ (recall µ | Z MwP = µ ′ | Z MwP ). This implies that h w ( λ ′ ) , β i = h µ, β i − X α ∈ S ( w P ) n α h α, β i ≥ β ∈ S \ S ( w P ) (as µ is dominant and h α, β i ≤ α = β ∈ S ). Since h w ( λ ′ ) , β i = h µ ′ , β i = 0 for β ∈ S ( w P ) (use again Lemma 2.2.2.2), we see that w ( λ ′ )is dominant.Now let us find such a w . First, pick w ′ ∈ W such that w ′ ( λ ′ ) is dominant, byLemma 2.2.2.1 applied to w ′ ( λ ′ ) the set of elements β in S such that s β fixes w ′ ( λ ′ )generate a root subsystem of R with corresponding Weyl group the subgroup of W ofelements that fix w ′ ( λ ′ ). This root subsystem has two natural bases of simple roots:namely the elements β above and the elements w ′ ( γ ) ∈ w ′ ( S ) such that s γ fixes λ ′ (they are usually distinct as W doesn’t preserve S ). This second basis obviouslycontains w ′ ( S ( P )). Therefore, there is w ′′ in the Weyl group of this root subsystem,43.e. w ′′ ∈ W fixing w ′ ( λ ′ ), that maps the second basis to the first. In particular wehave w ′′ w ′ ( S ( P )) ⊆ S and w ′′ w ′ ( λ ′ ) = w ′ ( λ ′ ) dominant, thus we can take w def = w ′′ w ′ .(ii) The positivity of the n α follows from the fact f θ G is the highest weight of L ⊗ | T (forthe diagonal embedding of T as in (35)). Let w , w as in (i) and λ ′ as in (36). Then w ( λ ′ ) = w ( λ ′ ) as these two weights are dominant (by the first part of the proof of(i)) and in a single W -orbit. Since λ ′ only depends on λ | Z MP by Remark 2.2.2.3(i), it istherefore enough to prove that the support of f θ G − w ( λ ′ ) is exactly the set of simpleroots (37) for one (any) w as in (i). Writing f θ G − w ( λ ′ ) = ( f θ G − w ( λ ))+( w ( λ ) − w ( λ ′ ))and recalling that w ( λ ) − w ( λ ′ ) is a sum of roots in w ( S ( P )) ⊆ S (as w ( λ ) , w ( λ ′ ) havesame restriction to Z M wP from the proof of (i)), we see that this support is containedin (37) and that it contains { α ∈ S \ w ( S ( P )) , n α = 0 } . It is thus enough to provethat this support also contains w ( S ( P )). Since f θ G ≥ w ( λ ′ ) (use f θ G ≥ ww ′ ( λ ) forany w ′ ∈ W and sum over w ′ ∈ W ( P )) and h β, α i ≤ α = β ∈ S , it is enough tocheck that h f θ G − w ( λ ′ ) , α i > Q ) for any α ∈ w ( S ( P )). But this follows fromLemma 2.2.2.2 and h f θ G − w ( λ ′ ) , α i = f h θ G , α i − h w ( λ ′ ) , α i = f − f . Remark 2.2.2.5.
Note that it is not true in general that, for λ as in Lemma2.2.2.4(ii), one can find w ∈ W such that w ( S ( P )) ⊆ S and w ( λ ) | Z MwP is the re-striction to Z M wP of a dominant weight of X ( T ) (one really needs X ( T ) ⊗ Z Q ).The proof of Lemma 2.2.2.4 also gives the following equivalent proposition thatwe will use repeatedly in the sequel. Proposition 2.2.2.6.
Let P be a standard parabolic subgroup of G . (i) Let λ ∈ X ( T ) and λ ′ as in (36) , there exists w ∈ W such that w ( S ( P )) ⊆ S and w ( λ ′ ) is a dominant weight of X ( T ) ⊗ Z Q . (ii) Let λ ∈ X ( T ) such that λ | Z MP occurs in L ⊗ | Z MP and let w as in (i). Then wehave f θ G − w ( λ ′ ) = P α ∈ S n α α for some n α ∈ Q ≥ and the support of f θ G − w ( λ ′ ) is S ( P ( C P )) . Let C P be an isotypic component of L ⊗ | Z MP associated to some λ P ∈ X ( Z M P ) =Hom Gr ( Z M P , G m ). We denote by P ( C P ) the unique standard parabolic subgroup of G whose associated set of simple roots S ( P ( C P )) is (37) for one (equivalently any)weight λ ∈ X ( T ) such that λ | Z MP = λ P . We also define W ( C P ) def = { w ∈ W as in Proposition 2 . . . λ ∈ X ( T ) , λ | Z MP = λ P } (38)( W ( C P ) doesn’t depend on the choice of such λ by the claim in the proof of Lemma2.2.2.4(i) and by Remark 2.2.2.3(i)). We see from (37) that for all w ∈ W ( C P ) wehave the inclusion w P ⊆ P ( C P ) . (39)44 ote that the set W ( C P ) is not in general a group, in particular it is distinct ingeneral from the Weyl group W ( P ( C P )) (see Lemma 2.2.2.10 below for the relationbetween the two). Remark 2.2.2.7.
The inclusion w P ⊆ P ( C P ) for some w ∈ W (such that w ( S ( P )) ⊆ S ) doesn’t imply w ∈ W ( C P ) (take P = B ). Also P ( C P ) doesn’t necessarily contain P , see e.g. the end of Example 2.2.2.9(ii) below. The subgroup generated by all w P for w ∈ W ( C P ) may also be strictly contained in P ( C P ) (see e.g. Example 2.2.2.9(i)below).The parabolic subgroups P ( C P ) respect inclusions. Lemma 2.2.2.8.
Let P ′ ⊆ P be two standard parabolic subgroups of G , C P anisotypic component of L ⊗ | Z MP and C P ′ an isotypic component of L ⊗ | Z MP ′ such that C P ′ ⊆ C P | Z MP ′ . Then P ( C P ′ ) ⊆ P ( C P ) .Proof. Let λ ∈ X ( T ) such that C P ′ is the isotypic component of λ | Z MP ′ . Then byassumption C P is the isotypic component of λ | Z MP . Define λ ′ P ∈ ( X ( T ) ⊗ Z Q ) W ( P ) , λ ′ P ′ ∈ ( X ( T ) ⊗ Z Q ) W ( P ′ ) as in (36) for respectively P and P ′ and let ( w P , w P ′ ) ∈ W × W such that w P ( S ( P )) ⊆ S and w P ( λ ′ P ) dominant, w P ′ ( S ( P ′ )) ⊆ S and w P ′ ( λ ′ P ′ )dominant ( w P , w P ′ exist by Proposition 2.2.2.6(i)). Then we have w P ( λ ′ P ) = 1 | W ( P ) | X w ′ ∈ W ( wP P ) w ′ w P ( λ ) , w P ( λ ′ P ′ ) = 1 | W ( P ′ ) | X w ′ ∈ W ( wP P ′ ) w ′ w P ( λ )and also w P ( λ ′ P ) = | W ( P ′ ) || W ( P ) | X σ ∈ W ( wP P ) /W ( wP P ′ ) σw P ( λ ′ P ′ ) . (40)Since w P ′ ( λ ′ P ′ ) is dominant, we have w P ′ ( λ ′ P ′ ) ≥ ww P ′ ( λ ′ P ′ ) for any w ∈ W and in par-ticular w P ′ ( λ ′ P ′ ) ≥ σw P ( λ ′ P ′ ) = ( σw P w − P ′ ) w P ′ ( λ ′ P ′ ). Summing up these inequalitiesover σ ∈ W ( w P P ) /W ( w P P ′ ) and multiplying by | W ( P ′ ) || W ( P ) | , one gets with (40): w P ′ ( λ ′ P ′ ) ≥ w P ( λ ′ P ) . (41)Now the result follows from f θ G − w P ( λ ′ P ) = (cid:16) f θ G − w P ′ ( λ ′ P ′ ) (cid:17) + (cid:16) w P ′ ( λ ′ P ′ ) − w P ( λ ′ P ) (cid:17) together with Proposition 2.2.2.6(ii) and (41). Example 2.2.2.9.
We give a few simple examples (beyond GL ( Q p )).(i) Assume n = 2 and P = B . Then L ⊗ | Z MB = L ⊗ | T has f + 1 isotypic components C ( λ i ) given by the characters λ i : diag( x , x ) x f − i x i for 0 ≤ i ≤ f . For i < f / i is dominant, W ( C ( λ i )) = { } and f θ G − λ i = i ( e − e ). For i = f / f is even), λ i = s e − e ( λ i ) is dominant, W ( C ( λ i )) = { , s e − e } and f θ G − w ( λ i ) = f / e − e )for w ∈ W ( C ( λ i )). For i > f / s e − e ( λ i ) is dominant, W ( C ( λ i )) = { s e − e } and f θ G − s e − e ( λ i ) = ( f − i )( e − e ). We see that w B = B ( P ( C ( λ i )) = G if i / ∈ { , f } and w B = P ( C ( λ i )) = B if i ∈ { , f } .(ii) Assume n = 3 and K = Q p .If P = B , then L ⊗ | T has 7 isotypic components given by the 6 characters λ w :diag( x , x , x ) x w − (1) x w − (2) for w ∈ S and the character det : diag( x , x , x ) x x x . If C P corresponds to some λ w , one gets that W ( C P ) is the singleton { w } and θ G − w ( λ w ) = 0, which implies w B = P ( C P ) = B . If C P corresponds to det, onegets W ( C P ) = W and θ G − w (det) = ( e − e ) + ( e − e ) for w ∈ W , which implies w B = B ( P ( C P ) = G .If P is the standard parabolic subgroup of Levi diag(GL , GL ), then L ⊗ | Z MP has 3isotypic components C P given by the characters λ : diag( x , x , x ) x , λ : diag( x , x , x ) x x , λ : diag( x , x , x ) x x . One has λ ′ = 3 / e + e ), λ ′ = e + e + e , λ ′ = 1 / e + e ) + 2 e from which onededuces for the three respective isotypic components C P (where w ∈ W ( C P )): W ( C P ) = { } θ G − w ( λ ′ ) = 1 / e − e ) W ( C P ) = { , s e − e s e − e } θ G − w ( λ ′ ) = ( e − e ) + ( e − e ) W ( C P ) = { s e − e s e − e } θ G − w ( λ ′ ) = 1 / e − e ) . If C P corresponds to λ one gets w P = P ( C P ) = P , if C P corresponds to λ onegets w P ( P ( C P ) = G ( w P being P if w = Id and the standard parabolic subgroupof Levi diag(GL , GL ) if w = s e − e s e − e ), and if C P corresponds to λ one gets w P = P ( C P ) = the standard parabolic subgroup of Levi diag(GL , GL ). In this lastcase we see that P ( C P ) doesn’t contain P .Finally, if M P = diag(GL , GL ), the situation is symmetric. Lemma 2.2.2.10.
We have W ( C P ) ⊆ W ( P ( C P )) w for any fixed element w ∈ W ( C P ) . Equivalently w ′ w − ∈ W ( P ( C P )) for any w, w ′ ∈ W ( C P ) .Proof. Let λ P ∈ X ( Z M P ) corresponding to C P , w C P ∈ W ( C P ), λ ∈ X ( T ) such that λ | Z MP = λ P and define λ ′ as in (36). Recall that an element w ∈ W is in W ( C P )if and only if w ( S ( P )) ⊆ S and w ( λ ′ ) is dominant (see Proposition 2.2.2.6(i)), andthat we have w ( λ ′ ) = w C P ( λ ′ ) for all w ∈ W ( C P ) (see the beginning of the proof ofLemma 2.2.2.4(ii)). We rewrite this ww − C P ( w C P ( λ ′ )) = w C P ( λ ′ ) ∀ w ∈ W ( C P ). By thedefinition of P ( C P ) and Proposition 2.2.2.6(ii), we know that S ( P ( C P )) is the set ofsimple roots in the support of f θ G − w C P ( λ ′ ). Since w C P ( λ ′ ) is dominant, by Lemma46.2.2.1 the subgroup of W fixing w C P ( λ ′ ) is generated by the simple reflections s β fixing w C P ( λ ′ ), or equivalently such that h w C P ( λ ′ ) , β i = 0. Since h f θ G − w C P ( λ ′ ) , β i = f − f , we see that these simple roots β are all in the support of f θ G − w C P ( λ ′ ).Therefore W ( P ( C P )) contains the subgroup of W fixing w C P ( λ ′ ). Since ww − C P fixes w C P ( λ ′ ), it follows that ww − C P ∈ W ( P ( C P )). Remark 2.2.2.11.
The inclusion in Lemma 2.2.2.10 is not an equality in general(take P = G ). L ⊗ We let P be a standard parabolic subgroup of G , we prove an important structuretheorem on the isotypic components of L ⊗ | Z MP (Theorem 2.2.3.9) as well as severaluseful technical results.Recall that W ( C P ) is defined in (38) and P ( C P ) is defined just before. Lemma 2.2.3.1. If P ( C P ) = w P for some w ∈ W ( C P ) then W ( C P ) has just oneelement.Proof. Let w C P ∈ W ( C P ) such that P ( C P ) = w CP P and let w ′ C P ∈ W ( C P ). Since P ( C P ) = w CP P we get S ( P ( C P )) = w C P ( S ( P )) and W ( P ( C P )) = w C P W ( P ) w − C P . ByLemma 2.2.2.10 applied to the element w C P , we deduce W ( C P ) ⊆ w C P W ( P ) andthus w − C P w ′ C P ∈ W ( P ). But since S ( P ( C P )) contains w ( S ( P )) for all w ∈ W ( C P ) bydefinition of W ( C P ) and (37), we have w ′ C P ( S ( P )) ⊆ S ( P ( C P )) = w C P ( S ( P )) whichimplies w ′ C P ( S ( P )) = w C P ( S ( P )) since the cardinalities are the same on both sides,that is, w − C P w ′ C P ( S ( P )) = S ( P ). Since w − C P w ′ C P ∈ W ( P ), this forces w ′ C P = w C P . Remark 2.2.3.2. (i) The converse to Lemma 2.2.3.1 is wrong in general (e.g. considerthe C ( λ i ) with i / ∈ { , f / , f } in Example 2.2.2.9(i)).(ii) For a general isotypic component C P , it is not true that one can find w ∈ W ( C P )such that w − M P ( C P ) w is the Levi subgroup of a standard parabolic subgroup of G . Proposition 2.2.3.3.
The isotypic components C P such that P ( C P ) = w P for some ( necessarily unique ) w ∈ W ( C P ) are those isotypic components which are associatedto f w − ( θ G ) | Z MP for the w ∈ W such that w ( S ( P )) ⊆ S .Proof. Let w ∈ W such that w ( S ( P )) ⊆ S and λ def = f w − ( θ G ) ∈ X ( T ). Since w ( λ ) = f θ G is dominant and f θ G − w ( λ ) = 0, the set (37) is w ( S ( P )). This implies P ( C P ) = w P .Conversely, let C P as in the statement, λ ∈ X ( T ) such that C P is the isotypic47omponent associated to the character λ | Z MP of Z M P and define λ ′ as in (36). Since S ( P ( C P )) = w ( S ( P )) by assumption, from Proposition 2.2.2.6(ii) we obtain f w − ( θ G ) − λ ′ = X α ∈ S ( P ) n α α (for some n α ∈ Q > ), which implies f w − ( θ G ) | Z MP = λ ′ | Z MP . Since λ | Z MP = λ ′ | Z MP (see the beginning of the proof of Lemma 2.2.2.4(i)), we deduce that C P is the isotypiccomponent associated to the character f w − ( θ G ) | Z MP .Note that if C P is associated to f w − ( θ G ) | Z MP (with w ( S ( P )) ⊆ S ), we have W ( C P ) = { w } by Lemma 2.2.3.1. Example 2.2.3.4.
Coming back to Example 2.2.2.9, the isotypic components as inProposition 2.2.3.3 are the isotypic components C ( λ ) , C ( λ f ) when n = 2, P = B ,the isotypic components associated to the six λ w when n = 3, K = Q p , P = B , andthe isotypic components associated to λ , λ when n = 3, K = Q p , M P = GL × GL .We set for α = e j − e j +1 ∈ S ( P ): λ α,P def = X e i − e j +1 ∈ R ( P ) + e i ∈ X ( T ) . (42)One easily checks that the λ α,P for α ∈ S ( P ) are fundamental weights for the reductivegroup M P and that h λ α,P , β i ≤ β ∈ S \ S ( P ). For any λ ∈ X ( T ), we define L P ( λ )as in (19) but with ( M P , M P ∩ B − ) instead of ( G, B − ). When S ( P ) = ∅ , we define L ⊗ P to be the trivial representation of T Gal( K/ Q p ) over F and, when S ( P ) = ∅ , we definesimilarly to (33) the algebraic representation of M Gal( K/ Q p ) P over F : L ⊗ P def = O Gal( K/ Q p ) (cid:18) O α ∈ S ( P ) L P ( λ α,P ) (cid:19) . (43)We also define θ P def = X α ∈ S ( P ) λ α,P ∈ X ( T ) and θ P def = θ G − θ P ∈ X ( T ) . (44)Since for α ∈ S ( P ) we have h θ P , α i = h θ G , α i − h θ P , α i = 1 − θ P extends to an element of Hom Gr ( M P , G m ). Likewise we have for α ∈ S ( P ) and w ∈ W such that w ( S ( P )) ⊆ S : h w − ( θ w P ) , α i = h θ w P , w ( α ) i = 0so that w − ( θ w P ) also extends to Hom Gr ( M P , G m ). Note that, since h θ P , β i ≤ β ∈ S \ S ( P ), we get h θ P , β i = h θ G , β i − h θ P , β i ≥
1, thus θ P is a dominant weight.48 xample 2.2.3.5. If G = GL and M P = GL × GL × GL , one gets θ P : diag( x , . . . , x ) ( x )( x x ) θ P : diag( x , . . . , x ) ( x x ) ( x x x ) . Lemma 2.2.3.6. If w ∈ W ( P ) , we have w ( θ P ) = θ P .Proof. The character θ P extends to M P and factors through M P /M der P . But conjuga-tion by W ( P ) is trivial on M P /M der P . Lemma 2.2.3.7.
Let λ ∈ X ( T ) be a dominant weight and denote by L ( λ ) µ ⊆ L ( λ ) for µ ∈ X ( T ) the isotypic component of L ( λ ) | T associated to µ ( i.e. the weight spaceof L ( λ ) for µ , see [Jan03, §I.2.11]) . Then M µ ∈ λ − P α ∈ S ( P ) Z ≥ α L ( λ ) µ ⊆ L ( λ ) is an M P -subrepresentation of L ( λ ) | M P which is isomorphic to L P ( λ ) .Proof. Since ⊕ µ ∈ λ − P α ∈ S ( P ) Z ≥ α L ( λ ) µ is the isotypic component of L ( λ ) | Z MP associ-ated to λ | Z MP (as λ | Z MP ∼ = µ | Z MP ⇐⇒ λ − µ ∈ P α ∈ S ( P ) Z α ), it is endowed with anaction of M P by the same proof as for Lemma 2.2.1.2. By [Jan03, II.2.2(1)], [Jan03,I.6.11(2)] and the transitivity of induction ([Jan03, I.3.5(2)]), we have an injection ofalgebraic representations of M P over F : H ( N P , L ( λ )) ֒ → L P ( λ ) (45)(recall N P is the unipotent radical of P ) and by [Jan03, II.2.11(1)] we have an iso-morphism of algebraic representations of M P over F : M µ ∈ λ − P α ∈ S ( P ) Z ≥ α L ( λ ) µ ∼ −→ H ( N P , L ( λ )) . It is therefore enough to prove that (45) is an isomorphism, or equivalently thatdim F M µ ∈ λ − P α ∈ S ( P ) Z ≥ α L ( λ ) µ ! = dim F L P ( λ ) . Let L ( λ ) def = (cid:16) ind GB − λ (cid:17) / Z ⊗ Z E , L P ( λ ) def = (cid:16) ind M P M P ∩ B − λ (cid:17) / Z ⊗ Z E and L ( λ ) µ ⊆ L ( λ ) theweight space associated to µ , we have dim F L ( λ ) µ = dim E L ( λ ) µ , and thusdim F M µ ∈ λ − P α ∈ S ( P ) Z ≥ α L ( λ ) µ ! = dim E M µ ∈ λ − P α ∈ S ( P ) Z ≥ α L ( λ ) µ ! . F L P ( λ ) = dim E L P ( λ ). It is therefore enough to havedim E M µ ∈ λ − P α ∈ S ( P ) Z ≥ α L ( λ ) µ ! = dim E L P ( λ ) . But now, we are over a field of characteristic 0, where it is well known that L ( λ ) and L P ( λ ) as defined above are simple modules with highest weight λ . Then the resultfollows from [Jan03, Prop.II.2.11].The following lemma is a special case of Lemma 2.2.3.7. Lemma 2.2.3.8.
Let λ ∈ X ( T ) be a dominant weight such that h λ, α i = 0 for all α ∈ S ( P ) ( equivalently λ extends to an element in Hom Gr ( M P , G m )) . Then any µ ∈ X ( T ) distinct from λ with L ( λ ) µ = 0 is such that λ − µ contains at least one rootof S \ S ( P ) in its support.Proof. Since λ ∈ Hom Gr ( M P , G m ), we have L P ( λ ) ∼ = λ by (19) applied with M P instead of G . By Lemma 2.2.3.7, we deduce L µ ∈ λ − P α ∈ S ( P ) Z ≥ α L ( λ ) µ ∼ = λ inside L ( λ ). This clearly implies the lemma.If R is any algebraic representation of M P or of M Gal( K/ Q p ) P and w ∈ W suchthat w ( S ( P )) ⊆ S , we define an algebraic representation of M w P = wM P w − orof M Gal( K/ Q p ) w P = wM Gal( K/ Q p ) P w − ( w acting diagonally via W ֒ → W Gal( K/ Q p ) ) by( g ∈ M w P or M Gal( K/ Q p ) w P ): w ( R )( g ) def = R ( w − gw ) . (46)If α ∈ S ( P ), one then easily checks that w ( λ α,P ) = λ w ( α ) , w P and w ( L P ( λ α,P )) = L w P ( λ w ( α ) , w P ), from which one gets w ( L ⊗ P ) = L ⊗ w P . (47) Theorem 2.2.3.9.
Let C P be an isotypic component of L ⊗ | Z MP , associated to λ | Z MP for λ ∈ X ( T ) . For any w ∈ W ( C P ) , there is an isomorphism of algebraic represen-tations of M Gal( K/ Q p ) P over F : C P ∼ = w − (cid:16) C P ( C P ) , w P (cid:17) ⊗ (cid:16) w − ( θ P ( C P ) ) ⊗ · · · ⊗ w − ( θ P ( C P ) ) | {z } Gal( K/ Q p ) (cid:17) , (48) where C P ( C P ) , w P is the isotypic component of L ⊗ P ( C P ) | Z MwP associated to ( w ( λ ) − f θ P ( C P ) ) | Z MwP ( thus an M Gal( K/ Q p ) w P -representation, recall w P ⊆ P ( C P )) and w − ( C P ( C P ) , w P ) is defined in (46) . roof. Step 1: Assuming the result holds if w = Id, we prove it for any w . For µ ∈ X ( T ) we have µ | Z MP = λ | Z MP if and only if w ( µ ) | Z MwP = w ( λ ) | Z MwP , thereforethe image w ( C P ) of C P for the diagonal action of w ∈ W on L ⊗ is the isotypic com-ponent of L ⊗ | Z MwP associated to w ( λ ) | Z MwP . Note that, as an algebraic M Gal( K/ Q p ) w P -subrepresentation of L ⊗ | M Gal( K/ Q p ) wP , w ( C P ) is indeed isomorphic to g C P ( w − gw )if g ∈ M Gal( K/ Q p ) w P , so the notation is consistent with (46). By Remark 2.2.2.3(ii) wehave w ( λ ′ ) = ( w ( λ )) ′ in ( X ( T ) ⊗ Z Q ) W ( w P ) . Recall that w ( λ ′ ), and hence ( w ( λ )) ′ , aredominant since w ∈ W ( C P ) (see Proposition 2.2.2.6(i)). Therefore Id ∈ W ( w ( C P ))and by the case w = Id for the parabolic subgroup w P and the isotypic compo-nent w ( C P ), we have w ( C P ) ∼ = C P ( w ( C P )) , w P ⊗ (cid:16) θ P ( w ( C P )) ⊗ · · · ⊗ θ P ( w ( C P )) (cid:17) . Moreover S ( P ( w ( C P ))), which is the support of f θ G − ( w ( λ )) ′ by Proposition 2.2.2.6(ii) (ap-plied to w = Id), is the same as S ( P C P ), which is the support of f θ G − w ( λ ′ ) by loc.cit. (applied to w ), i.e. we have P ( w ( C P )) = P ( C P ). We thus deduce w ( C P ) ∼ = C P ( C P ) , w P ⊗ (cid:16) θ P ( C P ) ⊗ · · · ⊗ θ P ( C P ) (cid:17) which gives (48) by applying w − .Step 2: From now on we assume w = Id (so in particular P ⊆ P ( C P )). Writing L ⊗ = O Gal( K/ Q p ) (cid:18) O α ∈ S ( P ( C P )) L ( λ α ) (cid:19)! O O Gal( K/ Q p ) (cid:18) O α ∈ S \ S ( P ( C P )) L ( λ α ) (cid:19)! , we prove that any ( µ , µ ) ∈ X ( T ) × X ( T ) such that(i) µ occurs in (cid:16) N Gal( K/ Q p ) (cid:16) N α ∈ S ( P ( C P )) L ( λ α ) (cid:17)(cid:17) | T (for the diagonal action of T );(ii) µ occurs in (cid:16) N Gal( K/ Q p ) (cid:16) N α ∈ S \ S ( P ( C P )) L ( λ α ) (cid:17)(cid:17) | T (idem);(iii) µ | Z MP + µ | Z MP = λ | Z MP must be such that µ = f P α ∈ S \ S ( P ( C P )) λ α (note that µ ≤ f P α ∈ S \ S ( P ( C P )) λ α and µ ≤ f P α ∈ S ( P ( C P )) λ α ). Let λ ′ , µ ′ , µ ′ as in (36) for P ( C P ) and the respectivecharacters λ , µ , µ , we have λ ′ = µ ′ + µ ′ from (iii) and Remark 2.2.2.3(i), and thus f θ G − λ ′ = f (cid:18) X α ∈ S ( P ( C P )) λ α (cid:19) − µ ′ + f (cid:18) X α ∈ S \ S ( P ( C P )) λ α (cid:19) − µ ′ . (49)Assume µ is not f P α ∈ S \ S ( P ( C P )) λ α . Then writing µ = P j,α µ ,j,α where ( j, α ) ∈ Gal( K/ Q p ) × S \ S ( P ( C P )) and µ ,j,α occurs in L ( λ α ) and applying Lemma 2.2.3.8with P = P ( C P ), λ = λ α and µ = µ ,j,α for α ∈ S \ S ( P ( C P )) (the assumptions inLemma 2.2.3.8 are satisfied since the λ α , α ∈ S are fundamental weights), we get that f P α ∈ S \ S ( P ( C P )) λ α − µ has at least one root of S \ S ( P ( C P )) in its support. Averagingover w ∈ W ( P ( C P )) as in (36) and using w ( λ α ) = λ α for w ∈ W ( P ( C P )) and α ∈ S \ S ( P ( C P )) (same proof as for Lemma 2.2.3.6), we get applying Lemma 2.2.2.251o P = P ( C P ) that f P α ∈ S \ S ( P ( C P )) λ α − µ ′ has still at least one root of S \ S ( P ( C P ))in its support (and that µ ′ ≤ f P α ∈ S \ S ( P ( C P )) λ α ). Since µ ′ ≤ f P α ∈ S ( P ( C P )) λ α by theproof of Step 3 below, this root doesn’t vanish in the sum (49). But by Proposition2.2.2.6(ii), S ( P ( C P )) is the support of (49), which is a contradiction. Therefore, wemust have µ = f P α ∈ S \ S ( P ( C P )) λ α and thus from (iii) that C P ∼ = C ′ P ( C P ) ,P ⊗ O Gal( K/ Q p ) X α ∈ S \ S ( P ( C P )) λ α ! , (50)where C ′ P ( C P ) ,P is the isotypic component of (cid:16) N Gal( K/ Q p ) (cid:16) N α ∈ S ( P ( C P )) L ( λ α ) (cid:17)(cid:17) | Z MP associated to (cid:16) λ − f P α ∈ S \ S ( P ( C P )) λ α (cid:17) | Z MP (= ( λ − µ ) | Z MP = µ | Z MP ).Step 3: We prove that f (cid:18) X α ∈ S ( P ( C P )) λ α (cid:19) − µ ∈ X α ∈ S ( P ( C P )) Z ≥ α (i.e. no root of S \ S ( P ( C P )) is in the support). Since λ α is dominant, we have λ α ≥ λ ′ α ,where λ ′ α is defined as in (36) for P = P ( C P ) and the character λ α . This implies(with obvious notation) f (cid:16) X α ∈ S ( P ( C P )) λ α (cid:17) − µ ′ ≥ f (cid:16) X α ∈ S ( P ( C P )) λ ′ α (cid:17) − µ ′ = (cid:18) f (cid:16) X α ∈ S ( P ( C P )) λ α (cid:17) − µ (cid:19) ′ ≥ , (51)where the last inequality follows from Lemma 2.2.2.2 (applied with P = P ( C P )). If f (cid:16)P α ∈ S ( P ( C P )) λ α (cid:17) − µ has roots of S \ S ( P ( C P )) in its support, then by Lemma 2.2.2.2again so is the case of (cid:16) f (cid:16)P α ∈ S ( P ( C P )) λ α (cid:17) − µ (cid:17) ′ , and thus of f (cid:16)P α ∈ S ( P ( C P )) λ α (cid:17) − µ ′ by (51). As in Step 2, this is again a contradiction by (49) and the definition of P ( C P ).Step 4: We prove the statement for w = Id. By Lemma 2.2.3.7 applied with P = P ( C P ) and the various L ( λ α ) for α ∈ S ( P ( C P )), we deduce from Step 3 that µ is aweight of N Gal( K/ Q p ) (cid:16)N α ∈ S ( P ( C P )) L P ( C P ) ( λ α ) (cid:17) inside N Gal( K/ Q p ) (cid:16)N α ∈ S ( P ( C P )) L ( λ α ) (cid:17) (see just after (42)). Let α ∈ S ( P ( C P )), for each β ∈ S ( P ( C P )) we have h λ α , β i = h λ α,P ( C P ) , β i (a straightforward check from (42)), thus λ α − λ α,P ( C P ) extends toHom Gr ( M P ( C P ) , G m ) which implies L P ( C P ) ( λ α ) ∼ = L P ( C P ) ( λ α,P ( C P ) ) ⊗ ( λ α − λ α,P ( C P ) ).Thus µ − f P α ∈ S ( P ( C P )) ( λ α − λ α,P ( C P ) ) is a weight of O Gal( K/ Q p ) (cid:18) O α ∈ S ( P ( C P )) L P ( C P ) ( λ α,P ( C P ) ) (cid:19) = L ⊗ P ( C P ) , or in other terms: C ′ P ( C P ) ,P ∼ = C P ( C P ) ,P ⊗ O Gal( K/ Q p ) X α ∈ S ( P ( C P )) ( λ α − λ α,P ( C P ) ) ! , C P ( C P ) ,P is the isotypic component of L ⊗ P ( C P ) | Z MP associated to (cid:16) λ − f P α ∈ S \ S ( P ( C P )) λ α − f P α ∈ S ( P ( C P )) ( λ α − λ α,P ( C P ) ) (cid:17) | Z MP . But by (44): X α ∈ S \ S ( P ( C P )) λ α + X α ∈ S ( P ( C P )) ( λ α − λ α,P ( C P ) ) = θ G − X α ∈ S ( P ( C P )) λ α,P ( C P ) = θ P ( C P ) , so together with (50) we are done. Remark 2.2.3.10.
The character w − ( θ P ( C P ) ) of M P doesn’t depend on w ∈ W ( C P ),as follows from Lemma 2.2.2.10 and Lemma 2.2.3.6 (the latter applied with P therebeing P ( C P )). In particular, by (48) we see that the representation w − (cid:16) C P ( C P ) , w P (cid:17) of M Gal( K/ Q p ) P is also independent of w ∈ W ( C P ).When C P is as in Proposition 2.2.3.3, its underlying M Gal( K/ Q p ) P -representationlooks like L ⊗ but for the reductive group M P instead of G . Corollary 2.2.3.11.
Let C P be an isotypic component of L ⊗ | Z MP such that P ( C P ) = w P for some ( unique ) w ∈ W such that w ( S ( P )) ⊆ S . Then there is an isomorphism C P ∼ = L ⊗ P ⊗ (cid:16) w − ( θ w P ) ⊗ · · · ⊗ w − ( θ w P ) | {z } Gal( K/ Q p ) (cid:17) of algebraic representations of M Gal( K/ Q p ) P over F .Proof. If P ( C P ) = w P , then L ⊗ P ( C P ) | Z MwP = L ⊗ w P | Z MwP has only one isotypic compo-nent, corresponding to f θ w P | Z MwP . So the corollary follows from Theorem 2.2.3.9 to-gether with (47). Note that, by Proposition 2.2.3.3, C P corresponds to λ = f w − ( θ G ),which is consistent with Theorem 2.2.3.9 since( w ( λ ) − f θ P ( C P ) ) | Z MwP = (cid:16) w ( f w − ( θ G )) − f θ w P (cid:17) | Z MwP = f ( θ G − θ w P ) | Z MwP = f θ w P | Z MwP . Remark 2.2.3.12.
In this remark, we use that we are working with G = GL n . Wewrite M P ( C P ) = diag( M , . . . , M d ) for some d > M i ∼ = GL n i , and correspond-ingly T = diag( T , . . . , T d ), where T i is the diagonal torus of GL n i , so that we have X ( T ) = ⊕ di =1 X ( T i ) and S ( P ( C P )) = ∐ di =1 S ( M i ), where X ( T i ) def = Hom Gr ( T i , G m ) and S ( M i ) def = S ( P ( C P )) ∩ X ( T i ) is the set of simple roots of M i (for the Borel subgroupof upper-triangular matrices). Note that S ( M i ) = ∅ if M i ∼ = GL . For i ∈ { , . . . , d } such that n i >
1, one easily checks that λ α,P ( C P ) ∈ X ( T i ) ⊆ X ( T ) if α ∈ S ( M i ) andthat the λ α,P ( C P ) ∈ X ( T i ) for α ∈ S ( M i ) are fundamental weights for the reductive53roup M i . For i ∈ { , . . . , d } and λ i ∈ X ( T i ), we define L M i ( λ i ) as in (19) but forthe reductive group M i instead of G . When n i = 1, we define L ⊗ i to be the trivialrepresentation of M Gal( K/ Q p ) i ∼ = G Gal( K/ Q p )m , and when n i >
1, we define as in (33) thealgebraic representation of M Gal( K/ Q p ) i over F (seeing λ α,P ( C P ) in X ( T i )): L ⊗ i def = O Gal( K/ Q p ) (cid:18) O α ∈ S ( M i ) L M i ( λ α,P ( C P ) ) (cid:19) . (52)We then clearly have L ⊗ P ( C P ) ∼ = N di =1 L ⊗ i . Likewise, we have θ P ( C P ) = ⊗ di =1 ( θ P ( C P ) ) i ,where ( θ P ( C P ) ) i ∈ X ( T i ) extends to Hom Gr ( M i , G m ) and where we denote by µ i theimage in X ( T i ) of a character µ ∈ X ( T ).For any w ∈ W ( C P ), we define ( w P ) i as the standard parabolic subgroup of M i which is the image of w P under w P ֒ → P ( C P ) ։ M P ( C P ) ։ M i (in particular its Levi M ( w P ) i is the image of M w P under M w P ֒ → M P ( C P ) ։ M i ).Applying w to (48), it is not difficult to deduce from Theorem 2.2.3.9 an isomorphismof algebraic representations of M Gal( K/ Q p ) w P ∼ = Q di =1 M Gal( K/ Q p )( w P ) i over F : w ( C P ) ∼ = d O i =1 (cid:18) C w,i ⊗ (cid:16) ( θ P ( C P ) ) i ⊗ · · · ⊗ ( θ P ( C P ) ) i | {z } Gal( K/ Q p ) (cid:17)(cid:19) , (53)where C w,i is the isotypic component of L ⊗ i | Z M ( wP ) i associated to ( w ( λ ) − f θ P ( C P ) ) i | Z M ( wP ) i (thus an M Gal( K/ Q p )( w P ) i -representation, note that C w,i is trivial if n i =1). If w ′ is another element in W ( C P ), writing w ′ = w P ( C P ) w with w P ( C P ) ∈ W ( P ( C P )) (Lemma 2.2.2.10), we have M w ′ P = w P ( C P ) M w P w − P ( C P ) , and thus w ′ ( C p ) ∼ = w P ( C P ) ( w ( C P )) and C P ( C P ) , w ′ P ∼ = w P ( C P ) (cid:16) C P ( C P ) , w P (cid:17) (as the twist by θ P ( C P ) ⊗ · · · ⊗ θ P ( C P ) doesn’t involve the choice of w ). Since w P ( C P ) M i w − P ( C P ) = M i for all i , weget M ( w ′ P ) i = w P ( C P ) M ( w P ) i w − P ( C P ) (inside M i ) and deduce for i ∈ { , . . . , d } an iso-morphism of algebraic representations of M Gal( K/ Q p )( w ′ P ) i over F (with notation similar to(46)): C w ′ ,i ∼ = w P ( C P ) ( C w,i ) . (54)We will avoid applying w − to C w,i since w − M P ( C P ) w is not in general the Levisubgroup of a standard parabolic subgroup of G (see Remark 2.2.3.2(ii)), although itindeed contains M P . We let P be a standard parabolic subgroup of G . We show that, if C P is an isotypiccomponent of L ⊗ | Z MP , then one can associate to C P in a natural way another isotypic54omponent w · C P of L ⊗ | Z MP for any w ∈ W such that w (cid:16) S ( P ( C P )) (cid:17) ⊆ S (seeProposition 2.2.4.2). Note that, on the contrary to w ( C P ), w · C P is an isotypiccomponent of L ⊗ | Z MP for the same standard parabolic subgroup P as C P . Lemma 2.2.4.1.
Let µ ∈ X ( T ) be a dominant weight. Then µ occurs in L ⊗ | T ( forthe diagonal embedding of T analogous to (35)) if and only if µ ≤ f θ G in X ( T ) .Proof. Since this statement only concerns weights, we can work in characteristic 0, i.e.with L ⊗ def = N Gal( K/ Q p ) (cid:16) N α ∈ S L ( λ α ) (cid:17) , where L ( λ α ) def = (cid:16) ind GB − λ α (cid:17) / Z ⊗ Z E (see (19)).Arguing as in the proof of [BH15, Lemma 2.2.3], it is equivalent to prove that µ is aweight of the algebraic representation L ( f θ G ) of G . The result then follows from theinequalities w ( µ ) ≤ µ ≤ f θ G for all w ∈ W (the left ones hold since µ is dominantand the right ones since f θ G is the highest weight) combined with [Hum78, Prop.21.3]. Proposition 2.2.4.2.
Let λ P ∈ X ( Z M P ) be a character of Z M P which occurs in L ⊗ | Z MP ( for the diagonal embedding, as usual ) with associated isotypic component C P of L ⊗ | Z MP , and let w ∈ W such that w (cid:16) S ( P ( C P )) (cid:17) ⊆ S . (i) For w C P ∈ W ( C P ) the character of Z M P : λ P − (cid:16) f w − C P ( θ G ) + f ( ww C P ) − ( θ G ) (cid:17) | Z MP (55) doesn’t depend on w C P . (ii) The character (55) corresponds to an isotypic component w · C P of L ⊗ | Z MP , i.e.occurs in L ⊗ | Z MP . (iii) We have P ( w · C P ) = w P ( C P ) .Proof. (i) For any α ∈ S ( P ( C P )) we have (since w ( α ) is still in S ) h w − ( θ G ) − θ G , α i = h θ G , w ( α ) i − h θ G , α i = 1 − s α ( w − ( θ G ) − θ G ) = w − ( θ G ) − θ G , and thus for all w ′ ∈ W ( P ( C P )): w ′ ( w − ( θ G ) − θ G ) = w − ( θ G ) − θ G . (57)Let w ′ C P ∈ W ( C P ), by Lemma 2.2.2.10 we have w ′ C P w − C P ∈ W ( P ( C P )) and thus by(57): ( w ′ C P w − C P )( w − ( θ G ) − θ G ) = w − ( θ G ) − θ G . Applying w ′ C P − we get in particular (cid:16) w − C P ( w − ( θ G ) − θ G ) (cid:17) | Z MP = (cid:16) w ′ C P − ( w − ( θ G ) − θ G ) (cid:17) | Z MP λ ∈ X ( T ) such that λ | Z MP = λ P . Applying ww C P to (55), it is sufficientto prove that f θ G − w (cid:16) f θ G − w C P ( λ ) (cid:17) occurs in L ⊗ | T (since L ⊗ | T is acted on bythe diagonal action of W ֒ → W Gal( K/ Q p ) ). Recall from Lemma 2.2.2.4(ii) (and thedefinition of P ( C P )) that f θ G − w C P ( λ ) ∈ X α ∈ S ( P ( C P )) Z ≥ α. (58)For β = w ( α ) ∈ w ( S ( P ( C P ))) and any w ′ ∈ W , we have h f θ G − w ( f θ G − w ′ ( λ )) , β i = h ww ′ ( λ ) , β i + f h θ G − w ( θ G ) , β i (59)= h ww ′ ( λ ) , β i + f h w − ( θ G ) − θ G , α i = h ww ′ ( λ ) , β i , where the last equality follows from (56). This can be rewritten as s β (cid:16) f θ G − w ( f θ G − w ′ ( λ )) (cid:17) = f θ G − w ( f θ G − w ′ ( λ )) − h ww ′ ( λ ) , β i β (60)= f θ G − w ( f θ G − s α w ′ ( λ )) . Iterating (60), we see that for any w P ( C P ) ∈ W ( P ( C P )), we have for w ′ ∈ W that ww P ( C P ) w − (cid:16) f θ G − w ( f θ G − w ′ ( λ )) (cid:17) = f θ G − w ( f θ G − w P ( C P ) w ′ ( λ )) . (61)Choose w P ( C P ) ∈ W ( P ( C P )) such that w P ( C P ) ( w C P ( λ )) is dominant for the rootsubsystem generated by S ( P ( C P )), equivalently h ww P ( C P ) w C P ( λ ) , β i ≥ ∀ β ∈ w ( S ( P ( C P ))) . (62)As λ occurs in L ⊗ | T , we get that w P ( C P ) ( w C P ( λ )) ∈ w C P ( λ ) + P α ∈ S ( P ( C P )) Z α occursin L ⊗ | T ( L ⊗ is stable under W ), and thus w P ( C P ) ( w C P ( λ )) ≤ f θ G . Since on the otherhand by (58): f θ G − w P ( C P ) ( w C P ( λ )) = ( f θ G − w C P ( λ )) + X α ∈ S ( P ( C P )) Z α ∈ X α ∈ S ( P ( C P )) Z α, we see that we must have f θ G − w P ( C P ) w C P ( λ ) ∈ X α ∈ S ( P ( C P )) Z ≥ α. (63)Since w ( S ( P ( C P ))) ⊆ S , we deduce h w ( f θ G − w P ( C P ) w C P ( λ )) , β i ≤ β ∈ S \ w ( S ( P ( C P ))). In particular we have for such β : h f θ G − w ( f θ G − w P ( C P ) w C P ( λ )) , β i = f − h w ( f θ G − w P ( C P ) w C P ( λ )) , β i (64) ≥ f. w ′ = w P ( C P ) w C P with (62) and (64), we obtain that f θ G − w ( f θ G − w P ( C P ) w C P ( λ )) is a dominant weight. Applying w to (63), we also getsince w ( S ( P ( C P ))) ⊆ S : f θ G − w ( f θ G − w P ( C P ) w C P ( λ )) ≤ f θ G . Lemma 2.2.4.1 then implies that f θ G − w ( f θ G − w P ( C P ) w C P ( λ )) occurs in L ⊗ | T . By(61) applied with w ′ = w C P , we finally deduce that f θ G − w ( f θ G − w C P ( λ )) also occursin L ⊗ | T .(iii) By definition S ( P ( w · C P )) ⊆ S is the union of w ′ ( S ( P )) and of the support of f θ G − w ′ (cid:16) λ − f w − C P ( θ G ) + f ( ww C P ) − ( θ G ) (cid:17) (65)for any w ′ ∈ W such that w ′ ( S ( P )) ⊆ S and w ′ (cid:16) λ − f w − C P ( θ G )+ f ( ww C P ) − ( θ G ) (cid:17) is therestriction to Z M w ′ P of a dominant weight of X ( T ) ⊗ Z Q . Consider the case w ′ def = ww C P ,since w C P ( S ( P )) ⊆ S ( P ( C P )) and w ( S ( P ( C P ))) ⊆ S , we get w ′ ( S ( P )) ⊆ S . Let uscheck that w ′ (cid:16) λ − f w − C P ( θ G ) + f ( ww C P ) − ( θ G ) (cid:17) = ww C P ( λ ) − f w ( θ G ) + f θ G is the restriction to Z M w ′ P of a dominant weight of X ( T ) ⊗ Z Q . Let λ ′ as in (36), since λ | Z MP = λ ′ | Z MP , we have w ′ ( λ ) | Z Mw ′ P = w ′ ( λ ′ ) | Z Mw ′ P and it is enough to prove that ww C P ( λ ′ ) − f w ( θ G ) + f θ G is dominant. As in (59) we have if α ∈ w ( S ( P ( C P ))): h ww C P ( λ ′ ) − f w ( θ G ) + f θ G , α i = h ww C P ( λ ′ ) , α i + f h θ G − w ( θ G ) , α i = h w C P ( λ ′ ) , w − ( α ) i ≥ w C P ( λ ′ ) is dominant in X ( T ) ⊗ Z Q by Proposition 2.2.2.6(i), and as in (64) wehave if α ∈ S \ w ( S ( P ( C P ))): h ww C P ( λ ′ ) − f w ( θ G ) + f θ G , α i = f − h w ( f θ G − w C P ( λ ′ )) , α i ≥ f since w (cid:16) f θ G − w C P ( λ ′ ) (cid:17) ∈ P β ∈ S ( P ( C P )) Q ≥ w ( β ) from Proposition 2.2.2.6(ii). Now allthat remains is to compute (65) for w ′ = ww C P , which gives w ( f θ G − w C P ( λ )), thesupport of which is w (support( f θ G − w C P ( λ ))). Therefore we obtain S ( P ( w · C P )) = w (cid:18) w C P ( S ( P )) ∪ support (cid:16) f θ G − w C P ( λ ) (cid:17)(cid:19) = w (cid:16) S ( P ( C P )) (cid:17) which finishes the proof. Remark 2.2.4.3. If C P is one of the isotypic components of Proposition 2.2.3.3, sayassociated to f w − C P ( θ G ) | Z MP for some w C P ∈ W such that w C P ( S ( P )) ⊆ S , and if w ∈ W is such that w ( S ( P ( C P ))) ⊆ S , i.e. ww C P ( S ( P )) ⊆ S , we see from (55) that w · C P is the isotypic component associated to f ( ww C P ) − ( θ G ) | Z MP .57 xample 2.2.4.4. Let us consider Example 2.2.2.9(ii) (Example 2.2.2.9(i) only pro-vides components C P which are either as in Remark 2.2.4.3 or such that P ( C P ) = G ).If P = B and C P is associated to λ Id = θ G , then w · C P for w ∈ S gives the iso-typic component associated to λ w (and there is no w · C P = C P if C P correspondsto det since P ( C P ) is the whole G ). If M P = GL × GL , consider C P associatedto λ and w ∈ S the unique permutation e e , e e , e e (so that w ( S ( P ( C P ))) = w ( e − e ) ⊆ S ). Then w · C P is the isotypic component associatedto λ (here again, there is no w · C P = C P for C P corresponding to λ ). ρ Following and generalizing the mod p variant of [BH15, §3.2], we define and study good conjugates of a continuous ρ : Gal( Q p /K ) → G ( F ) under a mild assumptionon ρ (see Definition 2.3.2.3) and still assuming K unramified. Though some of theresults might hold for more general split reductive groups, we use here in the proofsthat we work with GL n . We start with a few group-theoretic preliminaries.We fix a standard parabolic subgroup P of G . Recall that a subset C ⊆ R + isclosed if α ∈ C , β ∈ C with α + β ∈ R + implies α + β ∈ C . For instance R ( P ) + ⊆ R + is closed. Definition 2.3.1.1.
A subset X ⊆ R + is a closed subset relative to P if it satisfiesthe following three conditions:(i) it contains R ( P ) + ;(ii) X \ R ( P ) + is a closed subset of R + ;(iii) for any w ∈ W ( P ), w ( X \ R ( P ) + ) = X \ R ( P ) + .Note that a closed subset relative to B is the same thing as a closed subset andthat R + is the only closed subset relative to G . Lemma 2.3.1.2.
Let X ⊆ R + be a closed subset relative to P . Then X is a closedsubset of R + .Proof. Since we already know that both R ( P ) + and X \ R ( P ) + are closed, it remainsto show that if α ∈ R ( P ) + and β ∈ X \ R ( P ) + are such that α + β ∈ R + , then58 + β ∈ X . We work with GL n , and it is then easy to check that α + β = s α ( β ).Since s α ∈ W ( P ), we have α + β ∈ X \ R ( P ) + ⊆ X by Definition 2.3.1.1(iii). Remark 2.3.1.3.
Note that Lemma 2.3.1.2 doesn’t hold for an arbitrary split con-nected reductive algebraic group (for instance it doesn’t work for GSp ). An alterna-tive definition would be to consider closed subsets Y of R + \ R ( P ) + such that Y ∪ R ( P )is also closed.If X ⊆ R + is any closed subset, we let N X ⊆ N be the Zariski closed algebraicsubgroup generated by the root subgroups N α for α ∈ X (see [Jan03, §II.1.7]). Thanksto Lemma 2.3.1.2, we can thus consider N X for any X ⊆ R + closed relative to P . Lemma 2.3.1.4. (i)
Let X be a closed subset of R + relative to P . Then M P N X is a Zariski closedalgebraic subgroup of P containing M P . (ii) Let e P ⊆ P be a Zariski closed algebraic subgroup containing M P . Then thereexists a unique closed subset X relative to P such that e P = M P N X .Proof. (i) Since M P N X = M P N X \ R ( P ) + , it is enough to prove that M P normalizes N X \ R ( P ) + . Let α ∈ R ( P ) + , β ∈ X \ R ( P ) + and let n α ∈ N α , n β ∈ N β . Then n α n β n − α = (cid:16) Y i,j> n iα + jβ (cid:17) n β , (66)where the product is over all integers i, j > iα + jβ ∈ R + (see [Jan03,§II.1.2]). Since X ⊆ R + is closed, all these iα + jβ are in X , and since β / ∈ R ( P ) + , theyare all in X \ R ( P ) + . Therefore n α n β n − α ∈ N X \ R ( P ) + . Let w ∈ W ( P ), β ∈ X \ R ( P ) + and n β ∈ N β . Then w ( β ) ∈ X \ R ( P ) + implies wn β w − ∈ N X \ R ( P ) + . The Bruhatdecomposition for the reductive group M P then shows that M P normalizes N X \ R ( P ) + .(ii) Let e P ⊆ P be a closed algebraic subgroup containing M P . Then e P = M P ( e P ∩ B ) = M P ( e P ∩ N ) (since T ⊆ M P ⊆ e P ). By [BH15, Lemma 3.4.1] applied to e P ∩ B ⊆ B , wededuce e P ∩ N = N X for a (unique) closed subset X ⊆ R + . Since M P ∩ N ⊆ e P ∩ N ,the set X contains R ( P ) + . Since e P ∩ N P = N X \ R ( P ) + , the set X \ R ( P ) + is closed,and moreover e P = M P N X \ R ( P ) + . Since M P normalizes N P and e P , it normalizes e P ∩ N P = N X \ R ( P ) + , from which Definition 2.3.1.1(iii) easily follows. Remark 2.3.1.5. (i) The sets R ( P ) + and R + are closed with respect to P (theycorrespond respectively to e P = M P and e P = P in Lemma 2.3.1.4). In particular, if X is closed with respect to P , from w ( R + \ R ( P ) + ) = R + \ R ( P ) + and w ( X \ R ( P ) + ) = X \ R ( P ) + , we also get w ( R + \ X ) = R + \ X for all w ∈ W ( P ).(ii) If X ⊆ R + is a closed subset relative to P , it follows from the proof of Lemma2.3.1.4(i) that M P normalizes N X \ R ( P ) + . 59 emma 2.3.1.6. Let X ⊆ R + be a closed subset relative to P . Then there are roots α , . . . , α m ∈ R + \ X such that we have a partition R + = X ∐ { w ( α ) , w ∈ W ( P ) } ∐ · · · ∐ { w ( α m ) , w ∈ W ( P ) } and such that, for all i , α i is not in the smallest closed subset relative to P containing X and the α j for ≤ j ≤ i − .Proof. Since w ( R + \ X ) = R + \ X for all w ∈ W ( P ) (Remark 2.3.1.5(i)), we havea partition R + = X ∐ { w ( α ) , w ∈ W ( P ) } ∐ · · · ∐ { w ( α m ) , w ∈ W ( P ) } for some α , . . . , α m ∈ R + \ X . Denote by h ( · ) the height of a positive root (see e.g. [BH15,Rem.2.5.3]). Replacing each α i by a suitable w ( α i ) for w ∈ W ( P ), we can assume h ( α i ) maximal among the h ( w ( α i )), w ∈ W ( P ). Permuting the α i if necessary, wecan assume h ( α ) ≥ h ( α ) ≥ · · · ≥ h ( α m ). It is enough to prove that each set X ∐ { w ( α ) , w ∈ W ( P ) } ∐ · · · ∐ { w ( α i ) , w ∈ W ( P ) } for 1 ≤ i ≤ m is closed relativeto P , or equivalently that X i def = ( X \ R ( P ) + ) ∐ { w ( α ) , w ∈ W ( P ) } ∐ · · ·∐ { w ( α i ) , w ∈ W ( P ) } satisfies conditions (ii) and (iii) in Definition 2.3.1.1 for 1 ≤ i ≤ m . Since (iii)is clear, let us prove (ii), i.e. that each of the X i is closed in R + .This is obvious if i = m since R + \ R ( P ) + is closed, so we can assume i < m . If X i isnot closed for some i < m , then its complementary in R + contains an element x whichis the sum of at least two roots of X i , at least one being in { w ′ ( α j ) , w ′ ∈ W ( P ) , ≤ j ≤ i } (since R + \ R ( P ) + is closed). Such an element x is in R ( P ) + ∐ { w ( α j ) , w ∈ W ( P ) , i + 1 ≤ j ≤ m } and, since w ′ ( X i ) = X i for w ′ ∈ W ( P ), it also satisfies w ′ ( x ) ∈ R + for any w ′ ∈ W ( P ). In particular x can’t be in R ( P ) + , and is thus of theform x = w ( α k ) for some k ∈ { i + 1 , . . . , m } and some w ∈ W ( P ). Thus w ( α k ) is thesum of at least two roots of X i , one at least being in { w ′ ( α j ) , w ′ ∈ W ( P ) , ≤ j ≤ i } .Applying a convenient w ′ ∈ W ( P ) and using again w ′ ( X i ) = X i , we can modify w ifnecessary and assume that α j for some j ∈ { , . . . , i } appears in the sum of w ( α k ).This implies in particular h ( w ( α k )) > h ( α j ) for some j ≤ i (see the argument in theproof of [BH15, Lemma 3.2.1]), which is impossible since by assumption h ( w ( α k )) ≤ h ( α k ) ≤ h ( α j ). Hence X i is closed for all i . Lemma 2.3.1.7.
Let X ⊆ R + be a closed subset relative to P , e P def = M P N X and let w ∈ W such that w ( S ( P )) ⊆ S . Then the following assertions are equivalent: (i) w e P w − is contained in w P ; (ii) w ( X \ R ( P ) + ) ⊆ R + .Proof. We have w e P w − = ( wM P w − )( wN X \ R ( P ) + w − ) = ( wM P w − ) N w ( X \ R ( P ) + ) . As w P = ( wM P w − ) N , we deduce w e P w − ⊆ w P if and only if w ( X \ R ( P ) + ) ⊆ R + . 60 .3.2 Good conjugates of a generic ρ We define good conjugates of a Gal( Q p /K )-representation ρ under a mild genericityassumption and show how two good conjugates are related (Theorem 2.3.2.5). Theintuitive idea is that conjugating a good conjugate of ρ can only increase the imagein G ( F ).We fix a continuous homomorphism ρ : Gal( Q p /K ) −→ P ρ ( F ) ⊆ G ( F ) , (67)where P ρ ⊆ G is a standard parabolic subgroup. We consider ρ P ρ − ss : Gal( Q p /K ) ρ −→ P ρ ( F ) ։ M P ρ ( F ) , and assume that the image of ρ P ρ − ss is not contained in the F -points of a proper(not necessarily standard) parabolic subgroup of M P ρ . This implies in particularthat P ρ is uniquely determined by the homomorphism ρ . Finally we let ρ ss be thehomomorphism Gal( Q p /K ) → G ( F ) obtained by composing ρ P ρ − ss with the inclusion M P ρ ( F ) ⊆ G ( F ) (so ρ ss is the usual semisimplification of ρ ). We let X ρ be the smallestclosed subset of R + relative to P ρ such that e P ρ ( F ) def = M P ρ ( F ) N X ρ ( F ) contains all the ρ ( g ), g ∈ Gal( Q p /K ). By Lemma 2.3.1.4, e P ρ is the smallest closed algebraic subgroupof P ρ containing M P ρ such that ρ takes values in e P ρ ( F ), i.e. ρ : Gal( Q p /K ) → e P ρ ( F ) ֒ → P ρ ( F ) ֒ → G ( F ). Note that X ρ ss = R ( P ) + and e P ρ Pρ − ss = M P ρ . Lemma 2.3.2.1.
Assume that the irreducible constituents of ρ ss of dimension i.e.the characters of Gal( Q p /K ) occurring in ρ ss ) are all distinct. Let α ∈ R + \ X ρ and n α ∈ N α ( F ) \{ } . Then X n α ρn − α is the smallest closed subset relative to P ρ containing X ρ and α .Proof. The proof of this lemma is quite technical, but is no more than simple com-putations in GL n . We denote by X ρ,α ⊆ R + the smallest closed subset relative to P ρ containing X ρ and α and by f X ρ ⊆ X ρ the subset of roots which are not the sum ofat least two roots of X ρ,α . For g ∈ Gal( Q p /K ) we can write ρ ( g ) = ρ P ρ − ss ( g ) Y β ∈ X ρ \ R ( P ρ ) + n β ( g ) , (68)where ρ P ρ − ss ( g ) ∈ M P ρ ( F ) and n β ( g ) ∈ N β ( F ). Using (66), we see that n α (cid:18) Y β ∈ X ρ \ R ( P ρ ) + n β ( g ) (cid:19) n − α ∈ Y γ N γ ( F ) , (69)where γ runs among the roots in R + of the form Z ≥ α + Z > β + · · · + Z > β s for s ≥ β i ∈ X ρ \ R ( P ρ ) + . This clearly implies X n α ρn − α ⊆ X ρ,α . To prove the61everse inclusion, it is enough to prove f X ρ ⊆ X n α ρn − α and w ( α ) ∈ X n α ρn − α for some w ∈ W ( P ρ ) (as then α ∈ X n α ρn − α by Remark 2.3.1.5(i)).An easy explicit matrix computation in GL n (that we leave to the reader) gives that n α ρ P ρ − ss ( g ) n − α is of the form in GL n ( F ): n α ρ P ρ − ss ( g ) n − α ∈ ρ P ρ − ss ( g ) Y β ∈{ w ( α ) ,w ∈ W ( P ρ ) } m β ( g ) (70)with m β ( g ) ∈ N β ( F ) (note that, as w ∈ W ( P ρ ), w ( α ) is of the form α + n α + · · · + n t α t for some t ≥ α i ∈ S ( P ρ ), n i ∈ Z ). It then follows from (69) and (70) that, for β ∈ f X ρ \ ( f X ρ ∩ R ( P ρ ) + ), the entry n β ( g ) in (68) is not affected by the conjugation by n α . In particular, we have f X ρ ⊆ X n α ρn − α .We now prove that w ( α ) ∈ X n α ρn − α for some w ∈ W ( P ρ ). We first claim thatnone of the roots γ in (69) are in { w ( α ) , w ∈ W ( P ρ ) } . Indeed, assume w ( α ) = mα + m β + · · · + m s β s for some s ≥ m ≥ β i ∈ X ρ \ R ( P ρ ) + , m i >
0. If m = 0,then we get w ( α ) = m β + · · · + m s β s ∈ X ρ \ R ( P ρ ) + since X ρ \ R ( P ρ ) + is closed in R + ,which implies α ∈ X ρ \ R ( P ρ ) + by Definition 2.3.1.1(iii), a contradiction. If m > m − α + m β + · · · + m s β s = n α + · · · + n t α t (writing w ( α ) as in theabove form), which implies in particular all β i ∈ R ( P ρ ) + , a contradiction. We deducefrom this that for all g ∈ Gal( Q p /K ): n α ρ ( g ) n − α ∈ n α ρ P ρ − ss ( g ) n − α Y γ N γ ( F )with γ in R + \ (cid:16) R ( P ρ ) + ∐ { w ( α ) , w ∈ W ( P ρ ) } (cid:17) .We can see ρ P ρ − ss ( g ) as a block matrix diag( ρ ( g ) , . . . , ρ d ( g )), where ρ i : Gal( Q p /K ) → GL n i ( F ) is irreducible. Assume that { w ( α ) , w ∈ W ( P ρ ) } ) { α } . Then using that, forfixed i , the ρ i ( g ) for g ∈ Gal( Q p /K ) do not take all values in the F -points of a strict(not necessarily standard) parabolic subgroup of GL n i , one can check that at least one m β ( g ) in (70) is nontrivial for some g ∈ Gal( Q p /K ). If { w ( α ) , w ∈ W ( P ρ ) } = { α } ,then there are integers 1 ≤ i < j ≤ d such that n i = n j = 1 and the non-diagonalentry in m α ( g ) is ( ρ i ( g ) − ρ j ( g )) x α , where x α ∈ F × is the non-diagonal entry in n α .By assumption, there is at least one g ∈ Gal( Q p /K ) such that ρ i ( g ) = ρ j ( g ), whichimplies m α ( g ) = 1 for that g .Hence we finally deduce that n α ρ ( g ) n − α ∈ ρ P ρ − ss ( g ) Y β ∈{ w ( α ) ,w ∈ W ( P ρ ) } m β ( g ) ! Y γ N γ ( F )with γ in R + \ (cid:16) R ( P ρ ) + ∐ { w ( α ) , w ∈ W ( P ρ ) } (cid:17) and at least one m β ( g ) being nontrivialfor some g ∈ Gal( Q p /K ) and some β ∈ { w ( α ) , w ∈ W ( P ρ ) } . This implies that this β is in X n α ρn − α and finishes the proof. Proposition 2.3.2.2.
Let ρ : Gal( Q p /K ) → P ρ ( F ) and X ρ as below (67) , and assumethat the irreducible constituents of ρ ss of dimension are all distinct. Then there is h ∈ P ρ ( F ) ( non unique in general ) such that X h ρh − ⊆ X hρh − for all h ∈ P ρ ( F ) . roof. The proof is modelled on that of [BH15, Prop.3.2.3]. Since M P ρ normalizes N X ρ \ R ( P ρ ) + (Remark 2.3.1.5(ii)), it is enough to prove the same statement with h , h ∈ N P ρ ( F ). Using that ρ P ρ − ss ( g ) − hρ P ρ − ss ( g ) ∈ N X ρ \ R ( P ρ ) + ( F ) for h ∈ N X ρ \ R ( P ρ ) + ( F ) ⊆ N P ρ ( F ) by Remark 2.3.1.5(ii) again, and that N X ρ \ R ( P ρ ) + ( F ) is a group, we deduce X hρh − ⊆ X ρ for all h ∈ N X ρ \ R ( P ρ ) + ( F ). Replacing ρ by a suitable conjugate h ρh − with h ∈ N X ρ \ R ( P ρ ) + ( F ), we can assume X hρh − = X ρ for all h ∈ N X ρ \ R ( P ρ ) + ( F ). It isenough to prove X ρ ⊆ X hρh − for all h ∈ N P ρ ( F ). Choosing roots α , . . . , α m ∈ R + \ X ρ as in Lemma 2.3.1.6 (for P = P ρ and X = X ρ ), we can write any h ∈ N P ρ ( F ) as h = h m h m − · · · h h ρ , where h i ∈ Q β ∈{ w ( α i ) ,w ∈ W ( P ρ ) } N β ( F ) and h ρ ∈ N X ρ \ R ( P ρ ) + ( F ).We have X h ρ ρh − ρ = X ρ and a straightforward induction applying successively Lemma2.3.2.1 to X h ρ ρh − ρ and α = α , X h h ρ ρ ( h h ρ ) − and α = α , etc. (which we can dothanks to Lemma 2.3.1.6) gives that X hρh − is the smallest closed subset of R + relativeto P ρ containing X ρ and the α i , i = 1 , . . . , m . In particular X ρ ⊆ X hρh − for all h ∈ N P ρ ( F ). Definition 2.3.2.3.
Let ρ : Gal( Q p /K ) −→ G ( F ) be a continuous homomorphismsuch that the irreducible constituents of ρ ss of dimension 1 are all distinct. A goodconjugate of ρ is a conjugate ρ ′ of ρ in G ( F ) which satisfies the two conditions:(i) it is of the form ρ ′ : Gal( Q p /K ) → P ρ ′ ( F ) ⊆ G ( F ) for a standard parabolicsubgroup P ρ ′ of G such that the image of ρ ′ P ρ ′ − ss : Gal( Q p /K ) ρ ′ → P ρ ′ ( F ) ։ M P ρ ′ ( F ) is not contained in the F -points of a proper parabolic subgroup of M P ρ ′ ;(ii) X ρ ′ ⊆ X hρ ′ h − for all h ∈ P ρ ′ ( F ).From Proposition 2.3.2.2, we easily deduce that good conjugates always exist. If ρ is irreducible, then any conjugate of ρ in G ( F ) is a good conjugate.For ρ : Gal( Q p /K ) −→ e P ρ ( F ) ⊆ P ρ ( F ) as in (67), set W ρ def = { w ∈ W, w ( S ( P ρ )) ⊆ S and w ( X ρ \ R ( P ρ ) + ) ⊆ R + } = { w ∈ W, w ( S ( P ρ )) ⊆ S and w e P ρ w − ⊆ w P ρ } , (71)where the second equality follows from Lemma 2.3.1.7. Using the definition of X ρ wesee that, for any w ∈ W ρ , we have X wρw − = w ( X ρ ), where wρw − : Gal( Q p /K ) −→ w e P ρ ( F ) w − = e P wρw − ( F ) ⊆ ( w P ρ )( F ) . (and note that the set X wρw − is relative to w P ρ , while the set X ρ is relative to P ρ ). Lemma 2.3.2.4.
Let ρ : Gal( Q p /K ) → G ( F ) as in Definition 2.3.2.3 and ρ ′ :Gal( Q p /K ) → e P ρ ′ ( F ) ⊆ P ρ ′ ( F ) a good conjugate of ρ ( where e P ρ ′ def = M P ρ ′ N X ρ ′ = M P ρ ′ N X ρ ′ \ R ( P ρ ′ ) + ) . Then any hρ ′ h − for h ∈ e P ρ ′ ( F ) and any wρ ′ w − for w ∈ W ρ ′ is agood conjugate of ρ . Moreover we have X hρ ′ h − = X ρ ′ and X wρ ′ w − = w ( X ρ ′ ) . roof. Again, the proof is formally the same as that of [BH15, Lemma 3.2.5]. Thestatement is obvious for h ∈ e P ρ ′ ( F ) (as hN X \ R ( P ) + h − = N X \ R ( P ) + for any X closedsubset relative P and any h ∈ N X \ R ( P ) + ) and the very last equality follows from thediscussion just above. Following the argument in the proof of Proposition 2.3.2.2, itis enough to check X h ( wρ ′ w − ) h − = X wρ ′ w − for all h ∈ N X wρ ′ w − \ R ( P wρ ′ w − ) + ( F ) = N w ( X ρ ′ \ R ( P ρ ′ ) + ) ( F ). We have h ( wρ ′ w − ) h − = w ( w − hw ) ρ ′ ( w − h − w ) w − . Since w − hw ∈ N X ρ ′ \ R ( P ρ ′ ) + ( F ), we have X ( w − hw ) ρ ′ ( w − h − w ) ⊆ X ρ ′ and since ρ ′ is agood conjugate, we have X ρ ′ ⊆ X ( w − hw ) ρ ′ ( w − h − w ) , hence X ρ ′ = X ( w − hw ) ρ ′ ( w − h − w ) .Applying the discussion just before this lemma to ( w − hw ) ρ ′ ( w − h − w ) and then to ρ ′ , we thus get X h ( wρ ′ w − ) h − = w ( X ( w − hw ) ρ ′ ( w − h − w ) ) = w ( X ρ ′ ) = X wρ ′ w − .We now state and prove the main result of this section (see [BH15, Prop.3.2.6]). Theorem 2.3.2.5.
Let ρ : Gal( Q p /K ) → G ( F ) be a continuous homomorphism suchthat the irreducible constituents of ρ ss of dimension are all distinct. Let ρ ′ and ρ ′′ be two good conjugates of ρ . Then there exist h ∈ e P ρ ′ ( F ) and w ∈ W ρ ′ such that ρ ′′ = w ( hρ ′ h − ) w − . In particular we have X ρ ′′ = w ( X ρ ′ ) .Proof. By assumption there is x ∈ G ( F ) such that ρ ′′ ( g ) = xρ ′ ( g ) x − for all g ∈ Gal( Q p /K ). We can write x = h ′′ wh ′ with h ′ ∈ P ρ ′ ( F ), h ′′ ∈ P ρ ′′ ( F ) and w ∈ W suchthat w ( R ( P ρ ′ ) + ) ⊆ R + .Step 1: We prove that w ( S ( P ρ ′ )) = S ( P ρ ′′ ). We have wh ′ ρ ′ ( g ) h ′− w − ∈ P ρ ′′ ( F )for all g ∈ Gal( Q p /K ), which implies h ′ ρ ′ ( g ) h ′− ∈ ( w − P ρ ′′ w ∩ P ρ ′ )( F ) ⊆ P ρ ′ ( F )for all g ∈ Gal( Q p /K ). In particular, using for instance [DM91, Prop.2.1(iii)], theimage of h ′ ρ ′ h ′− in M P ρ ′ ( F ) is contained in the F -points of the parabolic subgroup w − P ρ ′′ w ∩ M P ρ ′ of M P ρ ′ . But since ( h ′ ρ ′ h ′− ) P ρ ′ − ss is conjugate to ρ ′ P ρ ′ − ss (recall h ′ ∈ P ρ ′ ( F )), the image of h ′ ρ ′ h ′− in M P ρ ′ ( F ) is not contained in the F -points of aproper parabolic subgroup of M P ρ ′ . Thus we must have w − P ρ ′′ w ∩ M P ρ ′ = M P ρ ′ whichimplies M P ρ ′ ⊆ w − M P ρ ′′ w . The same argument starting with w − h ′′− ρ ′′ ( g ) h ′′ w ∈ P ρ ′ ( F ) yields M P ρ ′′ ⊆ wM P ρ ′ w − , i.e. we have M P ρ ′ = w − M P ρ ′′ w . Since by assumption w ( R ( P ρ ′ ) + ) ⊆ R + , this forces w ( S ( P ρ ′ )) = S ( P ρ ′′ ) (and thus w ( R ( P ρ ′ ) + ) = R ( P ρ ′′ ) + ).Step 2: We choose roots α ′ , . . . , α ′ m ′ ∈ R + \ X ρ ′ as in Lemma 2.3.1.6 (for P = P ρ ′ and X = X ρ ′ ) and we write h ′ = h ′ m ′ h ′ m ′ − · · · h ′ h ′ ρ , where h ′ i ∈ Q β ∈{ w ′ ( α ′ i ) ,w ′ ∈ W ( P ρ ′ ) } N β ( F )and h ′ ρ ′ ∈ e P ρ ′ ( F ). By Lemma 2.3.2.4, we can replace ρ ′ by h ′ ρ ′ ρ ′ h ′− ρ ′ and thus assume h ′ ρ ′ = 1. By Lemma 2.3.2.1 and an induction as in the proof of Proposition 2.3.2.2, X h ′ ρ ′ h ′− is the smallest closed subset relative to P ρ ′ containing X ρ ′ and those α ′ i such64hat h ′ i = 1. Since w ( h ′ ρ ′ h ′− ) w − takes values in P ρ ′′ ( F ) and w ( R ( P ρ ′ )) = R ( P ρ ′′ )(by Step 1), we must also have w ( X h ′ ρ ′ h ′− \ R ( P ρ ′ ) + ) ⊆ R + \ R ( P ρ ′′ ) + . This implies ww ′ ( α ′ i ) ∈ R + if w ′ ∈ W ( P ρ ′ ) and h ′ i = 1, and w ( X ρ ′ \ R ( P ρ ′ ) + ) ⊆ R + . In particular w ∈ W ρ ′ together with Step 1.Step 3: We prove that X ρ ′′ = w ( X ρ ′ ). Setting h i def = wh ′ i w − ∈ Y β ∈{ ww ′ ( α ′ i ) ,w ′ ∈ W ( P ρ ′ ) } N β ( F ) ⊆ P ρ ′′ ( F )(we proved ww ′ ( α ′ i ) ∈ R + in Step 2), we have ρ ′′ = h ′′ ( h m ′ · · · h )( wρ ′ w − )( h − · · · h − m ′ ) h ′′− , (72)where h ′′ h m ′ · · · h ∈ P ρ ′′ ( F ) and where ρ ′′ and wρ ′ w − are good conjugates of ρ (thelatter by Lemma 2.3.2.4). Applying Definition 2.3.2.3 to both ρ ′′ and wρ ′ w − , we get X ρ ′′ = X wρ ′ w − = w ( X ρ ′ ) (and thus w − e P ρ ′′ w = e P ρ ′ ).Step 4 : We complete the proof. We choose again roots α ′′ , . . . , α ′′ m ′′ ∈ R + \ X wρ ′ w − as in Lemma 2.3.1.6 for P = P wρ ′ w − = P ρ ′′ (this latter equality from Remark 2.2.1.4)and X = X wρ ′ w − = X ρ ′′ and we write h ′′ ( h m ′ · · · h ) = h ′′ m ′′ h ′′ m ′′ − · · · h ′′ h ′′ X ρ ′′ , where h ′′ i ∈ Q β ∈{ w ′′ ( α ′′ i ) ,w ′′ ∈ W ( P ρ ′′ ) } N β ( F ) and h ′′ X ρ ′′ ∈ e P wρ ′ w − ( F ) = e P ρ ′′ ( F ). From (72)and Lemma 2.3.2.1, we see that we must have h ′′ i = 1 for all i ∈ { , . . . , m ′′ } otherwise X ρ ′′ would be strictly bigger that X wρ ′ w − . Thus we deduce ρ ′′ = h ′′ X ρ ′′ wρ ′ w − h ′′− X ρ ′′ = w ( w − h ′′ X ρ ′′ w ) ρ ′ ( w − h ′′− X ρ ′′ w ) w − . Setting h def = w − h ′′ X ρ ′′ w ∈ w − e P ρ ′′ ( F ) w = e P ρ ′ ( F ), this finishes the proof. Given a sufficiently generic n -dimensional representation of Gal( Q p /K ) over F (where K = Q p f is still unramified) and a good conjugate ρ of this representation as inDefinition 2.3.2.3, we define what it means for a smooth representation of G ( K ) over F to be compatible with e P ρ (Definition 2.4.1.5, see the beginning of §2.3.2 for e P ρ ) andto be compatible with ρ (Definition 2.4.2.7). e P We first define what it means for a smooth representation of G ( K ) over F to becompatible with a Zariski closed subgroup e P of a standard parabolic subgroup P asin Definition 2.2.1.3. We keep the notation of §§2.2, 2.3.65e fix a Zariski closed algebraic subgroup e P of a standard parabolic subgroup P of G as in Definition 2.2.1.3 (by Remark 2.2.1.4, P is in fact determined by e P ). Welet X be the unique closed subset of R + relative to P such that e P = M P N X (Lemma2.3.1.4) and define W e P def = { w ∈ W, w ( S ( P )) ⊆ S, w ( X \ R ( P ) + ) ⊆ R + } . Note that W e P is analogous to W ρ in (71) with e P ρ replaced by e P .Let Q be a parabolic subgroup containing w e P P for some w e P ∈ W e P , w Q an elementof W such that w Q ( S ( Q )) ⊆ S and Q ′ a parabolic subgroup containing w Q Q (note thatboth Q and Q ′ are standard). So we have inclusions of standard parabolic subgroups w Q w e P P ⊆ w Q Q ⊆ Q ′ and likewise for the Levi subgroups M wQw e P P = w Q w e P M P ( w Q w e P ) − ⊆ M wQ Q = w Q M Q w − Q ⊆ M Q ′ . Using that we work with GL n , we write M Q ′ = diag( M , . . . , M d )with M i ∼ = GL n i and we define the standard parabolic subgroup ( w Q Q ) i of M i as( w Q Q ) i def = Im (cid:16) w Q Q ֒ → Q ′ ։ M Q ′ ։ M i (cid:17) . We define a standard parabolic subgroup ( w Q w e P P ) Q of M wQ Q , resp. a standard parabo-lic subgroup ( w Q w e P P ) Q,i of M ( wQ Q ) i , as the image of w Q w e P P via w Q w e P P ⊆ w Q Q ։ M wQ Q ,resp. via w Q w e P P ⊆ w Q Q ։ M wQ Q ։ M ( wQ Q ) i . Equivalently,( w Q w e P P ) Q = w Q ( w e P P ∩ M Q ) w − Q ⊆ w Q M Q w − Q = M wQ Q ( w Q w e P P ) Q,i = Im (cid:16) w Q ( w e P P ∩ M Q ) w − Q ⊆ M wQ Q ։ M ( wQ Q ) i (cid:17) . Note that M ( wQw e P P ) Q = w Q M ( w e P P ) ∩ M Q w − Q = w Q w e P M P ( w Q w e P ) − . We finally define a Zariski closed algebraic subgroup ( w Q w e P e P ) Q of ( w Q w e P P ) Q contain-ing M ( wQw e P P ) Q , resp. a Zariski closed algebraic subgroup ( w Q w e P e P ) Q,i of ( w Q w e P P ) Q,i containing M ( wQw e P P ) Q,i , as( w Q w e P e P ) Q def = w Q (cid:16) ( w e P e P w − e P ) ∩ M Q (cid:17) w − Q ⊆ w Q ( w e P P ∩ M Q ) w − Q = ( w Q w e P P ) Q ( w Q w e P e P ) Q,i def = Im (cid:18) w Q (cid:16) ( w e P e P w − e P ) ∩ M Q (cid:17) w − Q ⊆ M wQ Q ։ M ( wQ Q ) i (cid:19) . We also define the continuous group homomorphism ω − ◦ θ Q ′ : Q ′− ( K ) −→ M Q ′ ( K ) θ Q ′ −→ K × ω − −→ F × p ֒ → F × , where θ Q ′ is defined in (44) (applied with P = Q ′ ).We need a quite formal and easy lemma.66 emma 2.4.1.1. Let Π be a smooth representation of a p -adic analytic group over F which has finite length and distinct absolutely irreducible constituents. Let H be asplit connected reductive algebraic group over Z , P H ⊆ H a parabolic subgroup withLevi M P H , e P H ⊆ P H a Zariski closed algebraic subgroup containing M P H and R a ( finite-dimensional ) algebraic representation of P Gal( K/ Q p ) H over F . Assume that thereexist(a) a filtration on R by good subrepresentations for the P Gal( K/ Q p ) H -action ( see Def-inition 2.2.1.3 ) such that the graded pieces exhaust the isotypic components of R | Z MPH ;(b) a bijection Φ of partially ordered finite sets between the set of subrepresentationsof Π and the set of good subrepresentations of R | e P Gal( K/ Q p ) H ( both being orderedby inclusion ) .Then the following hold: (i) The bijection Φ uniquely extends to bijections between subquotients of Π andgood subquotients of R | e P Gal( K/ Q p ) H , and between irreducible constituents of Π andisotypic components of R | Z MPH . (ii) If Π ′ is a subquotient of Π , then Φ induces a bijection of partially ordered finitesets between the set of subrepresentations of Π ′ and the set of good subrepresen-tations of Φ(Π ′ ) | e P Gal( K/ Q p ) H .Proof. Formal and left to the reader.
Remark 2.4.1.2. (i) Let Π and Φ as in Lemma 2.4.1.1, Π ′ a subquotient of Π andΠ ′′ ⊆ Π ′ a subrepresentation. Then the bijection Φ also induces a short exact sequence0 → Φ(Π ′′ ) → Φ(Π ′ ) → Φ(Π ′ / Π ′′ ) → e P Gal( K/ Q p ) H over F .(ii) By Lemma 2.2.1.5 applied with P there being the parabolic w e P P above, we seethat Lemma 2.4.1.1 can be applied with H = G , P H = w e P P , e P H = w e P e P w − e P and R = L ⊗ . Using moreover Lemma 2.2.1.6, one easily sees that Lemma 2.4.1.1 can alsobe applied with H = M Q , P H = w e P P ∩ M Q , e P H = ( w e P e P w − e P ) ∩ M Q and R any isotypiccomponent C Q of L ⊗ | Z MQ (recall from (the proof of) Lemma 2.2.1.5 applied with P there being Q that the action of Q Gal( K/ Q p ) on the subquotient C Q of L ⊗ | Q Gal( K/ Q p ) factors through Q Gal( K/ Q p ) ։ M Gal( K/ Q p ) Q ).(iii) Let Q as above, C Q an isotypic component of L ⊗ | Z MQ , Q ′ def = P ( C Q ) (see §2.2.2)and w Q ∈ W ( C Q ) (see (38) and note that w Q Q ⊆ Q ′ by (39)). Lemma 2.4.1.1 can67lso be applied with H = M ( wQ Q ) i , P H = ( w Q w e P P ) Q,i , e P H = ( w Q w e P e P ) Q,i and R = C w Q ,i , where C w Q ,i is the algebraic representation of M Gal( K/ Q p )( wQ Q ) i defined in Remark2.2.3.12 with P there being Q (it is an isotypic component of L ⊗ i | Z M ( wQQ ) i ). Toprove that assumption (a) of Lemma 2.4.1.1 is satisfied in that case, note that C w Q ,i is a good subquotient of L ⊗ i | ( wQ Q ) Gal( K/ Q p ) i , and thus a fortiori a good subquotient of L ⊗ i | ( wQw e P P ) Gal( K/ Q p ) Q ′ ,i (Lemma 2.2.1.6), where ( w Q w e P P ) Q ′ ,i ⊆ ( w Q Q ) i ⊆ M i is the standardparabolic subgroup of M i with the same Levi as ( w Q w e P P ) Q,i . We have( w Q w e P e P ) Q,i ⊆ ( w Q w e P P ) Q,i ⊆ ( w Q w e P P ) Q ′ ,i ⊆ M i and ( w Q w e P e P ) Q,i is a closed algebraic subgroup of ( w Q w e P P ) Q ′ ,i containing M ( wQw e P P ) Q ′ ,i = M ( wQw e P P ) Q,i . One then applies Lemma 2.2.1.5 with L ⊗ i and with( w Q w e P e P ) Q,i ⊆ ( w Q w e P P ) Q ′ ,i ⊆ M i instead of e P ⊆ P ⊆ G , which implies that there is a filtration on C w Q ,i | ( wQw e P e P ) Gal( K/ Q p ) Q,i (or on C w Q ,i | ( wQw e P P ) Gal( K/ Q p ) Q ′ ,i , and thus on C w Q ,i | ( wQw e P P ) Gal( K/ Q p ) Q,i ) by good subrepresen-tations such that the graded pieces exhaust the isotypic components of C w Q ,i | Z M ( wQw e PP ) Q,i = C w Q ,i | Z M ( wQw e PP ) Q ′ ,i . Lemma 2.4.1.3.
Let e P ⊆ P , w e P ∈ W e P and Q containing w e P P as above. Let C Q bean isotypic component of L ⊗ | Z MQ and Q ′ def = P ( C Q ) . (i) For any w Q ∈ W ( C Q ) , there is a canonical bijection of partially ordered finitesets between the set of good subrepresentations of C Q | ( w e P e P w − e P ) Gal( K/ Q p ) = C Q | (( w e P e P w − e P ) ∩ M Q ) Gal( K/ Q p ) ( where the equality follows from Remark 2.4.1.2(ii) ) and the set of good subrep-resentations of w Q ( C Q ) | ( wQw e P e P ) Gal( K/ Q p ) Q . (ii) For any w Q , w ′ Q ∈ W ( C Q ) and i ∈ { , . . . , d } , there is a canonical bijec-tion of partially ordered finite sets between the set of good subrep-resentations of C w Q ,i | ( wQw e P e P ) Gal( K/ Q p ) Q,i and the set of good subrepresentations of C w ′ Q ,i | ( w ′ Qw e P e P ) Gal( K/ Q p ) Q,i .Proof. (i) follows from the definition of w Q ( C Q ) in (46) and the fact that ( w Q w e P e P ) Q = w Q (cid:16) ( w e P e P w − e P ) ∩ M Q (cid:17) w − Q . 68ii) We have w ′ Q = w Q ′ w Q with w Q ′ ∈ W ( P ( C Q )) = W ( Q ′ ) by Lemma 2.2.2.10(applied with P there being Q ). In particular w Q ′ ( w Q ( S ( Q ))) ⊆ S which implies( w ′ Q w e P e P ) Q,i = w Q ′ ( w Q w e P e P ) Q,i w − Q ′ inside M ( w ′ Q Q ) i = w Q ′ M ( wQ Q ) i w − Q ′ (viewing w Q ′ as anelement in W ( M i ) by abuse of notation). By (54) (applied with P there being Q )we have C w ′ Q ,i = w Q ′ ( C w Q ,i ), where the conjugation by w − Q ′ intertwines the actions of( w ′ Q w e P e P ) Q,i and of ( w Q w e P e P ) Q,i . The result follows.
Remark 2.4.1.4.
The bijections in Lemma 2.4.1.3 all extend to bijections betweengood subquotients or isotypic components on both sides, as for Lemma 2.4.1.1.Let Π, H , P H , e P H , R and Φ as in Lemma 2.4.1.1. For any w H ∈ W H (theWeyl group of H ) such that w H e P H w − H is contained in a standard parabolic subgroupof H , we can define another bijection w H (Φ) between the set of subquotients of Πand the set of good subquotients of R | ( w H e P H w − H ) Gal( K/ Q p ) as follows: w H (Φ)(Π ′ ) is thealgebraic representation w H (cid:16) Φ(Π ′ ) (cid:17) of ( w H e P H w − H ) Gal( K/ Q p ) , where w H (cid:16) Φ(Π ′ ) (cid:17) ( g ) def =Φ(Π ′ )( w − H gw H ) if g ∈ ( w H e P H w − H ) Gal( K/ Q p ) , see (46).Here is now the first crucial definition. Definition 2.4.1.5.
An admissible smooth representation Π of G ( K ) over F whichhas finite length and distinct absolutely irreducible constituents is compatible with e P if there exists a bijection Φ of partially ordered finite sets between the set of subre-presentations of Π and the set of good subrepresentations of L ⊗ | e P Gal( K/ Q p ) (both beingordered by inclusion) which satisfies the following conditions (once extended to allsubquotients as in Lemma 2.4.1.1):(i) ( form of subquotients ) for any w e P ∈ W e P , any parabolic subgroup Q con-taining w e P P and any isotypic component C Q of L ⊗ | Z MQ , writing M P ( C Q ) = M × · · · × M d with M i ∼ = GL n i we have w e P (Φ) − ( C Q ) ∼ = Ind G ( K ) P ( C Q ) − ( K ) (cid:16) π ( C Q ) ⊗ ( ω − ◦ θ P ( C Q ) ) (cid:17) , (73)where P ( C Q ) is defined in §2.2.2, θ P ( C Q ) is defined in (44) and where π ( C Q ) isa M P ( C Q ) -representation of the form π ( C Q ) ∼ = π ( C Q ) ⊗ · · · ⊗ π d ( C Q ) for some(finite length) admissible smooth representations π i ( C Q ) of M i ( K ) over F ;(ii) ( compatibility between subquotients ) for any w e P ∈ W e P , any parabolicsubgroup Q containing w e P P , any isotypic component C Q of L ⊗ | Z MQ and any w ∈ W such that w (cid:16) S ( P ( C Q )) (cid:17) ⊆ S , let w ( π ( C Q )) be the representationof M w P ( C Q ) ( K ) = wM P ( C Q ) ( K ) w − defined by w ( π ( C Q ))( g ) def = π ( C Q )( w − gw )69or π ( C Q ) as in (73) and g ∈ M w P ( C Q ) ( K ). Then we have π (cid:16) w · C Q (cid:17) ∼ = w (cid:16) π ( C Q ) (cid:17) , where w · C Q is the isotypic component of L ⊗ | Z MQ in Proposition 2.2.4.2(ii) (ap-plied with P there being Q ) and where π ( w · C Q ) is as in (73) for the isotypiccomponent w · C Q instead of C Q (note that P ( w · C Q ) = w P ( C Q ) by Proposition2.2.4.2(iii));(iii) ( product structure ) for any w e P ∈ W e P , any parabolic subgroup Q con-taining w e P P , any isotypic component C Q of L ⊗ | Z MQ , and one, or equivalentlyany by Lemma 2.4.1.3(ii), element w Q ∈ W ( C Q ), writing M P ( C Q ) =diag( M , . . . , M d ) with M i ∼ = GL n i , the restriction of w e P (Φ) to the set of subquo-tients of w e P (Φ) − ( C Q ) comes from d bijections w e P (Φ) w Q ,i of partially orderedsets between the set of M i ( K )-subrepresentations of π i ( C Q ) (where π i ( C Q ) isas in (i)) and the set of good subrepresentations of C w Q ,i | ( wQw e P e P ) Gal( K/ Q p ) Q,i (where C w Q ,i is the isotypic component of L ⊗ i | Z M ( wQQ ) i with its M Gal( K/ Q p )( wQ Q ) i -action in (53)applied with P there being Q ) in the following sense: for any subquotient Π ′ ofΦ − ( C Q ) of the formΠ ′ ∼ = Ind G ( K ) P ( C Q ) − ( K ) (cid:16) ( π ′ ⊗ · · · ⊗ π ′ d ) ⊗ ( ω − ◦ θ P ( C Q ) ) (cid:17) with π ′ i a subquotient of π i ( C Q ), the good subquotient w e P (Φ)(Π ′ ) of C Q | ( w e P e P w − e P ) Gal( K/ Q p ) = C Q | (( w e P e P w − e P ) ∩ M Q ) Gal( K/ Q p ) corresponds via Lemma 2.4.1.3(i) and Remark 2.4.1.4 to the following algebraicrepresentation of ( w Q w e P e P ) Gal( K/ Q p ) Q = Q di =1 ( w Q w e P e P ) Gal( K/ Q p ) Q,i : d O i =1 (cid:18) w e P (Φ) w Q ,i ( π ′ i ) ⊗ (cid:16) ( θ P ( C Q ) ) i ⊗ · · · ⊗ ( θ P ( C Q ) ) i | {z } Gal( K/ Q p ) (cid:17)(cid:19) ;(iv) ( supersingular ) for any isotypic component C P of L ⊗ | Z MP , the (absolutely ir-reducible) M P ( C P ) ( K )-representation π ( C P ) of (73) is supersingular (cf. [Her11,Def.4.7, Def.9.12, Cor.9.13]).If (Π , Φ) is as in Definition 2.4.1.5, then we have in particular Φ(Π) = L ⊗ and w e P (Φ) w Q ,i ( π i ( C Q )) = C w Q ,i . If e P = G , then Π is compatible with e P if and only if Πis absolutely irreducible supersingular. Also it is clear from Definition 2.4.1.5 that,for a fixed w e P ∈ W e P , Π is compatible with e P if and only if Π is compatible with w e P e P w − e P (replace Φ by w e P (Φ)). 70 emark 2.4.1.6. (i) In Definition 2.4.1.5, we have used Lemma 2.4.1.1 everywhere(see Remark 2.4.1.2(ii)(iii)). In Definition 2.4.1.5(iii), we have used Remark 2.4.1.4.Also, Definition 2.4.1.5 is somewhat redundant since a parabolic subgroup Q cancontain w e P P for several w e P ∈ W e P , but we found it too tedious to make it “non-redundant”.(ii) The representations π ( C Q ) and π i ( C Q ) in Definition 2.4.1.5(i) are uniquely de-fined since there are no nontrivial intertwinings between parabolic inductions (by[Eme10a]).(iii) When Q = w e P P , π ( C w e P P ) in (73) is absolutely irreducible, and is thus automat-ically of the form π ( C w e P P ) ∼ = π ( C w e P P ) ⊗ · · · ⊗ π d ( C w e P P ). It is then not difficult todeduce from this, together with Lemma 2.2.2.8 and [Eme10a] (and the properties ofΦ), that each π i ( C Q ) as in (73) has distinct (absolutely) irreducible constituents andthat each irreducible constituent of (73) is of the form Ind G ( K ) P ( C Q ) − ( K ) (cid:16) ( π ′ ⊗ · · · ⊗ π ′ d ) ⊗ ( ω − ◦ θ P ( C Q ) ) (cid:17) , where π ′ i is an irreducible constituent of π i ( C Q ). This also justifiesthe terminology “comes from d bijections w e P (Φ) w Q ,i ” in Definition 2.4.1.5(iii).(iv) It is in fact possible that Definition 2.4.1.5(i) for parabolic subgroups Q strictly containing some w e P P and Definition 2.4.1.5(iii) both automatically follow from theother conditions in Definition 2.4.1.5. See for instance how the results of [Hau18] areused in Example 2, Example 4, Example 5 and Example 6 of §2.4.3 below to showthat several conditions of Definition 2.4.1.5 are automatic in special cases.(v) In Definition 2.4.1.5(iii), we have to use some element w Q of W ( C Q ) and “passthrough w Q ( C Q )” because of Remark 2.2.3.2(ii) (see also the end of Remark 2.2.3.12).Nothing in here and what follows depends on the choice of such a w Q .(vi) For a given Π compatible with e P , a bijection Φ as in Definition 2.4.1.5 is notunique in general (consider the case e P = M P ).(vii) In Definition 2.4.1.5, it is necessary in general to consider all elements w e P ∈ W e P ,note just w e P = 1, otherwise one misses some condition, see for instance (97) below(note that this is also quite natural in view of Theorem 2.3.2.5). Example 2.4.1.7.
Let us consider the case n = 3, K = Q p and e P = P with M P = GL × GL in the last part of Example 2.2.2.9(ii) (see also Example 2.2.4.4).We denote by P ′ the standard parabolic subgroup of Levi GL × GL . Then Π iscompatible with e P if and only Π has 3 irreducible constituents and the following form(a line means a nonsplit extension of length 2 as a subquotient and the constituenton the left-hand side is the socle ):Ind GL ( Q p ) P − ( Q p ) (cid:16) π · ( ω − ◦ det) ⊗ χ (cid:17) SS Ind GL ( Q p ) P ′− ( Q p ) (cid:16) χω − ⊗ π (cid:17) where χ : Q × p → F × is a smooth character, π is a supersingular representation ofGL ( Q p ) and SS is a supersingular representation of GL ( Q p ). The case e P = M P isanalogous but with a semisimple Π (instead of nonsplit extensions). See also §2.4.3below for more examples. 71he following proposition shows that a representation Π as in Definition 2.4.1.5has internal symmetries. Proposition 2.4.1.8.
Assume Π is compatible with e P and let Φ be a bijection as inDefinition 2.4.1.5. Let w e P ∈ W e P , Q a parabolic subgroup containing w e P P and C Q anisotypic component of L ⊗ | Z MQ such that P ( C Q ) = w Q Q for some ( unique ) w Q ∈ W with w Q ( S ( Q )) ⊆ S . Then π i ( C Q ) is compatible with ( w Q w e P e P ) Q,i for i ∈ { , . . . , d } ,where π i ( C Q ) is as in Definition 2.4.1.5(i).Proof. The proof is long but essentially formal. Replacing e P by w e P e P w − e P and Φ by w e P (Φ) (see the discussion following Definition 2.4.1.5), we can assume w e P = Id. Wewrite for simplicity w instead of w Q . Recall from Proposition 2.2.3.3 that C Q is the isotypic component of f w − ( θ G ) | Z MQ in L ⊗ | Z MQ . More precisely, by (47),Corollary 2.2.3.11 and Remark 2.2.3.12 (especially (53)), we have an isomorphism ofalgebraic representations of M Gal( K/ Q p ) w Q ∼ = Q di =1 M Gal( K/ Q p ) i ∼ = Q di =1 GL Gal( K/ Q p ) n i : w ( C Q ) ∼ = L ⊗ w Q ⊗ (cid:16) θ w Q ⊗ · · · ⊗ θ w Q (cid:17) ∼ = d O i =1 (cid:18) L ⊗ i ⊗ (cid:16) ( θ w Q ) i ⊗ · · · ⊗ ( θ w Q ) i (cid:17)(cid:19) . (74)Thus the map Φ w,i in Definition 2.4.1.5(iii) (recall w e P = Id and w = w Q ) is a bijectionof partially ordered sets between the set of M i ( K )-subrepresentations of π i ( C Q ) andthe set of good subrepresentations of C w,i | ( w e P ) Gal( K/ Q p ) Q,i = L ⊗ i | ( w e P ) Gal( K/ Q p ) Q,i (recall that( w P ) Q,i is here a standard parabolic subgroup of M i and ( w e P ) Q,i a Zariski closed sub-group of ( w P ) Q,i containing M ( w P ) Q,i ). We have to check that Φ w,i satisfies conditions(i) to (iv) in Definition 2.4.1.5 (with M i instead of G and ( w e P ) Q,i instead of e P ). Wewill only check condition (i) below, leaving the others, which are again essentiallyformal, to the (motivated) reader.We can assume i = 1. Let P def = ( w P ) Q, , e P def = ( w e P ) Q, (so M P ⊆ e P ⊆ P ⊆ M ⊆ M w Q ) and recall that T is the torus of diagonal matrices in M . Let w e P ∈ W e P ⊆ W ( M ), Q a parabolic subgroup of M containing w e P P and C Q an isotypiccomponent of L ⊗ | Z MQ , we have to prove that w e P (Φ w, ) − ( C Q ) is of the form (73).Step 1: Let e w def = w e P × Id × · · · × Id ∈ W ( M ) × · · · × W ( M d ) = W ( w Q ) ⊆ W andset w e P def = w − e w w ∈ W ( Q ). Then w e P ∈ W e P and Q contains w e P P . Indeed, since w e P ∈ W e P and the simple roots of P are contained in w ( S ( Q )) ⊆ S , we see that w e P = w − e w w sends the simple (resp. positive) roots of e P ∩ M Q to simple (resp.positive) roots of M Q and the roots of e P ∩ N Q to positive roots (using that W ( Q )normalizes N Q ). Moreover, one easily checks that w e P P = ( ww e P P ) Q, = ( w ( w e P P )) Q, .Replacing P by w e P P and Φ by w e P (Φ), we can thus assume w e P = Id.72tep 2: Let λ ∈ X ( T ) be a weight of L ⊗ | T such that C Q is the isotypic componentof λ | Z MQ and recall that λ | Z M = f θ M | Z M = f θ w Q | Z M , where θ M i for i ∈ { , . . . , d } is defined as in (34) replacing G = GL n by M i = GL n i . Let λ w Q ∈ X ( T ) be the uniquecharacter such that λ w Q | T = λ and λ w Q | T i = f θ M i = f θ w Q | T i if i ∈ { , . . . , d } (here,we use the convention in Remark 2.2.3.12 and recall that θ M i is trivial if M i = GL ).Then λ w Q is a weight of N di =1 L ⊗ i | T i . We set λ def = λ w Q + f θ w Q ∈ X ( T )which is a weight of L ⊗ | T (use (74)). We have λ | Z M = λ | Z M + f θ w Q | Z M = f θ M | Z M + f θ w Q | Z M = f ( θ w Q + θ w Q ) | Z M = f θ G | Z M (75)and if i ≥ λ | T i = f θ M i + f θ w Q | T i = f ( θ w Q + θ w Q ) | T i = f θ G | T i . (76)In particular λ | Z MwQ = f θ G | Z MwQ and thus w − ( λ ) | Z MQ = f w − ( θ G ) | Z MQ . (77)Let Q (1) ⊆ Q be the standard parabolic subgroup of G such that w Q (1) ⊆ w Q hasLevi M Q × M × · · · × M d . As P ⊆ Q by Step 1, we note that w Q (1) contains w P and hence Q (1) contains P , W ( w Q (1) ) = W ( Q ) × W ( M ) × · · · × W ( M d ) and w ( S ( Q (1) )) = S ( Q ) ∐ S ( M ) ∐ · · · ∐ S ( M d ). Let C Q (1) be the isotypic component of L ⊗ | Z MQ (1) associated to w − ( λ ) | Z MQ (1) . From (77) we get C Q (1) ⊆ C Q (inside L ⊗ | Z MQ (1) )and from (75), (76) an isomorphism of algebraic representations of M Gal( K/ Q p ) Q ⊗ Q di =2 M Gal( K/ Q p ) i : w ( C Q (1) ) ∼ = (cid:18) C Q ⊗ (cid:16) ( θ w Q ) ⊗· · ·⊗ ( θ w Q ) (cid:17)(cid:19) ⊗ d O i =2 (cid:18) L ⊗ i ⊗ (cid:16) ( θ w Q ) i ⊗· · ·⊗ ( θ w Q ) i (cid:17)(cid:19) . (78)Step 3: Define λ ′ , λ ′ w Q and θ ′ G by the formula (36) for P = w Q (1) and the respectivecharacters λ , λ w Q and θ G . Set λ ′ def = | W ( Q ) | P w ′ ∈ W ( Q ) w ′ ( λ ) ∈ ( X ( T ) ⊗ Z Q ) W ( Q ) .From (the proof of) Lemma 2.2.3.6, we easily get λ ′ = λ ′ w Q + f θ w Q with λ ′ w Q | T = λ ′ .Let w ∈ W ( M ) such that w ( S ( Q )) ⊆ S ( M ) and w ( λ ′ ) is dominant ( w exists byProposition 2.2.2.6(i)). We prove that w ( λ ′ ) = w ( λ ′ w Q ) + f θ w Q is also dominant (weconsider here w as an element of W ( w Q ) in the obvious way and use that W ( w Q )acts trivially on θ w Q ). From (76) we easily get λ ′ | T i = f θ ′ G | T i if i ≥
2. But θ ′ G isdominant since θ G is (see the proof of Lemma 2.2.2.4(i)), thus h w ( λ ′ ) , α i = h λ ′ , α i = h f θ ′ G , α i ≥ α ∈ { e j − e j +1 , n + 1 ≤ j ≤ n − } . Since w ( λ ′ w Q ) | T = w ( λ ′ )is dominant by assumption and h f θ w Q , α i = 0 if α ∈ { e j − e j +1 , ≤ j ≤ n − } h w ( λ ′ ) , e n − e n +1 i ≥
0. But an explicitcomputation gives h w ( λ ′ ) , e n − e n +1 i = h w ( λ ′ w Q ) , e n − e n +1 i + h f θ w Q , e n − e n +1 i = h w ( λ ′ w Q ) , e n i − h w ( λ ′ w Q ) , e n +1 i + f n = h w ( λ ′ w Q ) , e n i − f n −
12 + f n ≥ f n + 12 , where the last inequality follows h w ( λ ′ w Q ) , e n i ≥ L ⊗ | T (instead of L ⊗ | T ) together with formula (36).Step 4: By definition, S ( P ( C Q )) is the support of f θ M − w ( λ ′ ) (see Proposition2.2.2.6(ii)). By Remark 2.2.2.3(ii) we have w − ( λ ′ ) = ( w − ( λ )) ′ in ( X ( T ) ⊗ Z Q ) W ( Q (1) ) ,where the latter is given by (36) applied to the parabolic Q (1) and the character w − ( λ ). Since w w ( S ( Q (1) )) ⊆ S and w w (( w − ( λ )) ′ ) = w w ( w − ( λ ′ )) = w ( λ ′ ) isdominant (Step 4), S ( P ( C Q (1) )) is by definition the support of f θ G − w ( λ ′ ) = f θ G − (cid:16) w ( λ ′ w Q ) + f θ w Q (cid:17) = f θ w Q − w ( λ ′ w Q )= ( f θ M − w ( λ ′ )) + d X i =2 ( f θ M i − f θ ′ M i ) , (79)where θ ′ M i is defined by (36) applied to P = M i = G and the character θ M i of T i . Infact, θ ′ M i is the character det ni − of T i , from which we easily see that the support of(79) is exactly S ( P ( C Q )) ∐ S ( M ) ∐ · · · ∐ S ( M d ). This implies M P ( C Q (1) ) = diag( M P ( C Q ) , M , . . . , M d ) . (80)Step 5: We now finally prove that Φ w, satisfies condition (i) in Definition 2.4.1.5.Write M P ( C Q ) = M , × · · · × M ,d (for some d ≥ − ( C Q (1) ) ∼ = Ind G ( K ) P ( C Q (1) ) − ( K ) (cid:18)(cid:16) π ( C Q (1) ) ⊗ · · · ⊗ π d ( C Q (1) ) (cid:17) ⊗ ( ω − ◦ θ P ( C Q (1) ) ) (cid:19) , (81)where π ( C Q (1) ) = π , ( C Q (1) ) ⊗ · · · ⊗ π ,d ( C Q (1) ) (with obvious notation). Let π ′ def = Ind M ( K ) P ( C Q ) − ( K ) (cid:16) π ( C Q (1) ) ⊗ ( ω − ◦ θ P ( C Q ) ) (cid:17) , (82)it is enough to prove that π ′ is a subquotient of π ( C Q ) and thatΦ w, ( π ′ ) = C Q | e P Gal( K/ Q p )1 (= C Q | ( e P ∩ M Q ) Gal( K/ Q p ) ) . θ P ( C Q (1) ) = θ P ( C Q ) + θ P ( C Q ) , (83)where we view θ P ( C Q ) as a character of T (not just T ) by sending the coordinatesin T i to 1 for i ≥ − ( C Q (1) ) ∼ = Ind G ( K ) P ( C Q ) − ( K ) (cid:18)(cid:16) π ′ ⊗ π ( C Q (1) ) ⊗ · · · ⊗ π d ( C Q (1) ) (cid:17) ⊗ ( ω − ◦ θ P ( C Q ) ) (cid:19) . Since C Q (1) is a subquotient of C Q (both being good subquotients of L ⊗ | e P Gal( K/ Q p ) ),Φ − ( C Q (1) ) is a subquotient of Φ − ( C Q ). This implies in particular (using the ordinaryfunctor of [Eme10a] together with Remark 2.4.1.6(iii)) that π ′ (resp. π i ( C Q (1) ) for i ≥
2) is a subquotient of π ( C Q ) (resp. of π i ( C Q ) for i ≥ ′ = Φ − ( C Q (1) ) (together with P ( C Q ) = w Q ), wealso get an isomorphism of algebraic representations of Q di =1 ( w e P ) Gal( K/ Q p ) Q,i over F : w ( C Q (1) ) = (cid:18) Φ w, ( π ′ ) ⊗ (cid:16) ( θ w Q ) ⊗ · · · ⊗ ( θ w Q ) (cid:17)(cid:19) ⊗ d O i =2 (cid:18) Φ w,i ( π i ( C Q (1) )) ⊗ (cid:16) ( θ w Q ) i ⊗ · · · ⊗ ( θ w Q ) i (cid:17)(cid:19) , (84)where Φ w, ( π ′ ) and Φ w,i ( π i ( C Q (1) )) ( i ≥
2) are good subquotients of L ⊗ i | ( w e P ) Gal( K/ Q p ) Q,i .Since we have good subquotients of L ⊗ i | ( w e P ) Gal( K/ Q p ) Q,i in each factor of (78) and (84),(78) and (84) imply Φ w, ( π ′ ) = C Q | e P Gal( K/ Q p )1 and Φ w,i ( π i ( C Q (1) )) = L ⊗ i | ( w e P ) Gal( K/ Q p ) Q,i for i ≥ L ⊗ i | M Gal( K/ Q p )( wP ) Q,i tautology occur with multiplicity 1,so there is no multiplicity issue). This finishes the proof of condition (i) in Definition2.4.1.5 for Φ w, . Remark 2.4.1.9.
When P ( C Q ) is strictly bigger than w Q Q for one, or equivalentlyany by Lemma 2.2.3.1, w Q ∈ W ( C Q ), there is no real analogue of Proposition 2.4.1.8since L ⊗ i has to be replaced by C w Q ,i in (53) which is not L ⊗ i in general. ρ We define what it means for a representation of G ( K ) over F to be compatible witha good conjugate ρ : Gal( Q p /K ) → e P ρ ( F ) as in §2.3.2. Essentially, an admissiblesmooth representation Π is compatible with ρ if it is compatible with e P ρ in thesense of Definition 2.4.1.5 and if the bijection Φ of loc.cit. satisfies some naturalcompatibilities with the functor V G in (16) (see Definition 2.4.2.7).75e now fix a continuous homomorphism ρ : Gal( Q p /K ) −→ G ( F )and recall that ρ ss denotes the semisimplification of the associated representation ofGal( Q p /K ) (see §2.3.2). We assume that ρ is generic in the following sense:(a) ρ ss has distinct irreducible constituents;(b) the ratio of any two irreducible constituents of ρ ss of dimension 1 is not in { ω, ω − } .By Proposition 2.3.2.2, conjugating ρ by an element of G ( F ) if necessary, we canassume that ρ is a good conjugate in the sense of Definition 2.3.2.3, that is we have ρ : Gal( Q p /K ) −→ e P ρ ( F ) ⊆ P ρ ( F ) ⊆ G ( F ) , where P ρ is a standard parabolic subgroup of G such that ρ ss is given by the com-position Gal( Q p /K ) ρ −→ P ρ ( F ) ։ M P ρ ( F ) (see (67)), e P ρ ⊆ P ρ is the smallest closedalgebraic subgroup of P ρ containing M P ρ and the ρ ( g ) for g ∈ Gal( Q p /K ) (in its F -points), and where, for any h ∈ P ρ ( F ), if we define e P hρh − ⊆ P ρ as for ρ , then wehave e P ρ ⊆ e P hρh − . Good conjugates are not unique, see Theorem 2.3.2.5, but we fixsuch a good conjugate ρ (and the associated pair ( e P ρ , P ρ )) for the moment.For any e w ∈ W ρ = W e P ρ (see (71)) and any parabolic subgroup Q containing e w P ρ ,we define the Q -semisimplification ρ Q − ss of ρ as the continuous homomorphism ρ Q − ss : Gal( Q p /K ) e wρ e w − −→ e w P ρ ( F ) ֒ → Q ( F ) ։ M Q ( F )(strictly speaking, it also depends on e w ). More generally, for any w ∈ W such that w ( S ( Q )) ⊆ S , we define the continuous homomorphisms w ( ρ Q − ss ) : Gal( Q p /K ) wρ Q − ss w − −→ wM Q ( F ) w − = M w Q ( F )and note that w ( ρ Q − ss ) actually takes values in( w e w e P ρ ) Q ( F ) ⊆ ( w e w P ρ ) Q ( F ) ⊆ M w Q ( F )(recall from the beginning of §2.4.1 that ( w e w P ρ ) Q = w ( e w P ρ ∩ M Q ) w − and ( w e w e P ρ ) Q = w (cid:16) ( e w e P ρ e w − ) ∩ M Q (cid:17) w − ).Let e w ∈ W ρ , Q a parabolic subgroup containing e w P ρ , w ∈ W such that w ( S ( Q )) ⊆ S and Q ′ a parabolic subgroup containing w Q . We write M Q ′ = diag( M , . . . , M d )with M i ∼ = GL n i and we set for i ∈ { , . . . , d } : w ( ρ Q − ss ) i : Gal( Q p /K ) w ( ρ Q − ss ) −→ M w Q ( F ) ֒ → M Q ′ ( F ) ։ M i ( F ) .
76e also have (recall from §2.4.1 that ( w Q ) i is a standard parabolic subgroup of M i ): w ( ρ Q − ss ) i : Gal( Q p /K ) w ( ρ Q − ss ) −→ M w Q ( F ) ։ M ( w Q ) i ( F ) ֒ → M i ( F ) . (85)Composing w ( ρ Q − ss ) i with M i ( F ) ։ ( M i /M der i )( F ) ∼ = F × , we obtain by class fieldtheory for K a continuous group homomorphismdet( w ( ρ Q − ss ) i ) : K × −→ F × . (86) Lemma 2.4.2.1.
Let ρ , Q as above, C Q an isotypic component of L ⊗ | Z MQ and Q ′ def = P ( C Q ) . Then the characters (86) for i ∈ { , . . . , d } and w ∈ W ( C Q ) ( see (38)) don’t depend on the choice of w ∈ W ( C Q ) . Moreover, we have Q di =1 det( w ( ρ Q − ss ) i ) =det( ρ ) .Proof. This follows from Lemma 2.2.2.10 (applied to P = Q ) together with the factthat conjugation by W ( P ( C Q )) (seen in M P ( C Q ) ( F )) is trivial on M P ( C Q ) /M der P ( C Q ) , andthus on each M i /M der i . The last assertion is obvious.As previously, w ( ρ Q − ss ) i in (85) takes values in( w e w e P ρ ) Q,i ( F ) ⊆ ( w e w P ρ ) Q,i ( F ) ⊆ M ( w Q ) i ( F ) ⊆ M i ( F ) ∼ = GL n i ( F )(recall from the beginning of §2.4.1 that ( w e w P ρ ) Q,i is a standard parabolic subgroup of M ( w Q ) i and that ( w e w e P ρ ) Q,i is a Zariski closed algebraic subgroup of ( w e w P ρ ) Q,i containing M ( w e w P ρ ) Q,i ). Proposition 2.4.2.2.
Let ρ , Q as above, w ∈ W such that w ( S ( Q )) ⊆ S and Q ′ def = w Q . Then w ( ρ Q − ss ) i : Gal( Q p /K ) −→ M i ( F ) is a good conjugate with values in ( w e w e P ρ ) Q,i ( F ) for i ∈ { , . . . , d } .Proof. Note that e wρ e w − is a good conjugate (with values in e w e P ρ ( F ) e w − ⊆ e w P ρ ( F ))by Lemma 2.3.2.4. Since w ( ρ Q − ss ) is obtained from ρ Q − ss by permuting the blocs M i ∼ = GL n i of M Q , it is equivalent to prove the statement for w = Id. Assumethat ρ i def = ( ρ Q − ss ) i : Gal( Q p /K ) → M i ( F ) is not a good conjugate. Then it followsfrom Proposition 2.3.2.2 that there is h i ∈ ( e w P ρ ) Q,i ( F ) such that h i ρ i h − i is a goodconjugate, and thus X h i ρ i h − i ( X ρ i (with the notation of §2.3.2). Let α i be a positiveroot of GL n i in X ρ i \ X h i ρ i h − i and note that, if α i is a sum of roots in R + (viewing α i in R + ), then all of these roots are positive roots of GL n i . Set h j def = Id GL nj ∈ GL n j ( F )if j = i and define h = ( h , . . . , h d ) ∈ diag( M , . . . , M d ) = M Q ( F ) ⊆ Q ( F ). Ifwe had α i ∈ X h e wρ e w − h − , then from what we just said necessarily we would have α i ∈ X ( hρ Q − ss h − ) i = X h i ρ i h − i which is impossible. Therefore α i / ∈ X h e wρ e w − h − . Butsince α i ∈ X ρ i ⊆ X e wρ e w − (viewing the positive roots of GL n i as a subset of R + ) wededuce X h e wρ e w − h − ( X e wρ e w − which is impossible as e wρ e w − is a good conjugate.77or σ ∈ Gal( K/ Q p ) = Gal( Q p f / Q p ) consider ρ σ : Gal( Q p /K ) → e P ρ ( F ) ⊆ P ρ ( F ) ⊆ G ( F ) , where ρ σ ( g ) def = ρ ( σgσ − ). Here g ∈ Gal( Q p /K ) and σ is any lift of σ in Gal( Q p / Q p ).Since Gal( Q p /K ) is normal in Gal( Q p / Q p ), ρ σ ( g ) is well defined up to conjugation(by elements in e P ρ ( F )). If C is a good subquotient of L ⊗ | e P Gal( K/ Q p ) ρ (Definition 2.2.1.3),we can view in particular C as a continuous homomorphism e P ρ ( F ) × · · · × e P ρ ( F ) | {z } Gal( K/ Q p ) −→ Aut (cid:16) C ( F ) (cid:17) (87)(denoting by C ( F ) the underlying F -vector space of the algebraic representation C )and define a Gal( Q p /K )-representation C ( ρ ) asGal( Q p /K ) Q ρ σ −→ e P ρ ( F ) × · · · × e P ρ ( F ) C −→ Aut (cid:16) C ( F ) (cid:17) , where, in the first arrow, we choose any order on the elements σ of Gal( K/ Q p ). Lemma 2.4.2.3.
The
Gal( Q p /K ) -representation C ( ρ ) is well-defined up to isomor-phism and canonically extends to a Gal( Q p / Q p ) -representation.Proof. The algebraic representation C of e P Gal( K/ Q p ) ρ over F doesn’t depend up toisomorphism on the order of the copies of e P ρ , i.e. any permutation of the e P ρ ’s yields analgebraic representation which is conjugate by an element of Aut( C ( F )). Indeed, thisclearly holds when C is an isotypic component of L ⊗ | Z MPρ as Z M Pρ embeds diagonallyinto e P Gal( K/ Q p ) ρ . Thus, for a general good subquotient C , any permutation of the e P ρ ’sgives a representation C ′ which contains the same isotypic components of L ⊗ | Z MPρ asthose of C . Assume now that C is a good subrepresentation of L ⊗ | e P Gal( K/ Q p ) ρ . Then C ′ must be isomorphic to C since isotypic components of L ⊗ | Z MPρ tautologicallyoccur with multiplicity 1. In general, one writes C as the quotient of two goodsubrepresentations of L ⊗ | e P Gal( K/ Q p ) ρ . All this implies that C ( ρ ) is well-defined.We now prove that it extends to Gal( Q p / Q p ). First, if C = L ⊗ | e P Gal( K/ Q p ) ρ , then C ( ρ )is the tensor induction (20) and thus canonically extends to Gal( Q p / Q p ). Let usrecall explicitly how it extends. Fix σ , . . . , σ f some representatives in Gal( Q p / Q p )of the elements of Gal( K/ Q p ) = Gal( Q p f / Q p ) and define permutations w , . . . , w f on { , . . . , f } by σ i σ − j = σ − w i ( j ) h i,j , where h i,j ∈ Gal( K/ Q p ). The underlying F -vectorspace L ⊗ ( F ) of L ⊗ is f O i =1 (cid:18)(cid:16) O α ∈ S L ( λ α ) (cid:17) ( F ) (cid:19) , (cid:16) N α ∈ S L ( λ α ) (cid:17) ( F ) is the underlying vector space of N α ∈ S L ( λ α ), and the actionof σ i then sends v ⊗ v ⊗ · · · ⊗ v f ∈ L ⊗ ( F ) to u ⊗ u ⊗ · · · ⊗ u f , where: u w i ( j ) def = (cid:18)(cid:18) O α ∈ S L ( λ α ) (cid:19) ( ρ ( h i,j )) (cid:19) ( v j ) . (88)This yields an action of Gal( Q p / Q p ) which doesn’t depend on any choice (up toisomorphism). It is enough to prove that this action of Gal( Q p / Q p ) preserves thesubspaces C ( F ) ⊆ L ⊗ ( F ), where C is any good subrepresentation of L ⊗ | e P Gal( K/ Q p ) ρ .But this is clear from (88) since C ( F ) is preserved by the action of Gal( Q p /K ) and by any permutation of the v i (as we have seen at the beginning). Remark 2.4.2.4.
One could also use L -groups as in §2.1.4 in order to have moreintrinsic definitions (see Remark 2.2.1.1(i)). However the above pedestrian approachwill be sufficient for our purpose.The following lemma is in the same spirit as Lemma 2.4.2.1. Lemma 2.4.2.5.
Let ρ , Q as above, C Q an isotypic component of L ⊗ | Z MQ and Q ′ def = P ( C Q ) . For w ∈ W ( C Q ) and i ∈ { , . . . , d } , let • C w,i be the isotypic component of L ⊗ i | Z M ( wQ ) i defined in (53) ( applied with P there being Q ) ; • w ( ρ Q − ss ) i the representation of Gal( Q p /K ) with values in M ( w Q ) i ( F ) defined in (85) ( applied to Q ′ = P ( C Q )) ; • C w,i (cid:16) w ( ρ Q − ss ) i (cid:17) the representation of Gal( Q p / Q p ) defined in Lemma 2.4.2.3 ( applied to ρ = w ( ρ Q − ss ) i , L ⊗ i and C = C w,i ) .Then the Gal( Q p / Q p ) -representation C w,i (cid:16) w ( ρ Q − ss ) i (cid:17) doesn’t depend on w ∈ W ( C Q ) .Proof. Let w ′ be another element in W ( C Q ). Then w ′ = w P ( C Q ) w with w P ( C Q ) ∈ W ( P ( C Q )) by Lemma 2.2.2.10 (with P there being Q ). Since w P ( C Q ) respects M i , wehave w ′ ( ρ Q − ss ) i = w P ( C Q ) w ( ρ Q − ss ) i w − P ( C Q ) . The result then follows from (54) (applied with P = Q ). Remark 2.4.2.6.
Lemma 2.4.2.5 still holds replacing C w,i by any good subquotientof C w,i | ( w e w e P ρ ) Gal( K/ Q p ) Q,i and using the proof of Lemma 2.4.1.3(ii) and Remark 2.4.1.4 tocompare with the corresponding good subquotient of C w ′ ,i | ( w ′ e w e P ρ ) Gal( K/ Q p ) Q,i . The proofis the same as for Lemma 2.4.2.5 using that w ( ρ Q − ss ) i takes values in ( w e w e P ρ ) Q,i ( F ).79e now state the second crucial definition. We use the functor V H defined in§2.1.1 in the case H = GL m , m ≥ π of H ( K ) has a central character, we denote it by Z ( π ) (sowriting Z ( π ) in the sequel implicitly means that π has a central character). We alsodefine ω − ◦ θ M i : Z M i ( K ) = K × θ Mi | ZMi −→ K × ω − −→ F × p ֒ → F × (89)( θ M i as in (34) replacing G by M i ). Definition 2.4.2.7.
An admissible smooth representation Π of G ( K ) over F whichhas finite length and distinct absolutely irreducible constituents is compatible with ρ if there exists a bijection Φ as in Definition 2.4.1.5 for e P = e P ρ (in particular Π iscompatible with e P ρ ) which satisfies the following extra conditions:(i) for any subquotient Π ′ of Π, we have an isomorphism of Gal( Q p / Q p )-representa-tions over F : V G (Π ′ ) ∼ = Φ(Π ′ )( ρ ) , (90)where Φ(Π ′ )( ρ ) is the associated representation of Gal( Q p / Q p ) defined in Lem-ma 2.4.2.3;(ii) for any e w ∈ W ρ , any parabolic subgroup Q containing e w P ρ and any isotypic com-ponent C Q of L ⊗ | Z MQ , writing M P ( C Q ) = diag( M , . . . , M d ) with M i ∼ = GL n i wehave for one, or equivalently any, element w ∈ W ( C Q ) and for any subquotient π ′ i of π i ( C Q ): Z (cid:16) π ′ i (cid:17) ∼ = det( w ( ρ Q − ss ) i ) · ω − ◦ θ M i (91) V M i (cid:16) π ′ i (cid:17) ∼ = e w (Φ) w,i ( π ′ i ) (cid:16) w ( ρ Q − ss ) i (cid:17) , where• π i ( C Q ) is the admissible smooth representation of M i ( K ) over F in Defini-tion 2.4.1.5(i);• det( w ( ρ Q − ss ) i ) (resp. ω − ◦ θ M i ) is the character of K × defined in (86) (resp.in (89));• e w (Φ) w,i ( π ′ i ) is the good subquotient of C w,i | ( w e w e P ρ ) Q,i defined in Definition2.4.1.5(iii);• w ( ρ Q − ss ) i is the representation of Gal( Q p /K ) with values in ( w e w e P ρ ) Q,i ( F ) ⊆ M ( w Q ) i ( F ) defined in (85) (applied to Q ′ = P ( C Q ));• e w (Φ) w,i ( π ′ i ) (cid:16) w ( ρ Q − ss ) i (cid:17) is the representation of Gal( Q p / Q p ) defined inLemma 2.4.2.3 (applied to ρ = w ( ρ Q − ss ) i , L ⊗ i and C = e w (Φ) w,i ( π ′ i )).80f Π is compatible with ρ , then we have in particular V G (Π) ∼ = L ⊗ ( ρ )and V M i (cid:16) π i ( C Q ) (cid:17) ∼ = C w,i (cid:16) w ( ρ Q − ss ) i (cid:17) for Q, w, i as in Definition 2.4.2.7(ii) (recall that V M i ( π i ( C Q )) is always the trivial representation of Gal( Q p / Q p ) when n i = 1). If ρ is(absolutely) irreducible, then e P ρ = P ρ = G , W ρ = { Id } and Π is compatible with ρ if and only if Π is absolutely irreducible supersingular, Z (Π) ∼ = det( ρ ) · ω − ◦ ( θ G | Z G )and V G (Π) ∼ = L ⊗ ( ρ ). Remark 2.4.2.8. (i) The isomorphisms in (91) are consistent with Lemma 2.4.2.1,Lemma 2.4.2.5 and Remark 2.4.2.6 since their left-hand sides don’t depend on w ∈ W ( C Q ).(ii) Let Π be compatible with ρ . From (73) applied with w e P = 1 and Q = P , (91)applied with e w = 1 and Q = P ρ , the last assertion in Lemma 2.4.2.1, and from θ G | Z G = θ P ( C Q ) | Z G θ P ( C Q ) | Z G = θ P ( C Q ) | Z G (cid:18) d Y i =1 θ M i | Z Mi (cid:19) (which follows from (44)), we deduce that each irreducible constituent Π ′ of Π is suchthat Z (Π ′ ) = det( ρ ) · ω − ◦ ( θ G | Z G ). Since these irreducible constituents are all dis-tinct by assumption, we obtain that Π has a central character Z (Π) = det( ρ ) · ω − ◦ ( θ G | Z G ) = det( ρ ) · ω − n ( n − .(iii) Let Π be compatible with ρ , Π ′ a subquotient of Π and Π ′′ ⊆ Π ′ a subrepre-sentation. Then from Remark 2.4.1.2(i) we have an exact sequence of Gal( Q p / Q p )-representations:0 −→ Φ(Π ′′ )( ρ ) −→ Φ(Π ′ )( ρ ) −→ Φ(Π ′ / Π ′′ )( ρ ) −→ . Thus (90) implies that the sequence 0 → V G (Π ′′ ) → V G (Π ′ ) → V G (Π ′ / Π ′′ ) → V G behaves like anexact functor.(iv) Let χ : K × → F × be a smooth character. Then it easily follows from Remark2.1.1.4(ii) that Π is compatible with ρ if and only if Π ⊗ ( χ ◦ det) is compatible with ρ ⊗ χ .(v) For a given Π compatible with ρ , a bijection Φ as in Definition 2.4.2.7 is stillnot unique in general. For instance consider the case n = 4, K = Q p , e P ρ = M P ρ =diag(GL , GL ) and ρ = ρ ⊕ ρ with ρ i : Gal( Q p / Q p ) → GL ( F ) absolutely irreducibledistinct for i = 1 , ∧ F ρ ∼ = ∧ F ρ .Definition 2.4.2.7 doesn’t depend on the choice of a good conjugate. Proposition 2.4.2.9. If ρ ′ : Gal( Q p /K ) → e P ρ ′ ( F ) ⊆ P ρ ′ ( F ) is another good conjugateof ρ , then Π is compatible with ρ if and only if Π is compatible with ρ ′ .Proof. From Theorem 2.3.2.5, we have ρ ′ = whρh − w − for some h ∈ e P ρ ( F ) and some w ∈ W ρ . By symmetry, it is enough to prove that Π compatible with ρ implies Π81ompatible with ρ ′ . We have first that Π is compatible with hρh − . Indeed, e P hρh − = e P ρ and the conditions in Definition 2.4.2.7 for hρh − follow from the conditions for ρ since w ( ρ Q − ss ) i and w (( hρh − ) Q − ss ) i are conjugate in ( w e w e P ρ ) Q,i ( F ) (with e w, w hereas in Definition 2.4.2.7). Thus we can assume h = Id. But then, it is clear fromDefinition 2.4.2.7 that Π is compatible with ρ ′ = wρw − .Just as some statements in Definition 2.4.1.5 should follow from others (see Re-mark 2.4.1.6(iv)), we expect the isomorphisms (90) to follow in many cases from theisomorphisms (91): Proposition 2.4.2.10.
Assume Π is compatible with ρ and let Φ be a bijection asin Definition 2.4.2.7. Let e w ∈ W ρ , Q a parabolic subgroup containing e w P ρ , C Q anisotypic component of L ⊗ | Z MQ and Π ′ a subquotient of e w (Φ) − ( C Q ) of the form Π ′ ∼ = Ind G ( K ) P ( C Q ) − ( K ) (cid:16) ( π ′ ⊗ · · · ⊗ π ′ d ) ⊗ ( ω − ◦ θ P ( C Q ) ) (cid:17) , where π ′ i is a subquotient of the representation π i ( C Q ) of M i ( K ) over F definedin Definition 2.4.1.5(i) ( so that e w (Φ)(Π ′ ) is a good subquotient of C Q | e w e P Gal( K/ Q p ) ρ = C Q | (( e w e P ρ e w − ) ∩ M Q ) Gal( K/ Q p ) ) . Assume that V M P ( CQ ) ( π ′ ⊗ · · · ⊗ π ′ d ) ∼ = N di =1 V M P ( CQ ) ,i ( π ′ i )( with the notation used in Lemma 2.1.1.5 ) . Then the isomorphism (90) for Π ′ followsfrom the isomorphisms (91) .Proof. For i ∈ { , . . . , d } , we have (easy computation):( θ P ( C Q ) ) i = det n − P ij =1 n j . (92)Let π ′′ i def = π ′ i ⊗ ( ω − ◦ det) n − P ij =1 n j , we have by Lemma 2.1.1.5, (92) and Remark2.1.1.4(ii): V G (Π ′ ) = V G (cid:18) Ind G ( K ) P ( C Q ) − ( K ) ( ⊗ di =1 π ′ i ) ⊗ ( ω − ◦ θ P ( C Q ) ) (cid:19) ∼ = d O i =1 (cid:18) V M i ( π ′′ i ) ⊗ (cid:16) Z ( π ′′ i ) n − P ij =1 n j (cid:17) | Q × p δ − M i (cid:19)! ⊗ δ G ∼ = d O i =1 (cid:18) V M i ( π ′ i ) ⊗ (cid:18)(cid:16) Z ( π ′ i ) · ω ◦ θ M i (cid:17) n − P ij =1 n j (cid:19) | Q × p (cid:19)! ⊗ δ, δ def = (cid:16) δ G Q di =1 δ − M i (cid:17) ind ⊗ Q p K ( ω − P di =1 c i ) with (by an explicit computation): c i = n i ( n i − (cid:18) n − i X j =1 n j (cid:19) + n i (cid:18) n − i X j =1 n j (cid:19) = n i (cid:18) n − i X j =1 n j (cid:19)(cid:18) n i − n − i X j =1 n j (cid:19) = n i (cid:18) n − i X j =1 n j (cid:19)(cid:18) n − − i − X j =1 n j (cid:19) . (93)Now, assuming (91) we have for one, or equivalently any, w of W ( C Q ):Φ(Π ′ )( ρ ) ∼ = e w (Φ)(Π ′ )( ρ )= e w (Φ)(Π ′ )( ρ Q − ss ) ∼ = d O i =1 e w (Φ) w,i ( π ′ i ) (cid:16) w ( ρ Q − ss ) i (cid:17) ⊗ (cid:18)(cid:16) ( θ P ( C Q ) ) i ⊗ · · · ⊗ ( θ P ( C Q ) ) i (cid:17) ◦ (cid:16) ⊕ σ ( w ( ρ Q − ss ) i ) σ (cid:17)(cid:19)! ∼ = d O i =1 (cid:18) V M i ( π ′ i ) ⊗ (cid:18)(cid:16) det( w ( ρ Q − ss ) i ) (cid:17) n − P ij =1 n j (cid:19) | Q × p (cid:19) ∼ = d O i =1 (cid:18) V M i ( π ′ i ) ⊗ (cid:18)(cid:16) Z ( π ′ i ) · ω ◦ θ M i (cid:17) n − P ij =1 n j (cid:19) | Q × p (cid:19) , where the first isomorphism follows from ρ ∼ = e wρ e w − , the second equality is obvi-ous ( e w (Φ)(Π ′ ) being a representation of M Gal( K/ Q p ) Q as it is a subquotient of C Q ),the second isomorphism follows from Definition 2.4.1.5(iii), and the last two isomor-phisms from (91), (92) and local class field theory for Q p . So we have to prove( δ G Q di =1 δ − M i ) ind ⊗ Q p K ( ω − P di =1 c i ) = 1, which amounts to checking the following explicitidentity (using (93) and Example 2.1.1.3): n − X j =1 j = d X i =1 n i − X j =1 j + d X i =1 (cid:18) n i (cid:16) n − i X j =1 n j (cid:17)(cid:16) n − − i − X j =1 n j (cid:17)(cid:19) . This follows easily by induction on d using the case d = 2 and the identity( n − m ) + ( n − m + 1) + · · · + ( n − = 1 + 2 + · · · + ( m − + m ( n − m )( n − n ≥ m ≥ roposition 2.4.2.11. Assume Π is compatible with ρ and let Φ be a bijection as inDefinition 2.4.2.7. Let e w ∈ W ρ , Q a parabolic subgroup containing e w P ρ and C Q anisotypic component of L ⊗ | Z MQ such that P ( C Q ) = w Q for some ( unique ) w ∈ W with w ( S ( Q )) ⊆ S . Then π i ( C Q ) is compatible with w ( ρ Q − ss ) i for i ∈ { , . . . , d } , where π i ( C Q ) is as in Definition 2.4.1.5(i) and w ( ρ Q − ss ) i as in (85) .Proof. We use the notation in the proof of Proposition 2.4.1.8. Replacing ρ by e wρ e w − and Φ by e w (Φ), we can assume e w = Id. We have to prove that the map Φ w,i satisfiesconditions (i) and (ii) of Definition 2.4.2.7 with M i instead of G and w ( ρ Q − ss ) i insteadof ρ . Note that this makes sense thanks to Proposition 2.4.2.2. We can assume i = 1.Condition (i) clearly follows from the second equality in (91) applied to π ′ = π ( C Q ).Arguing as in Step 1 of Lemma 2.4.1.8, we need only consider a standard parabolicsubgroup Q of M containing ( w P ρ ) Q, and C Q an isotypic component of L ⊗ | Z MQ .Let C Q (1) be the isotypic component of L ⊗ | Z MQ (1) defined in Step 2 of the proof ofProposition 2.4.1.8. Then it is easy to check that condition (ii) for M , w ( ρ Q − ss ) , C Q and an element w ∈ W ( C Q ) follows from condition (ii) with G , ρ , C Q (1) and w w ∈ W ( C Q (1) ) (see Step 3, Step 4 and Step 5 of the proof of Proposition 2.4.1.8). We explicitly give the form of a representation Π compatible with ρ for various ρ .In the examples below, as in Example 2.4.1.7, a line means a nonsplit extensionbetween two irreducible constituents, the constituent on the left being the subobjectof the corresponding (length 2) subquotient. Example 1
We start with GL ( Q p f ) and e P ρ = P ρ = B as in Example 2.2.2.9(i), i.e. we have ρ ∼ = χ ∗ χ ! , where χ i are two smooth characters Q × p f → F × (via class field theory) with ratio = 1 , ω ± (and where ∗ is nonsplit). Let Π be compatible with ρ . Then Π has f + 1irreducible constituents and the following form:Ind GL ( Q pf ) B − ( Q pf ) ( χ ω − ⊗ χ ) SS SS · · · SS f − Ind GL ( Q pf ) B − ( Q pf ) ( χ ω − ⊗ χ )where the SS i for i ∈ { , . . . , f − } are distinct supersingular representations ofGL ( Q p f ) over F such that Z (SS i ) = det( ρ ) ω − and V G (SS i ) ∼ = M I ⊆ Gal( K/ Q p ) | I | = f − i (cid:18)(cid:16) O σ ∈ I χ σ (cid:17) ⊗ (cid:16) O σ / ∈ I χ σ (cid:17)(cid:19) χ σi def = χ i ( σ · σ − ) and V G (SS i ) is immediately checked to be a representation ofGal( Q p / Q p )). Moreover it follows from Example 2.1.1.6 that V G (cid:18) Ind GL ( Q pf ) B − ( Q p ) ( χ ω − ⊗ χ ) (cid:19) ∼ = ⊗ σ ∈ Gal( K/ Q p ) χ σ and likewise with Ind GL ( Q pf ) B − ( Q p ) ( χ ω − ⊗ χ ). Finally the conditions in (90) imply that V G behaves as an exact functor on the (not necessarily irreducible) subquotients of Π(see Remark 2.4.2.8(iii)).Still with GL ( Q p f ) but when e P ρ = T , i.e. ρ = χ ⊕ χ , then Π (compatible with ρ ) is semisimple, i.e. has the same form as above but with split extensions every-where. This is consistent with the discussion at the end of [BP12, §19]. Notethat, if we only require Π to be compatible with e P ρ (Definition 2.4.1.5), then Πhas the same form as above, but with arbitrary distinct supersingular representationsof GL ( Q p f ) and arbitrary distinct irreducible principal series Ind GL ( Q pf ) B − ( Q pf ) ( η ω − ⊗ η )and Ind GL ( Q pf ) B − ( Q pf ) ( η ω − ⊗ η ). See [HW, §10.6] and §3.4.4 for instances of represen-tations Π (coming from mod p cohomology) satisfying (special cases of) the aboveproperties. Example 2
We go on with GL ( Q p ) as in Example 2.2.2.9(ii) and e P ρ = P ρ = B , i.e. we have ρ ∼ = χ ∗ ∗ χ ∗ χ , where χ i are three smooth characters Q × p → F × (via class field theory) of ratio = 1 , ω ± . For τ ∈ W ∼ = S , we definePS χ τ (1) ,χ τ (2) ,χ τ (3) def = Ind GL ( Q p ) B − ( Q p ) ( χ τ (1) ω − ⊗ χ τ (2) ω − ⊗ χ τ (3) ) . Let Π be compatible with ρ . Then Π has 7 irreducible constituents and the followingform: PS χ ,χ ,χ PS χ ,χ ,χ PS χ ,χ ,χ SS PS χ ,χ ,χ PS χ ,χ ,χ PS χ ,χ ,χ rrrrrrrrrr ▲▲▲▲▲▲▲▲▲▲ ▲▲▲▲▲▲▲▲▲▲▲ rrrrrrrrrrr rrrrrrrrrrr ▲▲▲▲▲▲▲▲▲▲▲ rrrrrrrrrrr ▲▲▲▲▲▲▲▲▲▲ rrrrrrrrrr where SS is a supersingular representation of GL ( Q p ) over F such that Z (SS) =det( ρ ) · ω − and V G (SS) ∼ = ( χ χ χ ) ⊕ = det( ρ ) ⊕ . It follows from the proof of85Hau16, Thm.5.2.1], or from [Hau18, Thm.1.4(i)], combined with [Eme10a, Cor.4.3.5],that the nonsplit extensions between two principal series in subquotient are automat-ically parabolic inductions as required in condition (i) of Definition 2.4.1.5 (lookingat isotypic components of L ⊗ | Z MQ with M Q ∈ { GL × GL , GL × GL } , see Example2.2.2.9(ii)). Conditions (ii) to (iv) in Definition 2.4.1.5 are then easily checked. Con-cerning Definition 2.4.2.7, the subquotients involving only principal series do satisfy(90) and (91) by [Bre15, Rem.9.9]. The reader can then easily work out the remain-ing conditions in (90) which all involve the supersingular representation SS, and alsowork out the shape of a Π which is compatible with e P ρ = B only (but not necessarilywith ρ ). Example 3
We stay with GL ( Q p ) but where e P ρ = P ρ = P with M P = diag(GL , GL ), i.e. wehave ρ ∼ = ρ ∗ χ ! , where ρ : Gal( Q p / Q p ) → GL ( F ) is any absolutely irreducible representation and χ is any smooth character Q × p → F × (via class field theory). Note that such a ρ isalways generic (see the beginning of §2.4.2). Then Π is compatible with ρ if and onlyΠ has the same form as in Example 2.4.1.7:Ind GL ( Q p ) P − ( Q p ) (cid:16) π · ( ω − ◦ det) ⊗ χ (cid:17) SS Ind GL ( Q p ) P ′− ( Q p ) (cid:16) χ ω − ⊗ π (cid:17) and where moreover• π is the supersingular representation of GL ( Q p ) over F corresponding to ρ by the mod p local Langlands correspondence for GL ( Q p ), i.e. we have Z ( π ) = det( ρ ) ω − (via class field theory) and V GL ( π ) ∼ = ρ ;• Z (SS) = det( ρ ) ω − ;• V G (Π) ∼ = ρ ⊗ F ∧ F ρ ;• V G (cid:18) Ind GL ( Q p ) P − ( Q p ) (cid:16) π · ( ω − ◦ det) ⊗ χ (cid:17) SS (cid:19) ∼ = Ker( ρ ⊗ F ∧ F ρ ։ χ ⊗ ρ ).The properties of V G in §2.1.1 (in particular Lemma 2.1.1.5 which can be applied here86hanks to Remark 2.1.1.7) then automatically give the remaining conditions in (90): V G (cid:18) Ind GL ( Q p ) P − ( Q p ) (cid:16) π · ( ω − ◦ det) ⊗ χ (cid:17)(cid:19) ∼ = ρ ⊗ F ∧ F ρ ∼ = ρ ⊗ det( ρ ) V G (cid:18) SS Ind GL ( Q p ) P ′− ( Q p ) (cid:16) χ ω − ⊗ π (cid:17) (cid:19) ∼ = ( ρ ⊗ F ∧ F ρ ) / ( ρ ⊗ F ∧ F ρ ) V G (cid:18) Ind GL ( Q p ) P ′− ( Q p ) (cid:16) χ ω − ⊗ π (cid:17)(cid:19) ∼ = ρ ⊗ χ V G (SS) ∼ = ( ρ ⊗ ⊗ χ ) ⊕ det( ρ ) χ . The case e P ρ = M P , i.e. ρ ∼ = ρ χ ! , is analogous and easier since Π is then semisim-ple. Example 4
We consider GL ( Q p ) and e P ρ = P ρ = P , where M P = diag(GL , GL , GL ), that iswe have a good conjugate ρ ∼ = ρ ∗ ∗ χ ∗ χ , where ρ : Gal( Q p / Q p ) → GL ( F ) is any absolutely irreducible representation and χ i two smooth characters Q × p → F × (via class field theory) of ratio = 1 , ω ± . If 1 ≤ i ≤ P ij =1 n j = 4 with 1 ≤ n j ≤
4, we write P n ,...,n i for the standard parabolicsubgroup of GL of Levi diag(GL n , . . . , GL n i ) (so P , , = P , P , , , = B , etc.). Asin Example 3 above, we let π be the supersingular representation of GL ( Q p ) over F corresponding to ρ by the mod p local Langlands correspondence for GL ( Q p )(so Z ( π ) = det( ρ ) · ω − and V GL ( π ) ∼ = ρ ). We define the following parabolicinductions: PI π ,χ ,χ = Ind GL ( Q p ) P − , , ( Q p ) (cid:16) π · ( ω − ◦ det) ⊗ χ ω − ⊗ χ (cid:17) PI π ,χ ,χ = Ind GL ( Q p ) P − , , ( Q p ) (cid:16) π · ( ω − ◦ det) ⊗ χ ω − ⊗ χ (cid:17) PI χ ,π ,χ = Ind GL ( Q p ) P − , , ( Q p ) (cid:16) χ ω − ⊗ π · ( ω − ◦ det) ⊗ χ (cid:17) PI χ ,π ,χ = Ind GL ( Q p ) P − , , ( Q p ) (cid:16) χ ω − ⊗ π · ( ω − ◦ det) ⊗ χ (cid:17) PI χ ,χ ,π = Ind GL ( Q p ) P − , , ( Q p ) (cid:16) χ ω − ⊗ χ ω − ⊗ π (cid:17) PI χ ,χ ,π = Ind GL ( Q p ) P − , , ( Q p ) (cid:16) χ ω − ⊗ χ ω − ⊗ π (cid:17) and also, for ss , ss two (not necessarily distinct) supersingular representations of87L ( Q p ) over F : PI ss ,χ = Ind GL ( Q p ) P − , ( Q p ) (cid:16) ss · ( ω − ◦ det) ⊗ χ (cid:17) PI ss ,χ = Ind GL ( Q p ) P − , ( Q p ) (cid:16) ss · ( ω − ◦ det) ⊗ χ (cid:17) PI χ , ss = Ind GL ( Q p ) P − , ( Q p ) (cid:16) χ ω − ⊗ ss (cid:17) PI χ , ss = Ind GL ( Q p ) P − , ( Q p ) (cid:16) χ ω − ⊗ ss (cid:17) . We then let SS , SS , SS , SS be 4 distinct supersingular representations of GL ( Q p )over F . If Π is compatible with ρ , then it has the following form:PI π ,χ ,χ PI ss ,χ PI χ ,π ,χ PI π ,χ ,χ SS SS PI χ , ss PI ss ,χ SS SS PI χ ,χ ,π PI χ ,π ,χ PI χ , ss PI χ ,χ ,π tttttt ❏❏❏❏❏❏ tttttt ❏❏❏❏❏❏❏ ttttttt ttttttt ❏❏❏❏❏❏❏ ttttttt❏❏❏❏❏❏❏ ttttttt ❏❏❏❏❏❏❏ ttttttt ❏❏❏❏❏❏❏ttttttt ttttttt ttttttt❏❏❏❏❏❏❏ ttttttt ttttttt ❏❏❏❏❏❏❏ ❏❏❏❏❏❏tttttt tttttt where we havePI π ,χ ,χ PI ss ,χ PI χ ,π ,χ ∼ = Ind GL ( Q p ) P − , ( Q p ) (Π · ( ω − ◦ det) ⊗ χ )PI χ ,π ,χ PI χ , ss PI χ ,χ ,π ∼ = Ind GL ( Q p ) P − , ( Q p ) ( χ ω − ⊗ Π ) (94)for Π ∼ = Ind GL ( Q p ) P − , ( Q p ) (cid:16) π · ( ω − ◦ det) ⊗ χ (cid:17) ss Ind GL ( Q p ) P − , ( Q p ) (cid:16) χ ω − ⊗ π (cid:17) , andalsoPI π ,χ ,χ PI π ,χ ,χ ∼ =Ind GL ( Q p ) P − , ( Q p ) (cid:18) π · ( ω − ◦ det) ⊗ (cid:16) Ind GL ( Q p ) B − ( Q p ) ( χ ω − ⊗ χ ) Ind GL ( Q p ) B − ( Q p ) ( χ ω − ⊗ χ ) (cid:17)(cid:19) (95)and an analogous isomorphism for PI χ ,χ ,π PI χ ,χ ,π . It actually easily followsfrom [Hau18, Thm.1.4(i)] (together with [Eme10a, Cor.4.3.5]) that the isomorphism(95) and the analogous isomorphism with PI χ ,χ ,π PI χ ,χ ,π are automatic . It alsofollows from [Hau18, Thm.1.2(ii)] and [Hau18, Thm.1.2(ii)] that we automaticallyhave isomorphismsPI π ,χ ,χ PI ss ,χ ∼ = Ind GL ( Q p ) P − , ( Q p ) (cid:18) Ind GL ( Q p ) P − , ( Q p ) (cid:16) π · ( ω − ◦ det) ⊗ χ (cid:17) ss (cid:19) PI ss ,χ PI χ ,π ,χ ∼ = Ind GL ( Q p ) P − , ( Q p ) (cid:18) ss Ind GL ( Q p ) P − , ( Q p ) (cid:16) χ ω − ⊗ π (cid:17) (cid:19) χ ,π ,χ PI χ , ss PI χ ,χ ,π . It is likely thatthe full isomorphisms (94) are in fact also automatic.We must have moreover Z (ss ) = det( ρ ) χ ω − , Z (ss ) = det( ρ ) χ ω − , Z (SS i ) =det( ρ ) ω − for i ∈ { , , , } and V GL (ss ) ∼ = ( ρ ⊗ ⊗ χ ) ⊕ det( ρ ) χ V GL (ss ) ∼ = ( ρ ⊗ ⊗ χ ) ⊕ det( ρ ) χ V GL (SS ) ∼ = (cid:16) ρ ⊗ ⊗ det( ρ ) χ χ (cid:17) ⊕ ⊕ (cid:16) det( ρ ) χ χ (cid:17) ⊕ V GL (SS ) ∼ = (cid:16) ρ ⊗ det( ρ ) χ χ (cid:17) ⊕ ⊕ (cid:16) ρ ⊗ ⊗ χ χ (cid:17) V GL (SS ) ∼ = (cid:16) ρ ⊗ det( ρ ) χ χ (cid:17) ⊕ ⊕ (cid:16) ρ ⊗ ⊗ χ χ (cid:17) V GL (SS ) ∼ = (cid:16) ρ ⊗ ⊗ χ χ (cid:17) ⊕ ⊕ (cid:16) det( ρ ) χ χ (cid:17) ⊕ . The reader can work out all the other conditions of Definition 2.4.2.7 (applying V G tosubquotients of Π). Note that by Proposition 2.4.2.11 the GL ( Q p )-representation Π is compatible with the subrepresentation (cid:16) ρ ∗ χ (cid:17) of ρ (see the last part in Example 2). Example 5
We stay with GL ( Q p ) but where P ρ = P with M P = diag(GL , GL , GL ) and agood conjugate of the form ρ ∼ = χ ∗ ∗ ρ
00 0 χ , where the ∗ are nonzero, ρ : Gal( Q p / Q p ) → GL ( F ) is any absolutely irreduciblerepresentation and χ i are two smooth characters Q × p → F × (via class field theory) ofratio = 1 , ω ± . One has (see (71)) W ρ = { Id , s e − e s e − e } = the set of permutationsof the last two blocks GL and GL . Using the notation and conventions of the89revious case, we can check that any Π compatible with ρ has the following form:PI χ ,π ,χ PI ss ,χ PI π ,χ ,χ PI χ , ss SS SS PI π ,χ ,χ PI χ ,χ ,π SS SS PI ss ,χ PI χ ,χ ,π PI χ , ss PI χ ,π ,χ ❥❥❥❥❥❥❥❥ ❚❚❚❚❚❚❚❚❚❚ ❥❥❥❥❥❥❥❥ ❚❚❚❚❚❚❚❚❚❚❥❥❥❥❥❥❥❥❥❥ ❚❚❚❚❚❚❚❚❚❚ ❚❚❚❚❚❚❚ ❥❥❥❥❥❥❥❥❥❥❚❚❚❚❚❚❚❚❚❚❚❚ ❚❚❚❚❚❚❚❚❚❚❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ❚❚❚❚❚❚❚ ❥❥❥❥❥❥❥❥❥❥❥❥ ❥❥❥❥❥❥❥❥❥❥❚❚❚❚❚❚❚❚❚❚ ❚❚❚❚❚❚❚❚❚❚❥❥❥❥❥❥❥❥ ❥❥❥❥❥❥❥❥ (recall the socle is the first layer on the left), where condition (i) in Definition 2.4.1.5yields, when applied to a suitable C Q with M Q = diag(GL , GL ):PI χ ,π ,χ PI ss ,χ PI π ,χ ,χ ∼ = Ind GL ( Q p ) P − , ( Q p ) (Π · ( ω − ◦ det) ⊗ χ )PI χ ,χ ,π PI χ , ss PI χ ,π ,χ ∼ = Ind GL ( Q p ) P − , ( Q p ) ( χ ω − ⊗ Π ) (96)for Π ∼ = Ind GL ( Q p ) P − , ( Q p ) (cid:16) χ ω − ⊗ π (cid:17) ss Ind GL ( Q p ) P − , ( Q p ) (cid:16) π · ( ω − ◦ det) ⊗ χ (cid:17) , and yields, when applied to a suitable C Q with M Q = diag(GL , GL ) (that is, s e − e s e − e P ρ ⊆ Q , note that here P ρ Q , see Remark 2.4.1.6(vii)):PI χ ,χ ,π PI χ ,χ ,π ∼ =Ind GL ( Q p ) P − , ( Q p ) (cid:18)(cid:16) Ind GL ( Q p ) B − ( Q p ) ( χ ω − ⊗ χ ω − ) Ind GL ( Q p ) B − ( Q p ) ( χ ω − ⊗ χ ω − ) (cid:17) ⊗ π (cid:19) (97)and an analogous isomorphism for PI π ,χ ,χ PI π ,χ ,χ . As in Example 4, it followsfrom [Hau18, Thm.1.4(i)] that (97) and the analogous isomorphism are automatic,and from [Hau18, Thm.1.2(ii)], [Hau18, Thm.1.2(ii)] that isomorphisms as in (96) butfor every “half” only of the extensions on the left are also automatic.One can again work out all the conditions of Definition 2.4.2.7 (conditions on Z (ss i ), Z (SS i ) and on V GL (ss i ), V GL (SS i ) are the same as in Example 4).90 xample 6 We consider GL ( Q p ) and e P ρ = P ρ = B , i.e. ρ ∼ = χ ∗ ∗ χ ∗ χ , where χ i are three smooth characters Q × p → F × (via class field theory) of ratio =1 , ω ± . We let ss , ss , ss be 3 (not necessarily distinct) supersingular representationsof GL ( Q p ) over F and SS i , i ∈ { , . . . , } be 7 distinct supersingular representationsof GL ( Q p ) over F . We use without comment notation for GL ( Q p ) analogous tothe ones in Example 2, Example 4 and Example 5 to denote principal series andparabolic inductions. If Π is compatible with ρ , then it has the following form:PS χ ,χ ,χ PI ss ,χ PS χ ,χ ,χ PI χ , ss SS SS PI χ , ss PS χ ,χ ,χ SS SS SS PS χ ,χ ,χ PI ss ,χ SS SS PI ss ,χ PS χ ,χ ,χ PI χ , ss PS χ ,χ ,χ ✇✇✇✇✇✇ ●●●●●● ✇✇✇✇✇✇ ●●●●●● ✇✇✇✇✇✇ ✇✇✇✇✇✇ ●●●●●● ✇✇✇✇✇✇●●●●●● ✇✇✇✇✇✇ ●●●●●● ✇✇✇✇✇✇ ●●●●●●✇✇✇✇✇✇ ✇✇✇✇✇✇ ✇✇✇✇✇✇●●●●●● ✇✇✇✇✇✇ ✇✇✇✇✇✇ ●●●●●● ●●●●●●✇✇✇✇✇✇ ✇✇✇✇✇✇●●●●●● ✇✇✇✇✇✇ ●●●●●● ✇✇✇✇✇✇ ●●●●●● ●●●●●● ●●●●●●✇✇✇✇✇✇ ✇✇✇✇✇✇ where we havePS χ ,χ ,χ PI ss ,χ PS χ ,χ ,χ ∼ = Ind GL ( Q p ) P − , ( Q p ) (cid:16) Π · ( ω − ◦ det) ⊗ χ (cid:17) PS χ ,χ ,χ PI χ , ss PS χ ,χ ,χ ∼ = Ind GL ( Q p ) P − , ( Q p ) (cid:16) χ ω − ⊗ Π (cid:17) PS χ ,χ ,χ PI ss ,χ PS χ ,χ ,χ ∼ = Ind GL ( Q p ) P − , ( Q p ) (cid:16) Π · ( ω − ◦ det) ⊗ χ (cid:17) PS χ ,χ ,χ PI χ , ss PS χ ,χ ,χ ∼ = Ind GL ( Q p ) P − , ( Q p ) (cid:16) χ ω − ⊗ Π (cid:17) (98)for Π ∼ = Ind GL ( Q p ) B − ( Q p ) (cid:16) χ ω − ⊗ χ (cid:17) ss Ind GL ( Q p ) B − ( Q p ) (cid:16) χ ω − ⊗ χ (cid:17) Π ∼ = Ind GL ( Q p ) B − ( Q p ) (cid:16) χ ω − ⊗ χ (cid:17) ss Ind GL ( Q p ) B − ( Q p ) (cid:16) χ ω − ⊗ χ (cid:17) . By a straightforward induction, it follows from [Hau18, Thm.1.3] combined with[Eme10a, Cor.4.3.5] that all isomorphisms (98) are actually true!91e must have moreover Z (ss ) = χ χ ω − , Z (ss ) = χ χ ω − , Z (ss ) = χ χ ω − , Z (SS i ) = det( ρ ) ω − for i ∈ { , . . . , } and, denoting by σ the only nontrivial elementof Gal( Q p / Q p ): V GL (ss ) ∼ = χ χ σ ⊕ χ σ χ V GL (ss ) ∼ = χ χ σ ⊕ χ σ χ V GL (ss ) ∼ = χ χ σ ⊕ χ σ χ V GL (SS ) ∼ = (cid:18) χ χ det( ρ ) σ ⊕ ( χ χ ) σ det( ρ ) (cid:19) ⊕ ⊕ (cid:18) χ χ ( χ χ ) σ ⊕ ( χ χ ) σ χ χ (cid:19) V GL (SS i ) ∼ = analogous for i ∈ { , , , , } (left to reader) V GL (SS ) ∼ = (cid:16) det( ρ ) det( ρ ) σ (cid:17) ⊕ ⊕ (cid:18) χ χ ( χ χ ) σ ⊕ ( χ χ ) σ χ χ (cid:19) ⊕ (cid:18) χ χ ( χ χ ) σ ⊕ ( χ χ ) σ χ χ (cid:19) ⊕ (cid:18) χ χ ( χ χ ) σ ⊕ ( χ χ ) σ χ χ (cid:19) (all obviously representations of Gal( Q p / Q p ) over F ). The reader can then work outthe conditions in (90) involving the various subquotients of Π. Finally, by Proposition2.4.2.11 the GL ( Q p )-representation Π (resp. Π ) is compatible with the subrepre-sentation (cid:16) χ ∗ χ (cid:17) (resp. with the quotient (cid:16) χ ∗ χ (cid:17) ) of ρ (see Example 1). Example 7
We end up with GL ( Q p ) and e P ρ = P ρ = B , i.e. ρ ∼ = χ ∗ ∗ ∗ χ ∗ ∗ χ ∗ χ , where χ i are four smooth characters Q × p → F × of ratio = 1 , ω ± . The structure of a Πcompatible with ρ is given in the next 3D diagram. Just like the previous 2D diagramslook like stacked squares, this 3D diagram looks like stacked cubes: there are 8 cubes,one being entirely “behind”. As before, each vertex is an irreducible constituentwith PS (in green) meaning principal series, SS (in red) meaning supersingular andPI (resp. PI ) (in blue) meaning parabolic induction from the standard parabolicsubgroup of Levi GL × GL (resp. of Levi GL × GL ). The socle is the principalseries at the very bottom and the cosocle is the principal series at the very top. Likepreviously, each edge is a nonsplit extension between two irreducible constituents, thedashed edges being those which are “behind” in the 3D picture. Near each vertex wewrite the value of V GL applied to the corresponding irreducible constituent.The interested reader can then check all the other conditions and compatibilities inDefinition 2.4.1.5 and Definition 2.4.2.7, for instance the two left faces on the bottomcorrespond to the parabolic induction PI of Example 2 tensored by the character χ .92 SPS PS PSPI1 PI2PSSS PSPSPS PS PI1PI2 PSPS SS SSSS PSPSSS PSPS PI1PI2 SS PSPSPS PS PI2PI1 PS PSPSPS PS χ χ χ χ χ χ χ χ χ χ χ χ [( χ χ χ ) ] ⊕ [ χ ( χ χ χ )] ⊕ χ χ χ [( χ χ ) ( χ χ )] ⊕ χ χ χ χ χ χ χ χ χ χ χ χ [( χ χ χ ) ] ⊕ [ χ ( χ χ χ )] ⊕ χ χ χ χ χ χ [( χ χ ) ( χ χ )] ⊕ [( χ χ ) ( χ χ )] ⊕ [( χ χ ) ( χ χ )] ⊕ χ χ χ χ χ χ [( χ χ ) ( χ χ )] ⊕ χ χ χ χ χ χ [( χ χ χ ) ] ⊕ [ χ ( χ χ χ )] ⊕ [( χ χ ) ( χ χ )] ⊕ χ χ χ χ χ χ χ χ χ χ χ χ [ χ ( χ χ χ )] ⊕ [( χ χ χ ) ] ⊕ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ .5 Strong local-global compatibility conjecture Back to the setting of §2.1 but assuming that F + v is unramified and that r ˜ v (for ˜ v | v )is generic as at the beginning of §2.4.2, we conjecture that the G ( F ˜ v )-representationHom U v ( σ v , S ( V v , F )[ m Σ ]) is a direct sum of copies of a G ( F ˜ v )-representation which is(up to twist) compatible with any good conjugate of r ˜ v (Definition 2.4.2.7).We consider exactly the same global setting as in §2.1.2. We fix v | p in F + suchthat F + v is an unramified extension of Q p and consider a continuous representation r : Gal( F /F ) → GL n ( F ) such that(i) r c ∼ = r ∨ ⊗ ω − n (recall r c ( g ) = r ( cgc ) for g ∈ Gal(
F /F ));(ii) r is an absolutely irreducible representation of Gal( F /F );(iii) r ˜ v for ˜ v | v has distinct irreducible constituents and the ratio of any two irre-ducible constituents of dimension 1 is not in { ω, ω − } (note that condition (iii) doesn’t depend on the place ˜ v of F dividing v since r ˜ v c ∼ = r ∨ ˜ v ⊗ ω − n ).The following is the main conjecture of this paper. Conjecture 2.5.1.
Let r : Gal( F /F ) → GL n ( F ) be a continuous homomorphismthat satisfies conditions ( i ) to ( iii ) above and fix a place v of F + which divides p such that F + v is unramified. Assume that there exist compact open subgroups V v ⊆ U v ⊆ H ( A ∞ ,vF + ) with V v normal in U v , a finite-dimensional representation σ v of U v /V v over F and a finite set Σ of finite places of F + as in §2.1.3 suchthat Hom U v ( σ v , S ( V v , F )[ m Σ ]) = 0 , where m Σ is the maximal ideal of T Σ associ-ated to r . Let ˜ v | v in F and see Hom U v ( σ v , S ( V v , F )[ m Σ ]) as a representation of H ( F + v ) ∼ = GL n ( F ˜ v ) = G ( F ˜ v ) via ι ˜ v ( cf. §2.1.2 ) . Then there is an integer d ∈ Z > depending only on v , U v , V v , σ v and r and an admissible smooth representation Π ˜ v of G ( F ˜ v ) over F ( depending a priori on ˜ v , U v , V v , σ v and r ) such that Hom U v ( σ v , S ( V v , F )[ m Σ ]) ∼ = (cid:16) Π ˜ v ⊗ ( ω n − ◦ det) (cid:17) ⊕ d , where Π ˜ v is compatible with one ( equivalently any by Proposition 2.4.2.9 ) good con-jugate of r ˜ v in the sense of Definition 2.3.2.3. Remark 2.5.2. (i) Conjecture 2.5.1 implies in particular that the G ( F ˜ v )-represen-tation Hom U v ( σ v , S ( V v , F )[ m Σ ]) is of finite length with all constituents of multiplicity d (under assumptions (i) to (iii) on r ), which is already far from being known ingeneral. See however §3.4 below for nontrivial evidence in the case of GL . It alsoimplies that Hom U v ( σ v , S ( V v , F )[ m Σ ]) has a central character, but this is known (at94east under some extra assumptions), see Lemma 2.1.3.3.(ii) When F + v is unramified and r ˜ v is as in (iii) above, Conjecture 2.5.1 of courseimplies (and is in fact much stronger than) Conjecture 2.1.3.1.(iii) Assuming that p is unramified in F + and that r ˜ w is generic as in (iii) above forall w | p , an even stronger conjecture would be as follows. Conjecture 2.5.3.
For U p ⊆ H ( A ∞ ,pF + ) such that S ( U p , F )[ m Σ ] = 0 ( where Σ containsthe set of places of F + that split in F and divide pN , or at which U p is not unramified,or at which r ramifies, and where S ( U p , F )[ m Σ ] is defined as in §2.1.2 replacing U v by U p ) and for any ˜ w | w in F with w | p , there is an integer d ∈ Z > depending only on p , U p and r and admissible smooth representations Π ˜ w of G ( F ˜ w ) over F , where Π ˜ w iscompatible with one ( equivalently any ) good conjugate of r ˜ w such that S ( U p , F )[ m Σ ] ∼ = (cid:18) O w | p (cid:16) Π ˜ w ⊗ ( ω n − ◦ det) (cid:17)(cid:19) ⊕ d . As in §2.1.3, we prove that Conjecture 2.5.1 holds for ˜ v if and only if it holds for˜ v c (we do not need here extra assumptions). We start with two formal lemmas. Weuse the previous notation and denote by w ∈ W the unique element with maximallength. Lemma 2.5.4.
Let ρ : Gal( Q p /K ) → e P ρ ( F ) ⊆ P ρ ( F ) ⊆ G ( F ) be a good conjugateas in §2.3.2. Then the continuous homomorphism Gal( Q p /K ) → G ( F ) = GL n ( F ) defined by g w τ (cid:16) ρ ( g ) (cid:17) − w (99) is a good conjugate of the dual of the representation associated to ρ .Proof. Denote by w P ρ the standard parabolic subgroup of G with set of simple roots − w ( S ( P ρ )) ⊆ S . Using that W ( w P ρ ) = w W ( P ρ ) w , one checks that − w ( X ρ ) ⊆ R + is a closed subset relative to w P ρ (Definition 2.3.1.1) and thus corresponds to aZariski-closed algebraic subgroup w e P ρ def = w M P ρ w N − w ( X ρ ) of w P ρ (Lemma 2.3.1.4).Denote by w τ ( ρ ) − w the homomorphism (99), its associated representation is thedual of the representation associated to ρ . Moreover one has e P w τ ( ρ ) − w = w e P ρ and X hw τ ( ρ ) − w h − = − w ( X w τ ( h ) − w ρw τ ( h ) w ) for any h ∈ w P ρ ( F ) (note that w τ ( h ) − w ∈ P ρ ( F )). The result follows from Definition 2.3.2.3.As in §2.1.3, if π is a smooth representation of G ( K ) over F we denote by π ⋆ thesmooth representation of G ( K ) with the same underlying vector space as π but where g ∈ G ( K ) = GL n ( K ) acts by τ ( g ) − . 95 emma 2.5.5. Let ρ : Gal( Q p /K ) → G ( F ) be a continuous homomorphism suchthat ρ ss has distinct irreducible constituents and the ratio of any two irreducible con-stituents of dimension is not in { ω, ω − } . Let Π be a smooth representation of G ( K ) over F . Then Π is compatible with one ( equivalently any by Proposition 2.4.2.9 ) goodconjugate of ρ if and only if Π ⋆ is compatible with one ( ibid. ) good conjugate of ρ ∨ ⊗ ω n − ( denoting by ρ ∨ the dual of the representation associated to ρ ) .Proof. We use the notation in the proof of Lemma 2.5.4. Assuming ρ is a good conju-gate, it is enough to show that if Π is compatible with ρ , then Π ⋆ is compatible with w τ ( ρ ) − w ⊗ ω n − . If R is a (finite-dimensional) algebraic representation of G Gal( K/ Q p ) over F , let R ⋆ be the algebraic representation where g ∈ G Gal( K/ Q p ) acts by τ ( g ) − (in-verse transpose on each factor). Then one checks that L ⊗ ⋆ ∼ = L ⊗ ⊗ (det − ( n − ) ⊗ [ K : Q p ] .Let Φ be a bijection as in Definition 2.4.2.7 and define Φ ⋆ from the set of subquo-tients Π ′ ⋆ of Π ⋆ (where Π ′ is a subquotient of Π) to the set of good subquotients of L ⊗ | ( w e P ρ ) Gal( K/ Q p ) as follows: Φ ⋆ (Π ′ ⋆ ) is the algebraic representation of ( w e P ρ ) Gal( K/ Q p ) given by Φ ⋆ (Π ′ ⋆ )( g ) def = Φ(Π ′ )( w τ ( g ) − w )det( g ) n − for g ∈ ( w e P ρ ) Gal( K/ Q p ) (with obvi-ous notation). We leave to the reader the tedious but formal task to check that Φ ⋆ sat-isfies all conditions of Definitions 2.4.1.5 and 2.4.2.7 with w e P ρ and w τ ( ρ ) − w ⊗ ω n − instead of e P ρ and ρ using (for Q any standard parabolic subgroup of G ): (cid:18) Ind G ( K ) Q − ( K ) ( π ⊗ · · · ⊗ π d ) (cid:19) ⋆ ∼ = Ind G ( K )( w Q ) − ( K ) ( π d⋆ ⊗ · · · ⊗ π ⋆ )and Lemma 2.1.3.4. Proposition 2.5.6.
Conjecture 2.5.1 holds for ˜ v if and only if it holds for ˜ v c .Proof. This follows from Lemma 2.5.5 together with r ˜ v c ∼ = r ∨ ˜ v ⊗ ω − n , Remark2.4.2.8(iv) and an easy computation.There is an obvious analogous statement with Conjecture 2.5.3 instead of Conjec-ture 2.5.1. Remark 2.5.7.
Let π be an admissible smooth representation of G ( K ) over F witha central character. In [Koh17, Cor.3.15], Kohlhaase associates higher smooth duals S i ( π ), i ≥ π which are also admissible (smooth) representations of G ( K ) over F with a central character. In view of the results when n = 2 (see condition (iii) in§3.3.5 below and [HW, Thm.8.2]), it is natural to expect that, when K = F ˜ v and Π ˜ v is as in Conjecture 2.5.1, we have S i (Π ˜ v ) = 0 if and only if i = i def = [ K : Q p ] n ( n − and that S i (Π ˜ v ) is compatible with (a good conjugate of) r ∨ ˜ v ⊗ ω n − (when n = 2,this is indeed consistent with loc.cit. since r ∨ ˜ v ∼ = r ˜ v ⊗ det( r ˜ v ) − ). It is also natural toask if we have S i (Π ˜ v ) ∼ = Π ⋆ ˜ v (see Lemma 2.5.5).96rom the results of [BH15, §4.4] and [Enn], we can at least give some very weakevidence for Conjecture 2.5.1, more precisely for the stronger Conjecture 2.5.3 inRemark 2.5.2(iii), when p is totally split in F + and r ˜ w is upper-triangular sufficientlygeneric for all w | p in F + .If Π is an admissible smooth representation of G ( K ) over F , we denote by Π ord ⊆ Πthe maximal G ( K )-subrepresentation such that all its irreducible constituents are iso-morphic to irreducible subquotients of principal series of G ( K ) over F . The followinglemma is not difficult using Proposition 2.2.3.3, [BH15, Thm.2.2.4] and the results of[BH15, §3.3], [BH15, §3.4] (the proof is left to the reader). Lemma 2.5.8.
Assume K = Q p and let ρ : Gal( Q p / Q p ) → B ( F ) ⊆ G ( F ) be generic ( as at the beginning of §2.4.2 ) and a good conjugate ( as in Definition 2.3.2.3 ) . Let Π be compatible with ρ ( as in Definition 2.4.2.7 ) . Then Π ord ∼ = Π( ρ ) ord , where Π( ρ ) ord is the representation of G ( Q p ) over F defined in [BH15, §3.4] . Note that one can explicitly determine V G (Π( ρ ) ord ) inside L ⊗ ( ρ ), see [Bre15, §9].We let S p be the set of places of F + dividing p . Recall that an injection betweentwo representations of a group is called essential if it induces an isomorphism on therespective socles. Theorem 2.5.9 ([Enn]) . Assume that
F/F + is unramified at finite places, that H is defined over O F + with H × O F + F + quasi-split at finite places of F + , and that p is totally split in F . Assume that r : Gal( F /F ) → GL n ( F ) satisfies assumptionsA1 to A6 of [Enn, §3.1] , let v be a finite place of F + as in [Enn, Lemma 3.1.2] and Σ def = S p ∪ { v } . Choose f v | v in F and let U p = Q w ∤ p U w ⊆ H ( A ∞ ,pF + ) such that U w = H ( O F + w ) if w splits in F , U w is maximal hyperspecial in H ( F + w ) if w is inertin F and ι e v ( U v ) is the Iwahori subgroup of GL n ( F e v ) . Then for any ˜ w | w in F andany good conjugates r ˜ w ( where w ∈ S p ) , we have an essential injection of admissiblesmooth representations of Q w | p H ( F + w ) over F : (cid:18) O w | p (cid:16) Π( r ˜ w ) ord ⊗ ω n − ◦ det (cid:17)(cid:19) ⊕ n ! ֒ → S ( U p , F )[ m Σ ] ord , where S ( U p , F )[ m Σ ] ord ⊆ S ( U p , F )[ m Σ ] is defined as Π ord ⊆ Π above replacing G ( K ) by Q w | p H ( F + w ) .Proof. This follows from [Enn, Thm.3.3.3] (which itself improves [BH15, Thm.4.4.7])and its proof (see just before [Enn, Lemma 3.2.1] for the n !). Remark 2.5.10.
The cokernel of the injection in Theorem 2.5.9 is an admissiblesmooth representation of Q w | p H ( F + w ) over F , and its Q w | p H ( F + w )-socle is by con-struction a direct sum of finitely many irreducible subquotients of principal series. If97e could prove that all these irreducible subquotients are irreducible principal serieswhich do not appear in the Q w | p H ( F + w )-socle of N w | p (Π( r ˜ w ) ord ⊗ ω n − ◦ det), then itwould follow from the mod p version of [Hau19, Cor.1.4] that the essential injectionin Theorem 2.5.9 is an isomorphism. 98 The case of GL ( Q p f ) We give evidence for Conjecture 2.1.3.1 and Conjecture 2.5.1 when F + v is unramifiedand G = GL . We now assume K = Q p f and n = 2 till the end. We fix an embedding σ : F p f = F q ֒ → F and let σ i def = σ ◦ ϕ i for ϕ the arithmetic Frobenius and i ≥ ( ϕ, O × K ) -modules and ( ϕ, Γ) -modules We associate étale ( ϕ, O × K )-modules to certain admissible smooth representations ofGL ( K ) over F and relate them to the étale ( ϕ, Γ)-modules of §2.1.1.We assume that p >
2. We let I def = (cid:16) O × K O K p O K O × K (cid:17) the Iwahori subgroup of GL ( O K ), K def = (cid:16) p O K p O K p O K p O K (cid:17) the pro- p radical of GL ( O K ), I def = (cid:16) p O K O K p O K p O K (cid:17) the pro- p radical of I , N def = (cid:16) O K (cid:17) ⊆ I , N − def = (cid:16) p O K (cid:17) ⊆ I and T def = (cid:16) p O K
00 1+ p O K (cid:17) ⊆ I .We denote by Z the center of I . If C is a pro- p group then F J C K denotes its Iwasawaalgebra over F , which is a local ring, and m C the maximal ideal of F J C K . If R (resp. M ) is a filtered ring (resp. filtered module) in the sense of [LvO96, §I.2], we denoteby F n R (resp. F n M ) for n ∈ Z its ascending filtration and gr( R ) def = ⊕ n ∈ Z F n R/F n − R (resp. with M ) the associated graded ring (resp. module). When R = F J C K , we set F n R def = m − nR if n ≤ F n R def = R if n ≥
0. If M is an R -module, the filtration F n M = m − nR M if n ≤ F n M = M if n ≥ m R -adic filtration on M . For any ring R and any R -module M , we set E iR ( M ) def = Ext iR ( M, R ) for i ≥ A We describe some properties of a complete noetherian ring A which will be a coefficientring for some multivariable ( ψ, O × K )-modules and ( ϕ, O × K )-modules.Let v N be the m N -adic valuation on the ring F J N K so that F n F J N K = { x ∈ F J N K , v N ( x ) ≥ − n } for n ∈ Z . We use the same notation to denote the uniqueextension of v N to a valuation of the fraction field of F J N K . For i ∈ { , . . . , f − } let Y i def = X a ∈ F × q σ ( a ) − p i a ]0 1 ! ∈ m N \ m N (100)and denote by y i def = gr( Y i ) the image of Y i in m N / m N ⊆ gr( F J N K ). Then F J N K is isomorphic to the power series ring F J Y , . . . , Y f − K and gr( F J N K ) to the poly-nomial algebra F [ y , . . . , y f − ]. Let S be the multiplicative subset of F J N K whoseelements are the ( Y · · · Y f − ) n for n ≥ F J N K S the corresponding localization and99 n F J N K S def = { x ∈ F J N K S , v N ( x ) ≥ − n } . We define the ring A as the completion ofthe filtered ring F J N K S ([LvO96, §I.3.4]). Note that v N extends to A , which is thusa complete filtered ring. As A is complete, an element x ∈ A is invertible in A if andonly if gr( x ) is invertible in gr( A ) (as is easily checked, here gr( x ) is the “principalpart” of x as in [LvO96, §I.4.2]).Let M be a filtered F J N K -module. The tensor product A ⊗ F J N K M is then afiltered A -module for the tensor product filtration as defined in [LvO96, p.57]. Welet A b ⊗ F J N K M be its completion. This filtered A -module can also be described asthe completion of the localization M S endowed with the tensor product filtrationassociated to the isomorphism M S ∼ = F J N K S ⊗ F J N K M . Lemma 3.1.1.1.
We have an isomorphism gr( A b ⊗ F J N K M ) ∼ = gr( M S ) ∼ = gr( M )[( y · · · y f − ) − ] . (101) Proof. As A b ⊗ F J N K M is the completion of M S , it is sufficient to prove that gr( M S ) ∼ =gr( M ) T , where T = { ( y . . . y f − ) k , k ≥ } . Note that we have an isomorphism of F J N K -algebras F J N K S ∼ = F J N K [ T ] / (( Y · · · Y f − ) T − F J N K [ T ] with the filtration F n ( F J N K [ T ]) = X k ≥ m kf − nN T k (with the convention m iN = F J N K for i ≤ F J N K S is the quotientfiltration via F J N K [ T ] ։ F J N K S . Therefore the filtration on M S is the quotientfiltration of the tensor product filtration on M [ T ] def = F J N K [ T ] ⊗ F J N K M .As the filtered F J N K -module F J N K [ T ] is filtered-free by definition (see [LvO96,Def.I.6.1]), it follows from [LvO96, Lemma I.6.14] that gr( M [ T ]) ∼ = gr( M )[ T ] withdeg( T ) = f . We claim that the following sequence of filtered modules is strict exact: M [ T ] ( Y ··· Y f − ) T − −−−−−−−−−→ M [ T ] −→ M S −→ . Namely the exactness of the second arrow follows from the definition of the quotientfiltration. As ( Y · · · Y f − ) T and 1 have degree 0 in F J N K [ T ], the multiplication by( Y · · · Y f − ) T − y · · · y f − ) T − M [ T ]) ∼ =gr( M )[ T ] which is injective. It follows from [LvO96, Thm.I.4.2.4(2)] (applied with L = 0, M = N = M [ T ], f = 0 and g being the multiplication by ( Y · · · Y f − ) T − Y · · · Y f − ) T − M [ T ]) ( y ··· y f − ) T − −−−−−−−−→ gr( M [ T ]) −→ gr( M S ) −→ . (102)Finally, since gr( M [ T ]) ∼ = gr( M )[ T ], we have gr( M S ) ∼ = gr( M )[( y · · · y f − ) − ].100 orollary 3.1.1.2. We have an isomorphism gr( A ) ∼ = F [ y , . . . , y f − , ( y · · · y f − ) − ] .As a consequence the ring A is a regular domain, i.e. a noetherian domain which hasa finite global dimension ([Ser00, §IV.D]) .Proof. The first sentence is a direct consequence of Lemma 3.1.1.1 applied with M = F J N K . This implies that the ring gr( A ) is a noetherian domain. Then thenoetherianity of A follows from [LvO96, Thm.I.5.7] applied to the ideals of A , andthe fact that A is a domain follows easily from gr( x ) gr( y ) = gr( xy ) if x, y ∈ A \{ } (us-ing gr( x ) gr( y ) = 0). As gr( A ) is a regular commutative ring, it follows from [LvO96,Thm.III.2.2.5] that A is an Auslander regular ring (note that A is Zariskian by [LvO96,Prop.II.2.2.1]) and a fortiori has finite global dimension ([LvO96, Def.III.2.1.7]). Remark 3.1.1.3. (i) The ring A can also be defined as the microlocalization of F J N K along the set { ( y · · · y f − ) n , n ≥ } ⊆ gr( F J N K ) (see [LvO96, Cor.IV.1.20]). Thisshows that the ring A does not depend on our choice of elements Y i but rather on theelements y i .(ii) If M is a filtered F J N K -module, the filtration on M S is given explicitly by thefollowing formula: F n ( M S ) = X k ≥ ( Y · · · Y f − ) − k F n − kf ( M ) , n ∈ Z . As ( Y · · · Y f − ) m F n ( M ) ⊆ F n − mf ( M ) for all n ∈ Z and m ∈ N , we have( Y · · · Y f − ) − k F n − kf ( M ) ⊆ ( Y · · · Y f − ) − k − F n − ( k +1) f ( M )so that F n ( M S ) can also be described as the increasing union F n ( M S ) = [ k ≥ ( Y · · · Y f − ) − k F n − kf ( M ) . (iii) The ring A can also be defined as the set of series A = X d ≫−∞ P d ( Y · · · Y f − ) n d , P d ∈ ( Y , . . . , Y f − ) d + fn d , n d ≥ , d + f n d ≥ , equivalently, A is the set of infinite sums of monomials in the Y i with F -coefficientssuch that the total degree of the monomials tends to + ∞ .Let n ≥ N p n ⊆ N be the subgroup of p n -th powers (whichis p n O K under the identification N ∼ = O K ). Let S p n be the set of p n -th powersof S and let A p n be the completion of F J N p n K S pn for the filtration coming from thevaluation v N | F J N pn K = p n v N pn . As the saturation of S p n (see [LvO96, §IV.1]) contains S , we have by [LvO96, Cor.IV.1.20] F J N K S = F J N K S pn ∼ = F J N p n K S pn ⊗ F J N pn K F J N K . (103)101t is easy to check that F J N K is a filtered free F J N p n K -module with respect to the basis( Y i · · · Y i f − f − ) ≤ i j ≤ p n − ≤ j ≤ f − . Hence, by [LvO96, Lemma I.6.15] and (103), we concludethat F J N K S is a filtered free F J N p n K S pn -module with respect to the same basis, andthus by [LvO96, Lemma I.6.13(3)] that A is a filtered free A p n -module with respect tothe same basis again. Moreover, by [LvO96, Lemma I.6.14], we have an isomorphismof graded modules gr( A ) ∼ = gr( A p n ) ⊗ gr( F J N pn K ) gr( F J N K ) . (104)Note that the p n -power Frobenius map x x p n induces an isomorphism of fil-tered rings ( F J N K S , v N ) ∼ −→ ( F J N p n K , v N pn ) and thus, as v N | F J N pn K = p n v N pn , anisomorphism of topological rings ( F J N K S , v N ) ∼ −→ ( F J N p n K , v N | F J N pn K ). It inducesan isomorphism of complete topological rings A ∼ −→ A p n such that the composite map A ∼ −→ A p n ֒ → A is the p n -power Frobenius. This implies that the image of A p n in A is the subring of p n -th powers of A .The group O × K acts on the group N via a · ( b ) = ( ab ) and thus on F J N K ,preserving the valuation v N , and hence the filtration. This induces an action of O × K on the graded ring gr( F J N K ), where it is immediately checked that 1 + p O K actstrivially. Moreover if a ∈ F × q and 0 ≤ i ≤ f −
1, we have [ a ] · y i = σ i ( a ) y i . Lemma 3.1.1.4.
There is a unique continuous action of O × K on the ring A extendingthe action of O × K on F J N K .Proof. As O × K acts by ring endomorphisms on F J N K and as F J N K S is dense in A ,the uniqueness is clear.For the existence, let a ∈ O × K and consider the composition F J N K a −→ F J N K ⊆ A which extends to a ring homomorphism F J N K S → A since the elements of a ( S )are invertible in A (because they are invertible in gr( A ) as gr( a ( S )) = gr( S )). Theprecomposition of the valuation v N on A with this map is a valuation on F J N K S whichcoincides with v N on F J N K since the multiplication by a preserves the valuation on F J N K . Therefore the map F J N K S → A is isometric and extends to a filtered ringhomomorphism A → A ([LvO96, Thm.I.3.4.5]). This defines an action of O × K on A . We recall that ξ is the cocharacter x ( x
00 1 ) of GL . The conjugation by thematrix ξ ( p ) in GL ( K ) induces a group endomorphism of N and a continuous endo-morphism φ of F J N K . We have φ ( Y i ) = Y pi − for 1 ≤ i ≤ f − φ ( Y ) = Y pf − .This implies that φ is the composite of the (relative) Frobenius endomorphism witha permutation of the variables Y i . It follows that φ extends to a continuous injectiveendomorphism of the ring A with image A p . More generally, for n ≥
0, the subring A p n is the image of φ n . 102 roposition 3.1.1.5. Let a ⊆ A be an ideal of A which is O × K -stable. Then a iscontrolled by A p , which means a = A ( a ∩ A p ) . Proof.
The proof follows closely the strategy of [AW09].We note that the pair (
A, A p ) is a Frobenius pair in the sense of [AW09, Def.2.1](to see this use [AWZ08, Prop.6.6] applied to G = N together with [AW09, Lemma2.2.(a)] and Remark 3.1.1.3(i)). We endow A p with the filtration F n A p def = A p ∩ F n A induced by the filtration of A .Let F def = a /A ( a ∩ A p ). Endow A ( a ∩ A p ) and a with the filtration induced by A , and F with the quotient filtration. Then by [LvO96, Rk.I.5.2(2)] and [LvO96,Cor.I.5.5(1)] all these filtrations are good in the sense of [LvO96, Def.I.5.1]. Moreover a and A ( a ∩ A p ) are complete filtered A -modules by [LvO96, Cor.I.6.3(2))] and thusso is F by [LvO96, Prop.I.3.15].We want to prove that F = 0. Assume for a contradiction that F = 0, orequivalently gr( F ) = 0 by [LvO96, Prop.I.4.2(1)].Let Γ def = 1 + p O K (this not the Γ of the ( ϕ, Γ)-modules!). This is a uniform pro- p -group. Note that the action of Γ on N is uniform in the sense of [AW09, §4.1].In the notation of [AW09, §4.2], we have L N = O K , g = F q and the action of F q on L N /pL N is given by the multiplication in F q .Let P be a (homogeneous) prime ideal in the support of the gr( A )-module gr( F )(which exists since gr( F ) = 0).Let x ∈ F × q and γ x def = exp( p [ x ]) ∈ F J N K ⊆ A . It follows from [AW09, Prop.4.4]and [AW09, Prop.3.2(a)] that the familya( x ) def = ( γ x , γ px , γ p x , . . . )is a source of derivations of ( A, A p ) in the sense of [AW09, Def.3.2]. Let T P ⊆ gr( A ) be the set of homogeneous elements of gr( A ) which are not in P and let T ( p ) P def = T P ∩ gr( A p ). It follows again from [AW09, Prop.3.2(a)] that a( x ) induces on( Q T P ( A ) , Q T ( p ) P ( A p )) a source of derivations a T P ( x ), where Q T P ( A ) (resp. Q T ( p ) P ( A p ))is the microlocalization of A (resp. A p ) with respect to T P (resp. T ( p ) P ). Let S def = { a( x ) , x ∈ F × q } and S P def = { a T P ( x ) , x ∈ F × q } .As a is Γ-invariant, a it is also S -invariant, i.e. for all x ∈ F × q and r ≥
0, we have γ p r x a ⊆ a . Then a P def = Q T P ( a ) ∼ = Q T P ( A ) ⊗ A a ([LvO96, Cor.IV.1.18(2)], though hereeverything is simpler as all rings are commutative) is an ideal of Q T P ( A ) which is S P -invariant. 103et P def = P ∩ gr( F J N K ) (inside gr( A )). We prove that P contains L N /pL N ,where the latter is seen in gr − ( F J N K ) (recall L N ∼ = N ). Assume this is not true.Let J def = gr( a P ) ∼ = gr( a ) P ([AWZ08, Lemma 4.4]), which is a graded ideal of thelocalization gr( A ) P of gr( A ) with respect to the set of homogeneous elements whichare not in P , and let Y ∈ gr( A ) P such that Y ∈ J S P (see [AW09, Def.3.4] for thedefinition of J S P ). Noticing that gr( A ) P = gr( F J N K ) P and that L N /pL N is a 1-dimensional F q -vector space, we can apply [AW09, Prop.4.3] (together with [AW09,Prop.4.4(c)]) to the graded prime ideal P of B = gr( F J N K ) and the graded ideal J of gr( F J N K ) P . We deduce D P ( Y ) ⊆ J (see [AW09, §4.3] for the definition of D P ).It follows from [AW09, Thm.3.5] applied to the Frobenius pair ( Q T P ( A ) , Q T ( p ) P ( A p ))and the ideal a P that a P is controlled by Q T ( p ) P ( A p ). Then [AW09, Lemma 2.3] showsthat gr( F ) P = 0. This is a contradiction.As L N /pL N generates the F -vector space gr − ( F J N K ) = ⊕ f − i =0 F y i , it follows that y i ∈ P for all 0 ≤ i ≤ f − A ) = P . This is a contradiction so that F = 0 i.e. a = A ( a ∩ A p ). Lemma 3.1.1.6.
Let a ( A be a proper ideal of A . Then ∩ n ≥ ( A ( a ∩ A p n )) = 0 . Inparticular, if φ ( a ) ⊆ a we have ∩ n ≥ Aφ n ( a ) = 0 .Proof. Let a n def = A ( a ∩ A p n ). We endow a ∩ A p n with the induced filtration of A p n (orequivalently A ). As A is a finite free A p n -module, we have a n ∼ = A ⊗ A pn ( a ∩ A p n ). Weendow this A -module with the tensor product filtration. Since A is a filtered free A p n -module, it follows from [LvO96, Lemma I.6.14] that gr( a n ) ∼ = gr( A ) ⊗ gr( A pn ) gr( a ∩ A p n ).Since gr( A ) is a finite free gr( A p n )-module, the natural map gr( a n ) → gr( A ) is injective(and the filtration on a n is in fact the one induced from A ). Moreover from (104) wededuce gr( a n ) ∼ = gr( F J N K ) ⊗ gr( F J N pn K ) gr( a ∩ A p n ) . (105)Assume that a = A . Then as both a and A are complete and the injection a ֒ → A is strict, it follows as for the A -module F in the proof of Proposition 3.1.1.5 thatgr( A/ a ) = 0 (with the quotient filtration on A/ a ), hence by [LvO96, Thm.I.4.4(1)]that gr( a ) = gr( A ), and a fortiori gr( a n ) = gr( A ).Using (105) and the fact gr( F J N K ) ∼ = F [ y , . . . , y f − ] is free of finite rank overgr( F J N p n K ) ∼ = F [ y p n , . . . , y p n f − ], we have inside gr( A ) thatgr( a n ) ∩ gr( F J N K ) ∼ = gr( F J N K ) ⊗ gr( F J N pn K ) (gr( a ∩ A p n ) ∩ gr( F J N p n K )) . (106)The ideal gr( a n ) ∩ gr( F J N K ) is therefore generated by homogeneous elements ofgr( F J N K ) which are of degree ≤ − p n since homogeneous elements of F [ y p n , . . . , y p n f − ]of degree zero are invertible and gr( a n ) does not contain invertible elements (asgr( a n ) = gr( A )). We conclude thatgr( a n ) ∩ gr( F J N K ) ⊆ F − p n (gr( F J N K )) . T n ≥ a n has the induced filtration from A )gr (cid:16) \ n ≥ a n (cid:17) ∩ gr( F J N K ) ⊆ \ n ≥ (gr( a n ) ∩ gr( F J N K )) = 0 . (107)As gr( T n ≥ a n ) is an ideal in gr( A ) ∼ = F [ y , . . . , y f − , ( y · · · y f − ) − ], it follows from(107) that we must have gr( T n ≥ a n ) = 0, and hence that T n ≥ a n = 0 by [LvO96,Prop.I.4.2(1)]. Corollary 3.1.1.7.
The only ideals of A which are O × K -stable are and A .Proof. Let a be such an ideal and assume that a = A . It follows from Proposition3.1.1.5 applied recursively with A , A p , etc. that a = A ( a ∩ A p n ) for all n ≥
0. ThenLemma 3.1.1.6 implies a = 0. ( ψ, O × K ) -modules We define a functor from a certain abelian category of admissible smooth represen-tations of GL ( K ) over F to a category of multivariable ( ψ, O × K )-modules.Let R be a noetherian commutative ring of characteristic p endowed with aninjective ring endomorphism F R such that R is a finite free F R ( R )-module. If M isan R -module, we define F ∗ R ( M ) def = R ⊗ F R ,R M . Examples of such pairs ( R, F R ) aregiven by ( F J N K , φ ) and ( A, φ ) in §3.1.1.A ψ -module over R is a pair ( M, β ), where M is an R -module and β is an R -linear homomorphism M → F ∗ R ( M ). When R is a regular ring, F R is the Frobeniusendomorphism of R and β is an isomorphism, we recover the notion of F R -module of[Lyu97, Def.1.1]. We say that a ψ -module ( M, β ) is étale if β is injective.If ( M, β ) is a ψ -module, the exact functor F ∗ R gives us, for each n ≥
0, an R -linearmap ( F ∗ R ) n ( β ) : ( F ∗ R ) n ( M ) → ( F ∗ R ) n +1 ( M ) and we can define β n def = ( F ∗ R ) n − ( β ) ◦ · · · ◦ ( F ∗ R )( β ) ◦ β : M −→ ( F ∗ R ) n ( M ) . The inductive limit of the system (( F ∗ R ) n ( M ) , ( F ∗ R ) n ( β )) n gives rise to a ψ -module( M , β ) with β an isomorphism. Then ( M, β ) generates ( M , β ) in the sense of [Lyu97,Def.1.9]. Let M ´et be the image of M in M and M the kernel of M → M ´et . The map β induces a structure of ψ -module on M and M ´et and M ´et is an étale ψ -module.The ψ -module M ´et is called the étale part of M and M the nilpotent part of M . Wenote that ( M, β ) and ( M ´et , β ´et ) generate the same F R -module and ( M , β ) generatesthe trivial F R -module whose underlying module is zero. Note that the constructions( M, β ) ( M ´et , β ´et ) and ( M, β ) ( M , β ) are functorial in ( M, β ) and that, if β isinjective, we have M = 0. This implies that if f : ( M, β ) → ( M ′ , β ′ ) is a morphismof ψ -modules with ( M ′ , β ′ ) étale, then f factors through M ´et .105e are mainly interested in a special kind of ψ -module that we call ( ψ, O × K )-module over A . If M is a finitely generated A -module, we always endow it withthe topology defined by any good filtration (note that good filtrations generate thesame topologies, cf. [LvO96, Lemma I.5.3]). It is also the quotient topology givenby any surjection A ⊕ d ։ M (as follows from [LvO96, Rk.I.5.2(2)]), and we call itthe canonical topology on M . The group O × K acts continuously on A and this actioncommutes with the endomorphism φ . If M is an A -module which is endowed with anaction of O × K , we consider the diagonal action on φ ∗ ( M ), which is well defined since φ commutes with O × K . Definition 3.1.2.1.
A ( ψ, O × K ) -module over A is a ψ -module ( M, β ) over A suchthat M is a finitely generated A -module with a continuous semilinear action of O × K such that β is O × K -equivariant (here, continuity means that the map O × K × M → M iscontinuous). We say that a ( ψ, O × K )-module over A is étale if the underlying ψ -moduleover A is. Proposition 3.1.2.2.
Let ( M, β ) be an étale ( ψ, O × K ) -module over A . Then β is anisomorphism and M is a finite projective A -module.Proof. We note that the two A -modules M and φ ∗ ( M ) = A ⊗ φ,A M have the samegeneric rank. As β is an injective A -linear map between two finitely generated modulesof the same generic rank over a noetherian domain, its cokernel is torsion. Theannihilator of this cokernel is then a nonzero ideal of A which is O × K -stable, it followsfrom Corollary 3.1.1.7 that this is A . Then the cokernel of β is zero and β is anisomorphism.By Corollary 3.1.1.2, the ring A has finite global dimension so that M has finiteprojective dimension. By [Gro05, Lemma VIII.1.2] (applied to the functor F whichtakes an A -module to itself), we have a biduality spectral sequence E i,j = E iA (E − jA ( M )) ⇒ H i − j ( M ) , (108)where i, j ∈ Z , H ( M ) def = M , H s ( M ) def = 0 if s = 0 (and recall E sA ( − ) = 0 for s < j ∈ Z , the A -module E jA ( M ) is finitely generated. Let us prove that its annihilator isan ideal of A which is O × K -stable. Let γ ∈ O × K , we first have a canonical isomorphismof A -modules E jA ( M ) ⊗ A,γ A ∼ −→ E jA ( M ⊗ A,γ A ) . (109)The action of γ on M induces an A -linear isomorphism M ⊗ A,γ A ∼ −→ M and thus an A -linear isomorphism E jA ( M ) ∼ −→ E jA ( M ⊗ A,γ A ). Composing its inverse with (109)we obtain an A -linear isomorphism E jA ( M ) ⊗ A,γ A ∼ −→ E jA ( M ) (in particular an actionof γ on E jA ( M )) which proves that the annihilator of E jA ( M ) is a γ -stable ideal of A . It follows then from Corollary 3.1.1.7 that all the A -modules E jA ( M ) are zero orfaithful. 106et Frac( A ) be the fraction field of A , we haveE jA ( M ) ⊗ A Frac( A ) ∼ −→ E j Frac( A ) ( M ⊗ A Frac( A )) . As the category of Frac( A )-vector spaces is semisimple, we have E jA ( M ) ⊗ A Frac( A ) =0 for j >
0. As the A -modules E jA ( M ) are finitely generated and A is a domain, theyare killed by a nonzero element of A . Hence they must be 0 for j = 0 by the previousparagraph.Consequently the spectral sequence (108) degenerates at E and thus we haveE A (E A ( M )) = Hom A (Hom A ( M, A ) , A ) ∼ = M and E iA (E A ( M )) = E iA (Hom A ( M, A )) =0 for i >
0. Using that Ext rA ( − , − ) = 0 for some r ≥
0, as A has finite globaldimension, and thatExt iA (Hom A ( M, A ) , A ⊕ d ) ∼ = E iA (Hom A ( M, A )) ⊕ d = 0for i > d ≥
0, we see by a descending induction on i using the long exactsequences of Ext • A (Hom A ( M, A ) , − ) coming from Hom A (Hom A ( M, A ) , − ) applied toshort exact sequences 0 → N ′ → A ⊕ d → N → N finitely generated over A , that Ext iA (Hom A ( M, A ) , N ) = 0 for all i > N . It follows from[Bou07, Prop.X.8.2] that Hom A ( M, A ) has projective dimension 0, i.e. is a projec-tive A -module. Finally M ∼ = Hom A (Hom A ( M, A ) , A ) is projective too.We now define a functor from certain representations of GL ( K ) over F to ( ψ, O × K )-modules over A .Let π be an admissible smooth representation of GL ( K ) over F . Its ( F -linear)dual π ∨ is then a finitely generated F J I K -module. We fix a good filtration on π ∨ . Asabove, we endow A ⊗ F J N K π ∨ with the tensor product filtration and define the filtered A -module D A ( π ) def = A b ⊗ F J N K π ∨ ∼ = \ ( π ∨ ) S . (110)As all the good filtrations on π ∨ are equivalent ([LvO96, Lemma I.5.3]), the underlyingtopological A -module does not depend on the choice of the good filtration on π ∨ . Anexample of a good filtration on π ∨ is given by the m I -adic filtration, as follows directlyfrom the definition. It is very important to note that the topology used on π ∨ is not the m N -adic topology but the m I -adic topology, which is actually coarser. Proposition 3.1.2.3.
The functor π D A ( π ) is exact.Proof. Let 0 → π ′ → π → π ′′ → ( K ) over F . The sequence 0 → ( π ′′ ) ∨ → π ∨ → ( π ′ ) ∨ → π ∨ with a good filtration, ( π ′ ) ∨ with the quotient filtration and( π ′′ ) ∨ with the induced filtration (which are again good by e.g. [LvO96, Prop.II.1.2.3]).With these choices, the exact sequence remains exact after applying the functor gr107see for example [LvO96, Thm.I.4.2.4(1)]). It follows from Lemma 3.1.1.1, from theexactness of localization and from [LvO96, Thm.I.4.2.4(2)]) that the sequence 0 → ( π ′′ ) ∨ S → ( π ∨ ) S → ( π ′ ) ∨ S → strict . The exactness of 0 → D A ( π ′′ ) → D A ( π ) → D A ( π ′ ) → O × K on π ∨ as follows, for f ∈ π ∨ , γ ∈ O × K wehave ( γ · f )( x ) def = f γ −
00 1 ! x ! ∀ x ∈ π. As O × K normalizes I , the action of O × K on π ∨ is continuous for the m I -adic topology.We use the continuous action of O × K on A to extend this action diagonally to A ⊗ F J N K π ∨ and, by continuity, to D A ( π ). The action of O × K is continuous and A -semilinearin the sense that γ · ( af ) = ( γ · a )( γ · f ) ∀ ( γ, a, f ) ∈ O × K × A × D A ( π ) . We define an F -linear endomorphism ψ of π ∨ by the formula ψ ( f )( x ) = f ( ξ ( p ) x ) ∀ ( f, x ) ∈ π ∨ × π. (111)This endomorphism is continuous, clearly commutes with the action of O × K and sat-isfies the relation ψ ( φ ( a ) f ) = a ( ψ ( f ))for all a ∈ F J N K , f ∈ π ∨ . Lemma 3.1.2.4.
Let M be some F J N K -module and let ψ be an F -linear endomor-phism of M satisfying the relation ψ ( φ ( a ) m ) = aψ ( m ) ∀ ( a, m ) ∈ F J N K × M. Then for all integers n ≥ , we have ψ ( m pf − ( f − pnN M ) ⊆ m n +1 N M. As a consequence, for n ≥ pf − ( f − , we have ψ ( m nN M ) ⊆ m ⌈ np ⌉− fN M. Proof.
For n = 0, the result follows from the fact that, if Y i · · · Y i f − f − ∈ m pf − ( f − N ,there exists some 0 ≤ j ≤ f − i j ≥ p . Then, for all m ∈ M , we have ψ ( Y i · · · Y i f − f − m ) = Y j +1 ψ ( Y i · · · Y i j − pj · · · Y i f − f − m ) ∈ m N M. The general statement follows from a simple induction on n .108or the last statement, we choose m such that pm + pf − ( f − ≤ n < p ( m + 1) + pf − ( f − ψ ( m nN M ) ⊆ ψ ( m pm + pf − ( f − N M ) ⊆ m m +1 N M ⊆ m ⌈ np ⌉− fN M. We extend ψ to an F -linear map ( π ∨ ) S → ( π ∨ ) S (recall ( π ∨ ) S = F J N K S ⊗ F J N K π ∨ )by the formula ψ m ( Y · · · Y f − ) pn ! = ψ ( m )( Y · · · Y f − ) n (112)for all m ∈ π ∨ and n ≥
0. Each element of ( π ∨ ) S can be written as ( Y · · · Y f − ) − pn m for some m ∈ π ∨ and n ≥
0, and it follows from the properties of ψ on π ∨ that theright-hand side of (112) does not depend on this choice. For any element g in I , wedenote by δ g the corresponding element [ g ] in F J I K . Lemma 3.1.2.5.
The map ψ : ( π ∨ ) S → ( π ∨ ) S is continuous.Proof. As all the good filtrations on π ∨ are equivalent, we choose the m I -adic filtra-tion on π ∨ for this proof, i.e. F n π ∨ = m − nI π ∨ for n ≤ F n π ∨ = π ∨ for n > + , Prop.5.3.3] we have an equality for n ≥ m nI = X r,s,t ≥ r +2 s + t = n m rN m sT m tN − . (113)As ξ ( p ) commutes with each element in T , and ξ ( p ) − ( z ) ξ ( p ) = ( z ) p for any( z ) ∈ N − , it is easily checked from the definition of ψ and the F J I K -action on π ∨ that ψ ( δ h δ z · f ) = δ h δ z p ψ ( f ) (114)for all h ∈ T , z ∈ N − . In particular, ψ ( m sT m tN − π ∨ ) ⊆ m sT m ptN − π ∨ , and it follows from Lemma 3.1.2.4 that if r ≥ pf − ( f −
1) we have ψ ( m rN m sT m tN − π ∨ ) ⊆ m ⌈ rp ⌉− fN m sT m ptN − π ∨ ⊆ m ⌈ rp ⌉ +2 s + pt − fI π ∨ ⊆ m ⌈ r +2 s + tp ⌉− fI π ∨ . (115)If r < pf − ( f − Lemma 3.1.2.6.
Let M ⊆ π ∨ be a closed F J N − K -submodule. Then ψ ( F J N K m N − M ) ⊆ m I ψ ( F J N K M ) . As a consequence, for all t ≥ , ψ ( F J N K m tN − π ∨ ) ⊆ m tI π ∨ . roof. Note that m I × F J N K × M is compact, as M is closed, hence so is the image m I ψ ( F J N K M ) of the continuous map m I × F J N K × M → π ∨ , ( a, b, m ) aψ ( bm ).As m N − is generated as right F J N − K -module by the δ y − y ∈ N − and as ψ iscontinuous on π ∨ , it is thus sufficient to prove that, for y ∈ N − , x ∈ N and m ∈ M ,we have ψ ( δ x ( δ y − m ) ∈ m I ψ ( F J N K M ). As N − ⊆ K , K is normalized by N and K = N p T N − , we can write xy = x p t y x with ( x , t , y ) ∈ N × T × N − .Therefore ψ ( δ x ( δ y − m ) = ψ ( δ x p δ t δ y δ x m ) − ψ ( δ x m )= δ x t y p ψ ( δ x m ) − ψ ( δ x m ) = ( δ x t y p − ψ ( δ x m ) ⊆ m I ψ ( F J N K M ) . For the second statement, inductively apply the first to M = m t − N − π ∨ , M = m t − N − π ∨ ,etc.When r < pf − ( f −
1) = ( p − f + 1, we have 2 s + t ≥ r + 2 s + t − ( p − f sothat, using Lemma 3.1.2.6 and the fact that T normalizes N , we obtain ψ ( m rN m sT m tN − π ∨ ) ⊆ m sT ψ ( F J N K m tN − π ∨ ) ⊆ m s + tI π ∨ ⊆ m r +2 s + t − ( p − fI π ∨ . (116)We deduce from (112), (115) and (116) that, for all n ∈ Z , r ≥ s ≥ t ≥ k ≥ r + 2 s + t ≥ pf , we have ψ Y · · · Y f − ) pk m rN m sT m tN − π ∨ ! ⊆ Y · · · Y f − ) k m ⌈ r +2 s + tp ⌉− fI π ∨ so that, for n ≥ pf by (113) we have ψ Y · · · Y f − ) pk m nI π ∨ ! ⊆ Y · · · Y f − ) k m ⌈ np ⌉− fI π ∨ ⊆ F kf + f −⌈ np ⌉ (( π ∨ ) S ) . From Remark 3.1.1.3(ii), we know that, for n ∈ Z , F n (( π ∨ ) S ) is the increasing unionover k ≥ max { , npf } of the subspaces1( Y · · · Y f − ) pk m − n + pkfI π ∨ , hence we deduce for all n ∈ Z that ψ ( F n (( π ∨ ) S )) ⊆ [ k ≥ max { , npf } F kf + f −⌈ − n + pkfp ⌉ (( π ∨ ) S ) ⊆ F f + ⌊ np ⌋ (( π ∨ ) S ) . This proves the continuity of ψ . 110e can therefore extend ψ to a continuous F -linear map ψ : D A ( π ) → D A ( π ) suchthat ψ ( φ ( a ) m ) = aψ ( m ) ∀ ( a, m ) ∈ A × π ∨ . We fix { a , . . . , a q − } a system of representatives of the cosets of N p ∼ = p O K in N ∼ = O K , so that F J N K = L q − i =0 δ a i F J N p K . As φ ( F J N K ) = F J N p K and A = L q − i =0 δ a i φ ( A ), we have a canonical isomorphism for any A -module M : φ ∗ ( M ) ∼ = q − M i =0 ( F δ a i ⊗ F M ) . We define an F -linear map β : D A ( π ) → φ ∗ ( D A ( π )) = A ⊗ φ,A D A ( π ) by D A ( π ) −→ L q − i =0 ( F δ a i ⊗ F D A ( π )) m P q − i =0 δ a i ⊗ φ ψ ( δ − a i m ) (117)(we write x ⊗ φ y instead of just x ⊗ y in order not to forget the map φ in the tensorproduct). Remark 3.1.2.7.
The definition of the map β does not depend on the choice of thesystem { a i } , namely, replacing a i with a i b p for some b ∈ N , we have δ a i b p ⊗ φ ψ ( δ − a i b p m ) = δ a i b p ⊗ φ ψ ( φ ( δ b ) − δ − a i m ) = δ a i b p ⊗ φ δ − b ψ ( δ − a i m )= δ a i b p δ − b p ⊗ φ ψ ( δ − a i m ) = δ a i ⊗ φ ψ ( δ − a i m ) . Using Remark 3.1.2.7, we easily check that β is actually an A -linear map (notethat it is enough to check it for an element in δ a i φ ( A ) using A = L q − i =0 δ a i φ ( A ), andthus for δ a i and for an element in φ ( A )), hence β : D A ( π ) → φ ∗ ( D A ( π )) can be seenas a “linearization” of ψ : D A ( π ) → D A ( π ). Moreover, letting O × K act diagonally on A ⊗ φ,A D A ( π ), the map β is then O × K -equivariant. Indeed, for a ∈ O × K and m ∈ D A ( π ),we have a · β ( m ) = a · q − X i =0 δ a i ⊗ φ ψ ( δ − a i m ) = q − X i =0 δ a · a i ⊗ φ a · ψ ( δ − a i m )= q − X i =0 δ a · a i ⊗ φ ψ ( a · δ − a i m ) = q − X i =0 δ a · a i ⊗ φ ψ ( δ − a · a i ( a · m ))= β ( a · m ) , the last equality coming from Remark 3.1.2.7 and the fact that { a · a , . . . , a · a q − } is another system of representatives of N p in N .It is convenient to assume that the admissible smooth representation π has acentral character, in which case Z acts trivially on π and π ∨ is a finitely generated111 J I /Z K -module. We recall from [BHH + , §5.3] that the graded ring gr( F J I /Z K ) of F J I /Z K is isomorphic to a tensor product of (noncommutative) graded rings f − O i =0 F [ y i , z i , h i ] , (118)where variables with different indices commute, where [ y i , z i ] = h i , [ h i , y i ] = [ h i , z i ] =0, where y i , z i are homogeneous of degree −
1, and h i is homogeneous of degree −
2. Note that the m I /Z -adic topology on F J I /Z K induces the m N -adic topol-ogy on F J N K via the inclusion F J N K ⊆ F J I /Z K . Therefore the map gr( F J N K ) → gr( F J I /Z K ) is injective and its image is F [ y , . . . , y f − ] in gr( F J I /Z K ). Remark 3.1.2.8.
The A -module D A ( π ) can also be defined as the microlocaliza-tion of π ∨ with respect to the multiplicative subset T def = { ( y · · · y f − ) k , k ∈ N } ⊆ gr( F J I /Z K ). This shows that D A ( π ) can be promoted to a module over the noncom-mutative ring which is the microlocalization of F J I /Z K with respect to T .We now let C be the category of admissible smooth representations π of GL ( K )over F with a central character and such that there exists a good filtration on the F J I /Z K -module π ∨ such that gr( D A ( π )) is a finitely generated gr( A )-module, orequivalently by Lemma 3.1.1.1 and Corollary 3.1.1.2 gr( π ∨ )[( y · · · y f − ) − ] is finitelygenerated over gr( F J N K )[( y · · · y f − ) − ]. By [LvO96, Thm.I.5.7] this is also equiva-lent to require that D A ( π ) is finitely generated over A and that its natural filtrationin (110) is good (equivalently gives the canonical topology). In particular, if thisholds for one good filtration on π ∨ , then this holds for all good filtrations. It easilyfollows from the proof of Proposition 3.1.2.3 and the noetherianity of gr( A ) (Corollary3.1.1.2) that C is an abelian subcategory stable under extensions in the category ofsmooth representations of GL ( K ) over F with a central character.For π in C , the pair ( D A ( π ) , β ) is an example of ( ψ, O × K )-module over A as inDefinition 3.1.2.1. We can in particular consider its étale part D A ( π ) ´et . The actionof O × K on D A ( π ) preserves its nilpotent part D A ( π ) and thus induces a continuousaction of O × K on D A ( π ) ´et . In particular, D A ( π ) ´et is an étale ( ψ, O × K )-module over A .Note that the canonical topology on the finitely generated A -module D A ( π ) ´et is alsothe quotient topology of D A ( π ) ։ D A ( π ) ´et . Corollary 3.1.2.9.
Let π in C . Then the étale ( ψ, O × K ) -module D A ( π ) ´et is finiteprojective over A and the map β ´et : D A ( π ) ´et → φ ∗ D A ( π ) ´et is an isomorphism.Proof. This is a special case of Proposition 3.1.2.2.
Remark 3.1.2.10. If π is 1-dimensional (a character of GL ( K )), then D A ( π ) = D A ( π ) ´et = 0. 112e give an important condition on an admissible smooth representation π (witha central character) which ensures that π is in C . Let J be the following graded idealof gr( F J I /Z K ): J def = ( y i z i , h i , ≤ i ≤ f − . (119)From the definition of equivalent filtrations (see [LvO96, §I.3.2]), one easily sees (using[LvO96, Lemma I.5.3]) that if gr( π ∨ ) is annihilated by some power of J for one goodfiltration on π , then it is so for all good filtrations (but note that the power of J which annihilates gr( π ∨ ) may depend on the fixed good filtration). Proposition 3.1.2.11.
Assume that gr( π ∨ ) is annihilated by some power of J . Thenthe A -module D A ( π ) is finitely generated and the gr( A ) -module gr( D A ( π )) is finitelygenerated.Proof. As the hypothesis does not depend on the choice of the good filtration on π ∨ , we are free to work with the m I /Z -adic topology on π ∨ . Let us first prove thatgr( D A ( π )) is a finitely generated gr( A )-module. It follows from the admissibilityof π and from the hypothesis that gr( π ∨ ) is a finitely generated gr( F J I /Z K ) /J N -module for some N ≥
1. Lemma 3.1.1.1 then implies that gr( D A ( π )) is a finitelygenerated (gr( F J I /Z K ) /J N )[( y · · · y f − ) − ]-module. It is therefore sufficient to provethat (gr( F J I /Z K ) /J N )[( y · · · y f − ) − ] is a finitely generated gr( A )-module. Sincegr( F J I /Z K ) is noetherian, we are reduced by dévissage to the case N = 1, where wehave (cid:16) gr( F J I /Z K ) /J (cid:17) [( y · · · y f − ) − ] ∼ = ( F [ y i , z i , h i ] / ( y i z i , h i ))[( y · · · y f − ) − ]= F [ y ± i ] ∼ = gr( A ) . Finally, as D A ( π ) is a complete filtered A -module, it then follows from [LvO96,Thm.I.5.7] that D A ( π ) is finitely generated over A .It follows from Proposition 3.1.2.11 that the admissible smooth representations π (with a central character) such that gr( π ∨ ) is annihilated by some power of J for atleast one good filtration is a full subcategory of the category C . Moreover this fullsubcategory is abelian and stable under extensions in C . Namely, for a short exactsequence 0 → π ′ → π → π ′′ → C , the filtrations induced on ( π ′′ ) ∨ and ( π ′ ) ∨ bya good filtration of π ∨ are good. For these filtrations we have a short exact sequence0 → gr(( π ′′ ) ∨ ) → gr( π ∨ ) → gr(( π ′ ) ∨ ) → π ∨ ) is annihilated bya power of J if and only if gr(( π ′ ) ∨ ) and gr(( π ′′ ) ∨ ) are. Remark 3.1.2.12.
It is natural to consider the image D ♮A ( π ) of π ∨ in D A ( π ) = A b ⊗ F J N K π ∨ . Indeed, as the map π ∨ → D A ( π ) is continuous and π ∨ is compact, itfollows that D ♮A ( π ) is a compact F J N K -submodule of D A ( π ). However, the F J N K -module D ♮A ( π ) is not finitely generated when π is an irreducible admissible supersin-gular representation and [ K : Q p ] > D A ( π ) is finitely generated over A ).113amely, if this was the case, this would give us the existence of a nontrivial finitelygenerated F J N K [( p
00 1 )]-submodule of π that is admissible as F J N K -module and thiswould contradict the results of [Sch15] and [Wu]. Likewise, the image of π ∨ in thequotient D A ( π ) ´et of D A ( π ) won’t be finitely generated over F J N K in general (seeRemark 3.3.5.4(ii)). ( ϕ, O × K ) -modules Using the results of §3.1.2, we promote the functor π D A ( π ) ´et to an exact functorfrom C to a category of étale multivariable ( ϕ, O × K )-modules (Theorem 3.1.3.3) andwe compare D A ( π ) ´et with the functor D ∨ ξ ( π ) of §2.1.1 (Theorem 3.1.3.6).Let R be a noetherian commutative ring of characteristic p endowed with aninjective ring endomorphism F R such that R is a finite free F R ( R )-module (as atthe beginning of §3.1.2). A ϕ -module ( D, ϕ ) over R is an R -module D with an F R -semilinear map ϕ : D → D . We say that a ϕ -module ( D, ϕ ) is étale if the R -linearmap F ∗ R ( D ) → D defined by a ⊗ d aϕ ( d ) is an isomorphism. Definition 3.1.3.1.
A ( ϕ, O × K ) -module over A is a ϕ -module ( D, ϕ ) over A suchthat D is a finitely generated A -module, the endomorphism ϕ is continuous (for thecanonical topology on D as at the beginning of §3.1.2) and D is endowed with acontinuous A -semilinear action of O × K commuting with ϕ . We say that a ( ϕ, O × K )-module over A is étale if the underlying ϕ -module over A is.If ( D, β ) is an étale ( ψ, O × K )-module over A as in Definition 3.1.2.1, by the firststatement in Proposition 3.1.2.2 we can define a φ -semilinear endomorphism ϕ of D such that Id ⊗ ϕ = β − , so that ( D, ϕ ) is an étale ( ϕ, O × K )-module over A .We now go back to representations π of GL ( K ), but we first need some morenotation. The trace map tr : N ∼ = O K → Z p induces a ring homomorphism tr : F J N K → F J Z p K ∼ = F J X K , where we recall that X = ( ) −
1. Moreover, for Y i as in(100), we have tr( Y i ) ≡ − X mod X (see Lemma 3.2.2.2 and the last statement inLemma 3.2.2.4 below) and the universal property of the ring A shows that this mapextends to a continuous ring homomorphism tr : A → F (( X )). We let p def = Ker(tr : A → F (( X ))) . Then p is a closed maximal ideal of A . Note that p ∩ F J N K = Ker(tr : F J N K → F J X K ) = m N F J N K = ( Y − Y , . . . , Y − Y f − ) , where N ⊆ N is as in (11) (for the second isomorphism write N ∼ = N ⊕ Z p e , wheretr( e ) = 1, noting that tr : O K → Z p is surjective, as K is unramified, and for thethird use the first statement of Lemma 3.2.2.4 below).114 emark 3.1.3.2. Let B be the completion of F J N K S along the prime ideal generatedby ( Y − Y , . . . , Y − Y f − ) (see the beginning of §3.1.1 for S ). Expanding Y ni =( Y − ( Y − Y i )) n if n ≥
0, and writing Y ni = ( P + ∞ m =0 ( Y − Y i ) m Y m +10 ) − n and expandingeverything if n <
0, one can see using Remark 3.1.1.3(iii) that the ring A embedsinto B . The endomorphism φ on A extends to B but only the action of Z × p ⊆ O × K extends to B , as ( Y − Y , . . . , Y − Y f − ) is not preserved by all of O × K . Then fromCorollary 3.1.2.9 and as B is a local ring, we see that D A ( π ) ´et ⊗ A B is a finite free étale ( ϕ, Z × p ) -module over B , which is similar to the generalized ( ϕ, Γ)-modules defined in[SV11] (though loc.cit. only considers split algebraic groups over Q p ).Let π be in the category C . Using Corollary 3.1.2.9, we can define a φ -semilinearendomorphism ϕ of D A ( π ) ´et such that Id ⊗ ϕ = ( β ´et ) − , so that D A ( π ) ´et is an étale( ϕ, O × K )-module over A . As p is a φ -stable ideal of A , we deduce that D A ( π ) ´et / p ∼ = D A ( π ) ´et ⊗ A F (( X )) is an étale ( ϕ, Z × p )-module over F (( X )). Theorem 3.1.3.3. (i)
The functor π D A ( π ) ´et is exact from the category C to the category of étale ( ϕ, O × K ) -modules over A . (ii) The functor π D A ( π ) ´et ⊗ A F (( X )) is exact from the category C to the categoryof étale ( ϕ, Z × p ) -modules over F (( X )) .Proof. (i) is a consequence of Proposition 3.1.2.3, of the exactness of φ ∗ and of theexactness of direct limits, together with the description (see the beginning of §3.1.2) D A ( π ) ´et ∼ = lim −→ ( φ ∗ ) n ( β ´et ) ( φ ∗ ) n ( D A ( π ) ´et ) ∼ = lim −→ ( φ ∗ ) n ( β ) ( φ ∗ ) n ( D A ( π )) . (ii) is a consequence of (i), of Corollary 3.1.2.9 and of the exactness of ( − ) ⊗ A F (( X ))on short exact sequences of finite projective A -modules.We now compare the étale ( ϕ, Z × p )-module D A ( π ) ´et / p with D ∨ ξ ( π ) (15).Let ψ be the F -linear endomorphism of π ∨ / m N ∼ = ( π N ) ∨ defined by ψ ( x ) def = X b ∈ N /N p ψ ( δ ˜ b ˜ x ) mod m N , (120)where ˜ b ∈ N is a lift of b , ˜ x ∈ π ∨ is a lift of x and ψ is as in (111) (it is easy tocheck that the definition of ψ does not depend on the choice of these lifts). We have ψ ( S ( X p ) m ) = S ( X ) ψ ( m ) for all S ( X ) ∈ F J X K and m ∈ π ∨ / m N , and ψ is the dualof the endomorphism F of π N in §2.1.1. We define an endomorphism ψ of D A ( π ) / p D A ( π ) ´et / p ) by the same formula replacing π ∨ by D A ( π ) (resp. D A ( π ) ´et ) and m N by p , it is then clear that the following diagram commutes: π ∨ / m N π ∨ / m N D A ( π ) / p D A ( π ) / p , ψψ (121)together with an analogous diagram with D A ( π ) / p ։ D A ( π ) ´et / p that we leave to thereader.Let β : D A ( π ) / p → φ ∗ ( D A ( π ) / p ) ∼ = F J X K ⊗ ϕ, F J X K ( D A ( π ) / p ) ∼ = φ ∗ ( D A ( π )) / p bethe F (( X ))-linear map defined by β ( m ) def = p − X i =0 (1 + X ) − i ⊗ φ ψ ((1 + X ) i m ) . Lemma 3.1.3.4.
The following diagram is commutative ( where the horizontal mapsare the canonical surjections ) : D A ( π ) D A ( π ) / p φ ∗ ( D A ( π )) φ ∗ ( D A ( π ) / p ) . β β Proof.
We choose a system of representatives ( g − i b j ) ≤ i ≤ p − ≤ j ≤ p f − of N /N p such that g def = ( ) ∈ N and b , . . . , b p f − are in N . We then have for m ∈ D A ( π ) that β ( m ) = p − X i =0 p f − X j =1 (cid:16) δ − g i δ b j ⊗ φ ψ ( δ − b j δ g i m ) (cid:17) ≡ p − X i =0 (cid:16) δ − g i ⊗ φ p f − X j =1 ψ ( δ − b j ( δ g i m )) (cid:17) mod p φ ∗ ( D A ( π )) ≡ p − X i =0 δ − g i ⊗ φ ψ ( δ g i m ) mod p φ ∗ ( D A ( π )) , where the first equality follows from (117), the second from δ b j − ∈ p ⊆ A (and thecommutativity of N ), and the third from the analog of (120) for D A ( π ) / p . Notingthat the image of δ g i in F J X K is (1 + X ) i , we obtain the desired compatibility. Lemma 3.1.3.5.
Let M ⊆ π N be an F J X K -submodule that is admissible as F J X K -module. Then the surjective map π ∨ ։ M ∨ is continuous for the m I -adic topologyon π ∨ and the X -adic topology on M ∨ . roof. Since M is an admissible F J X K -module, the F J X K -module M ∨ is finitely gen-erated. As a consequence, M ∨ /X n M ∨ is finite for every n ≥ X n M ∨ in π ∨ is a cofinite F -vector subspace. As π ∨ is a compact F J I K -module,its m I -adic topology is separated, which implies that there exists N ≥ m NI π ∨ is contained in the inverse image of X n M ∨ . This gives the continuity.Recall that we defined in (15) a projective limit D ∨ ξ ( π ) of étale ( ϕ, Z × p )-modulesover F (( X )) associated to π . Theorem 3.1.3.6.
We have an isomorphism of étale ( ϕ, Z × p ) -modules over F (( X )) : D A ( π ) ´et / p ∼ −→ D ∨ ξ ( π ) . In particular, D ∨ ξ ( π ) is finite-dimensional over F (( X )) and the functor π D ∨ ξ ( π ) is exact on C .Proof. As a first step we construct the map. Let M ⊆ π N be a finitely generated F J X K [ F ]-submodule that is admissible as F J X K -module and Z × p -stable. By Lemma3.1.3.5, the map π ∨ ։ M ∨ is continuous. It extends to a surjection of F J N K S -modules ( π ∨ ) S ։ M ∨ [ X − ]. By definition of the tensor product filtration on ( π ∨ ) S ,this surjection is continuous if M ∨ [ X − ] is endowed with its natural topology offinite-dimensional F (( X ))-vector space. As M ∨ [ X − ] is complete for this topology, bycompletion we obtain a continuous surjection of topological A -modules ζ M : D A ( π ) ։ M ∨ [ X − ]. Since N acts trivially on M , ζ M factors through a surjection of F (( X ))-vector spaces ζ M : D A ( π ) / p ։ M ∨ [ X − ]. By definition of ψ , we obtain a commutativediagram (where F ∨ is the F -linear dual of F : M → M that we extend to M ∨ [ X − ]using F ∨ ( X − i f ) = X − i F ( X i ( p − f )) D A ( π ) / p M ∨ [ X − ] D A ( π ) / p M ∨ [ X − ] . ζ M ψ F ∨ ζ M It then follows from Lemma 3.1.3.4 that, identifying φ ∗ ( M ∨ ) ∼ = F J X K ⊗ ϕ, F J X K M ∨ with( F J X K ⊗ ϕ, F J X K M ) ∨ via (14), the following diagram is commutative: D A ( π ) D A ( π ) / p M ∨ [ X − ] φ ∗ ( D A ( π )) φ ∗ ( D A ( π ) / p ) φ ∗ ( M ∨ [ X − ]) , β ζ M β (Id ⊗ F ) ∨ Id ⊗ ζ M (122)where (Id ⊗ F ) ∨ comes from F -linear dual of Id ⊗ F : F J X K ⊗ ϕ, F J X K M → M . As(Id ⊗ F ) ∨ is an isomorphism (see just after (14)), the map ζ M : D A ( π ) ։ M ∨ [ X − ]117actors through D A ( π ) ´et and the map ζ M : D A ( π ) / p ։ M ∨ [ X − ] factors through D A ( π ) ´et / p . The map ζ M : D A ( π ) ´et / p ։ M ∨ [ X − ] clearly commutes with the actionof Z × p and the commutative diagram (122) shows that it is a morphism ϕ -modules.These maps are obviously compatible when M is varying among the finitely generated F J X K [ F ]-submodules of π N that are admissible as F J X K -modules and Z × p -stable sothat we obtain a map ζ : D A ( π ) ´et / p −→ lim ←− M M ∨ [ X − ] = D ∨ ξ ( π ) . We prove that the map ζ is surjective. Since D A ( π ) ´et / p is a finite-dimensional F (( X ))-vector space, the dimension of the vector spaces M ∨ [ X − ] when M is varyingis bounded. This implies that there exists some M such that D ∨ ξ ( π ) = M ∨ [ X − ] andthat the map ζ : D A ( π ) ´et / p → D ∨ ξ ( π ) is surjective. In particular, dim F (( X )) D ∨ ξ ( π ) < + ∞ .We prove that the map ζ is an isomorphism. Let D ♮ ( π ) ´et be the image of π ∨ in D A ( π ) ´et / p . This is a compact F J X K -module in the finite-dimensional F (( X ))-vectorspace D A ( π ) ´et / p , hence a finite free F J X K -module. Since the maps π ∨ → D A ( π ) / p ։ D A ( π ) ´et / p commute with the action of Z × p , D ♮ ( π ) ´et is preserved by Z × p . The imageof ( π ∨ ) S in D A ( π ) ´et / p coincides with D ♮ ( π ) ´et [ X − ]. As ( π ∨ ) S has a dense image in D A ( π ) by definition, D ♮ ( π ) ´et [ X − ] is a dense F (( X ))-vector subspace of D A ( π ) ´et / p and thus equal to D A ( π ) ´et / p by finiteness of the dimension. The surjective map π ∨ ։ D ♮ ( π ) ´et factors through π ∨ / m N ∼ = ( π N ) ∨ so that the topological F -linear dual( D ♮ ( π ) ´et ) ∨ of D ♮ ( π ) ´et is identified with an F J X K -submodule of π N (endowed with thediscrete topology) preserved by Z × p . As D ♮ ( π ) ´et is stable by ψ by (121), ( D ♮ ( π ) ´et ) ∨ is actually an F J X K [ F ]-submodule of π N . Since β ´et : D A ( π ) ´et ∼ −→ φ ∗ ( D A ( π ) ´et ) is anisomorphism, it easily follows from Lemma 3.1.3.4 that the map β induces a surjectivemap of finite-dimensional F (( X ))-vector spaces β ´et : D A ( π ) ´et / p ։ φ ∗ ( D A ( π ) ´et / p ).As these spaces have the same dimension, β ´et is actually an isomorphism, and inparticular β ´et | D ♮ ( π ) ´et : D ♮ ( π ) ´et → F J X K ⊗ ϕ, F J X K D ♮ ( π ) ´et is an injection. As β ´et | D ♮ ( π ) ´et coincides with (Id ⊗ F ) ∨ (this is analogous to (122) using (14) with M = ( D ♮ ( π ) ´et ) ∨ ),we conclude that the mapId ⊗ F : F J X K ⊗ ϕ, F J X K ( D ♮ ( π ) ´et ) ∨ −→ ( D ♮ ( π ) ´et ) ∨ is surjective. It then follows from [Emeb, Lemma 1.1] that ( D ♮ ( π ) ´et ) ∨ is a finitely gen-erated F J X K [ F ]-module. Moreover it is admissible as an F J X K -module since D ♮ ( π ) ´et is a finitely generated F J X K -module. Hence ( D ♮ ( π ) ´et ) ∨ is one of the modules M ⊆ π N in §2.1.1, in particulardim F (( X )) D ∨ ξ ( π ) ≥ dim F (( X )) ( D ♮ ( π ) ´et [ X − ]) = dim F (( X )) ( D A ( π ) ´et / p ) . This implies that the map ζ is an isomorphism (and that D A ( π ) ´et / p = D ♮ ( π ) ´et [ X − ] ∼ = D ∨ ξ ( π )). The very last statement follows from Theorem 3.1.3.3(ii).118 .1.4 An upper bound for the ranks of D A ( π ) ´et and D ∨ ξ ( π )For π in C we bound the dimension of D ∨ ξ ( π ) in terms of gr( π ∨ ). When gr( π ∨ ) iskilled by some J n , we give an interpretation of this bound as a certain multiplicity.We keep all previous notation. We start by the following lemma. Lemma 3.1.4.1.
Let M be a finitely generated A -module endowed with a good filtra-tion. Then the generic rank of the A -module M and the generic rank of the gr( A ) -module gr( M ) coincide.Proof. We first note that if N is an A -module of generic rank 0, then N ⊗ A Frac( A ) = 0and N is a torsion module. This implies that gr( N ) is a torsion module and that itsgeneric rank is 0.Let d be the generic rank of M and f : A ⊕ d → M ⊗ A Frac( A ) be a morphism of A -modules sending an A -basis of the left-hand side to a Frac( A )-basis of the right-hand side. The kernel of f is then a torsion A -submodule of A ⊕ d and is zero since A is a domain. Moreover there exists a ∈ A \{ } such that the image of af is containedin M . As Frac( A ) is a flat A -module, the generic rank is an additive map on theabelian category of finitely generated A -modules. As af is injective and A ⊕ d and M have identical generic ranks, this implies that the cokernel Q of af has generic rank0. We fix a good filtration on M : it induces good filtrations on af ( A ⊕ d ) and on Q .For these filtrations we have a short exact sequence0 −→ gr( af ( A ⊕ d )) −→ gr( M ) −→ gr( Q ) −→ . As Q has generic rank 0, so does gr( Q ) so that it suffices to prove that gr( af ( A ⊕ d ))has generic rank d . It follows from the second paragraph after [Bjö89, Def.4.2] that,for a finitely generated A -module N , the generic rank of gr( N ) does not depend onthe choice of good filtration. We can thus choose a good filtration af ( A ⊕ d ) ∼ = A ⊕ d which is filtered free with respect to the canonical basis of A ⊕ d , for which the resultis obvious.Let π be in the category C and choose a good filtration on the F J I /Z K -module π ∨ .Since the finitely generated A -module D A ( π ) doesn’t depend up to isomorphism onthe choice of this good filtration (see §3.1.2), it follows from Lemma 3.1.4.1 (appliedto M = D A ( π )) and Lemma 3.1.1.1 (applied to M = π ∨ ) that the generic rank ofgr( A ) ⊗ gr( F J N K ) gr( π ∨ ) also doesn’t depend on this choice. Proposition 3.1.4.2.
Let π ∈ C . Then rk A ( D A ( π ) ´et ) = dim F (( X )) D ∨ ξ ( π ) is boundedby the generic rank of the gr( A ) -module gr( A ) ⊗ gr( F J N K ) gr( π ∨ ) .Proof. As D A ( π ) ´et is a quotient of D A ( π ), the result follows from Lemma 3.1.4.1,Lemma 3.1.1.1 and Theorem 3.1.3.6. 119hen gr( π ∨ ) is moreover killed by the ideal J n for some n ≥ J is asin (119) and recall this doesn’t depend on the good filtration), the generic rank ofgr( A ) ⊗ gr( F J N K ) gr( π ∨ ) has a nice and useful interpretation that we give now.We define R def = gr( F J I /Z K ) /J . Recall using (118) that we have R ∼ = F [ y i , z i , ≤ i ≤ f − / ( y i z i , ≤ i ≤ f − . (123)Therefore R has 2 f minimal prime ideals which are the ideals ( y i , z j , i ∈ J , j / ∈ J )with J a subset of { , . . . , f − } . Let p def = ( z j , ≤ j ≤ f − J = ∅ .If N is a finitely generated module over R and q is a minimal prime ideal of R , wedenote by m q ( N ) the length of N q over R q . More generally, if N is a finitely generatedgr( F J I /Z K )-module annihilated by J n for some n ≥
1, we define the multiplicity of N at q to be m q ( N ) = n − X i =0 m q ( J i N/J i +1 N ) . (124) Lemma 3.1.4.3. If → N → N → N → is a short exact sequence of finitelygenerated gr( F J I /Z K ) /J n -modules, then m q ( N ) = m q ( N ) + m q ( N ) .Proof. This is checked by a standard dévissage. If n = 1, the statement is obvioussince gr( F J I /Z K ) /J = R is commutative (and noetherian). Assume n ≥ N is annihilated by J n − .Assume first that N and N are both annihilated by J n − (but not necessarily N ). Then N is a quotient of N/J n − N . Let Ker def = Ker( N/J n − N ։ N ) be thecorresponding kernel. Then we have two short exact sequences0 → Ker → N/J n − N → N → → J n − N → N → Ker → . (125)By definition of m q ( N ) and the inductive hypothesis, we then obtain m q ( N ) = m q ( J n − N ) + m q ( N/J n − N ) = m q ( N ) + m q ( N ) . Assume now that N is annihilated by J n − (but not necessarily for N ). Thenthe surjection N ։ N factors through the quotient N/J n − N of N . Again letKer def = Ker( N/J n − N ։ N ). Then m q ( N/J n − N ) = m q (Ker) + m q ( N ) by theinductive hypothesis. On the other hand, both J n − N and Ker are annihilated by120 n − , thus m q ( · ) is additive for the short exact sequence (125) by the discussion inlast paragraph. The result also holds in this case.To finish the proof it suffices to decompose further N as 0 → Ker ′ → N → N /J n − N → N is a finitely generated module over gr( F J I /Z K ) /J n for some n ≥ A )-module gr( A ) ⊗ gr( F J N K ) N is finitely generated by Proposition 3.1.2.11. Lemma 3.1.4.4.
Let N be a finitely generated module over gr( F J I /Z K ) /J n for some n ≥ . Then the generic rank of the gr( A ) -module gr( A ) ⊗ gr( F J N K ) N is equal to m p ( N ) .Proof. By Corollary 3.1.1.2, gr( A ) is flat over gr( F J N K ), so gr( A ) ⊗ gr( F J N K ) N has afinite filtration with graded pieces given by gr( A ) ⊗ gr( F J N K ) ( J i N/J i +1 N ) for 0 ≤ i ≤ n −
1. Since taking generic rank and taking m p ( · ) are both additive in short exactsequences (by Lemma 3.1.4.3 for the latter), we are reduced to the case where N iskilled by J .In that case we havegr( A ) ⊗ gr( F J N K ) N ∼ = (gr( A ) ⊗ gr( F J N K ) R ) ⊗ R N. Since the image of gr( F J N K ) in R is F [ y , . . . , y f − ], we havegr( A ) ⊗ gr( F J N K ) R ∼ = R [( y · · · y f − ) − ] ∼ = gr( A ) . Since the fraction field of R [( y · · · y f − ) − ] is just R p , we see that the generic rankof the R [( y · · · y f − ) − ]-module gr( A ) ⊗ gr( F J N K ) N is equal to m p ( N ).We finally deduce from Proposition 3.1.4.2 and Lemma 3.1.4.4: Corollary 3.1.4.5.
Let π be an admissible smooth representation of GL ( K ) over F with a central character having at least one good filtration such that the gr( F J I /Z K ) -module gr( π ∨ ) is killed by some power of J . Then we have rk A ( D A ( π ) ´et ) = dim F (( X )) D ∨ ξ ( π ) ≤ m p (gr( π ∨ )) . GL ( Q p f ) We prove that V GL ( π ) (as defined in (16)) contains some copies of a tensor inductionas in Example 2.1.2.1 for certain admissible smooth representations π of GL ( K ) over F (Theorem 3.2.1.1). 121 .2.1 Lower bound for V GL ( π ) : statement We state the main theorem of this section on V GL ( π ) for certain admissible smoothrepresentations π of GL ( K ) over F (Theorem 3.2.1.1). After some simple reductions,this theorem will be proved in §§3.2.2 to 3.2.4.We keep all the previous notation and denote by I K the inertia subgroup ofGal( Q p /K ). We fix an embedding σ ′ : F p f ֒ → F such that σ ′ | F pf = σ (see thevery beginning of §3), and denote by ω f , ω f : I K → F × Serre’s corresponding funda-mental characters of level f and 2 f .We consider ρ : Gal( Q p /K ) → GL ( F ) of the following form up to twist : ρ | I K ∼ = ω P f − j =0 ( r j +1) p j f ⊕ ρ is reducible, ω P f − j =0 ( r j +1) p j f ⊕ ω P f − j =0 ( r j +1) p j + f f if ρ is irreducible, (126)where the integers r i satisfy the following (strong) genericity condition:2 f − ≤ r j ≤ p − − f if j > ρ is reducible,2 f ≤ r ≤ p − − f if ρ is irreducible (127)(note that this implies in particular p ≥ f + 1). Let χ : Gal( Q p /K ) → F × such that( ρ ⊗ χ ) | I K is as in (126) and moreover det( ρ ⊗ χ ) = ω P j ( r j +1) p j f .We refer to [Paš04] and [BP12, §§9,13] (and the references therein) for the back-ground and definitions about diagrams .We choose one diagram D ( ρ ⊗ χ ) = ( D ֒ → D ) associated to ρ ⊗ χ in [Bre11, §5],and we set D ( ρ ) = ( D ( ρ ) ֒ → D ( ρ )) def = (cid:16) D ⊗ ( χ − ◦ det) ֒ → D ⊗ ( χ − ◦ det) (cid:17) , (128)where the actions of GL ( O K ) and the center K × on D ( ρ ) (resp. of I , (cid:16) p (cid:17) and K × on D ( ρ )) are multiplied by χ − ◦ det via local class field theory for K (note that χ istrivial on K and I and recall that (cid:16) p (cid:17) normalizes I and I ). Recall that the actionof GL ( O K ) on D ( ρ ) factors through GL ( O K ) ։ GL ( F q ). More precisely, denotingby W ( ρ ) the set of Serre weights of ρ defined in [BDJ10, §3], D ( ρ ) is the (unique)maximal finite-dimensional representation of GL ( F q ) over F with socle isomorphicto ⊕ σ ∈ W ( ρ ) σ such that each σ ∈ W ( ρ ) occurs with multiplicity 1 in D ( ρ ). Finally K × acts on D ( ρ ) by the character det( ρ ) ω − .If π is an admissible smooth representation of GL ( K ) over F , recall that ( π I ֒ → π K ) is naturally a diagram. We aim to prove the following theorem.122 heorem 3.2.1.1. Let π be an admissible smooth representation of GL ( K ) over F . Assume that there exists an integer r ≥ such that one has an isomorphism ofdiagrams D ( ρ ) ⊕ r ∼ −→ ( π I ֒ → π K ) . Then one has an I Q p -equivariant injection (cid:16) ind ⊗ Q p K ( ρ ) (cid:17) | ⊕ rI Q p ֒ → V GL ( π ) | I Q p . If weassume moreover that the constants ν i associated to D ( ρ ⊗ χ ) at the beginning of [Bre11, §6] are as in [Bre11, Thm.6.4] , then one has a Gal( Q p / Q p ) -equivariant injec-tion (cid:16) ind ⊗ Q p K ( ρ ) (cid:17) ⊕ r ֒ → V GL ( π ) . Let us first make some straightforward reductions. In order not to repeat argu-ments, we assume from now on that the constants ν i associated to D ( ρ ⊗ χ ) in [Bre11,§6] are as in [Bre11, Thm.6.4] and we will prove the last statement of Theorem 3.2.1.1(the proof for the first one being the same up to some trivial modifications). It isenough to prove Theorem 3.2.1.1 for the GL ( K )-subrepresentation of π generated by D ( ρ ) ⊕ r . Hence we can assume that π has a central character which is χ π def = det( ρ ) ω − .Using Remark 2.1.1.4(ii) (for n = 2), it is also enough to prove Theorem 3.2.1.1 for ρ ⊗ χ as above, i.e. we can assume ρ | I K is as in (126) and det( ρ ) = ω P j ( r j +1) p j f . Lemma 3.2.1.2.
With the notation in §2.1.1, it is enough to prove that ( π ⊗ χ − π ) N contains a finite type F J X K [ F ] -submodule M which is admissible as F J X K -module andstable under Z × p such that V ( M ∨ [1 /X ]) ∼ = (cid:16) ind ⊗ Q p K ( ρ ) (cid:17) ⊕ r .Proof. For any F J X K [ F ]-submodule M of π N which is stable under Z × p , denote by M ⊗ χ − π the same F J X K -module but where the action of F is multiplied by χ π ( p ) − and the action of x ∈ Z × p is multiplied by χ π ( x ) − . As ( π ⊗ χ − π ) N = π N as F -vectorsubspaces of π , we can assume that π N contains a finite type F J X K [ F ]-submodule M which is admissible as F J X K -module and stable under Z × p such that V (( M ⊗ χ − π ) ∨ [1 /X ]) ∼ = (cid:16) ind ⊗ Q p K ( ρ ) (cid:17) ⊕ r . From the definition of V G in (16), it is enough toprove V ∨ ( M ∨ [1 /X ]) ⊗ δ G ∼ = (cid:16) ind ⊗ Q p K ( ρ ) (cid:17) ⊕ r . From Example 2.1.1.3 and as in Remark2.1.1.4(ii) (both for n = 2), we have V ∨ ( M ∨ [1 /X ]) ⊗ δ G = V (cid:16) ( M ⊗ χ − π ) ∨ [1 /X ] (cid:17) ∨ ⊗ ( χ π ω ) | Q × p = (cid:16)(cid:16) ind ⊗ Q p K ( ρ ) (cid:17) ⊕ r (cid:17) ∨ ⊗ ind ⊗ Q p K (cid:16) det( ρ ) (cid:17) ( ) = (cid:16) ind ⊗ Q p K ( ρ ) (cid:17) ⊕ r which finishes the proof.The sections that follow will be devoted to the proof that there exists a certainfinite type F J X K [ F ]-submodule M π of π N which is admissible as F J X K -module and123table under Z × p such that V (( M π ⊗ χ − π ) ∨ [1 /X ]) ∼ = (cid:16) ind ⊗ Q p K ( ρ ) (cid:17) ⊕ r (see Proposition3.2.4.6). Note that the assumption det( ρ ) = ω P j ( r j +1) p j f implies χ π ( p ) = 1, so thatthe operator F on M π ⊗ χ − π is the same as on M π , but the action of γ ∈ Z × p nowcomes from the action of (cid:16) γ − (cid:17) on π N . We give some technical results on F J N K , F J N /N K and on certain modules over theserings coming from Serre weights.We let H def = (cid:16) F × q F × q (cid:17) ∼ = I/I ⊆ GL ( F q ) (this finite group H shouldn’t be confusedwith the algebraic group H in §2.1.1 or in §2.1). Note that the trace Tr K/ Q p : O K → Z p is surjective (using that K is unramified) hence directly induces an isomorphism N /N ∼ → Z p . Recall we defined the elements Y i for i ∈ { , . . . , f − } in (100). Wedefine analogously Y def = X a ∈ F × p a − (cid:16) a ]0 1 (cid:17) ∈ F J Z p K = F J N /N K . We write i for an element ( i , . . . , i f − ) in Z f , Y i for Y i · · · Y i f − f − and set k i k def = P f − j =0 i j . We also write i ≤ i ′ to mean i j ≤ i ′ j for 0 ≤ j ≤ f − Lemma 3.2.2.1.
We have the following isomorphisms and equalities: (i) F J N K ∼ = F J Y , . . . , Y f − K and F [ N /N p ] ∼ = F J Y , . . . , Y f − K / ( Y p , . . . , Y pf − ) ; (ii) Y pi (cid:16) p
00 1 (cid:17) = (cid:16) p
00 1 (cid:17) Y i +1 and (cid:16) ˜ λ
00 ˜ µ (cid:17) Y i = ( λµ − ) p i Y i (cid:16) ˜ λ
00 ˜ µ (cid:17) for λ, µ ∈ F × q ; (iii) F J N /N K ∼ = F J Y K and (cid:16) ˜ λ
00 ˜ µ (cid:17) Y = ( λµ − ) Y (cid:16) ˜ λ
00 ˜ µ (cid:17) for λ, µ ∈ F × p .Proof. Note that F [ N /N p ] ∼ = F h(cid:16) F q (cid:17)i . The first equality in (i) and the explicitaction of (cid:16) ˜ λ
00 ˜ µ (cid:17) on Y i in (ii) are immediately obtained from [Mor17, Lemma 3.2](after conjugating by the element (cid:16) p (cid:17) ). The second equality in (i) follows from thefirst by dimension reasons, as Y pi = 0 in F [ N /N p ]. The action of (cid:16) p
00 1 (cid:17) on Y i +1 in(ii) is a direct computation (see also [Mor17, Lemma 5.1]). Finally, (iii) is a specialcase of (i) and (ii).Note that F J N /N K ∼ = F J X K ∼ = F J Y K with X = (cid:16) (cid:17) − H -eigenvariable” Y rather than thevariable X . To compare them the following lemma will be useful.124 emma 3.2.2.2. We have X ∈ − Y (1 + Y F J Y K ) and Y ∈ − X (1 + X F J X K ) in F J N /N K .Proof. Equivalently, we have to prove Y = − X in m / m , where m is the maximalideal of F J N /N K . We can work modulo m p , i.e. in F [ N /N N p ] ∼ = F h(cid:16) F p (cid:17)i . In thatgroup ring we have Y = X a ∈ F × p a − (cid:16) a (cid:17) = p − X a =1 a − (1 + X ) a = − X. For λ, µ ∈ F × q we set α (cid:16)(cid:16) ˜ λ
00 ˜ µ (cid:17)(cid:17) def = λµ − ∈ F . Remark 3.2.2.3.
By Lemma 3.2.2.1(ii), if V is a representation of GL ( F q ) and v ∈ V H = χ , then Y i v ∈ V H = χα i , where α i def = α P f − j =0 i j p j . Lemma 3.2.2.4.
Assume p > . The kernel of the map h : F J N K ։ F J N /N K isgenerated by the elements Y i − Y j ( i = j ) . Moreover, there exists f ( Y ) ∈ F J N /N K ∼ = F J Y K such that h ( Y i ) = Y + Y p f ( Y ) .Proof. Note that Tr K/ Q p (˜ λ p i ) = Tr K/ Q p (˜ λ ) for all λ ∈ F × q and i ∈ Z , hence Y i − Y j ∈ Ker( h ). As F J N K / ( Y i − Y j , i = j ) and F J N /N K are both power series rings inone variable, the quotient map F J N K / ( Y i − Y j , i = j ) ։ F J N /N K has to be anisomorphism. To establish the final claim it suffices to prove that the image of Y in F J Y K / ( Y p ) ∼ = F [ N /N N p ] ∼ = F h(cid:16) F p (cid:17)i is Y . We compute X λ ∈ F × q λ − (cid:16) F q/ F p ( λ )0 1 (cid:17) = X a ∈ F p X λ ∈ F × q Tr F q/ F p ( λ )= a λ − !(cid:16) a (cid:17) . (129)If a = 0, we sum over the distinct roots of Y p f − + Y p f − + · · · + Y − a = 0, so theinside sum on the right hand side of (129) equals 1 /a (from the last two coefficients).If a = 0 we sum over the distinct roots of Y p f − − + · · · + Y p − + 1 = 0, so the insidesum in (129) equals 0 as p >
2. Hence the right-hand side of (129) is just Y .By Lemma 3.2.2.4, if V is a representation of GL ( F q ), then Y i = Y on V N .For 0 ≤ i ≤ q −
1, we set θ i def = X λ ∈ F q λ i (cid:16) λ (cid:17) ∈ F [ N /N p ] ∼ = F h(cid:16) F q (cid:17)i . So Y i = θ q − − p i in F [ N /N p ]. 125 emma 3.2.2.5. Suppose i ∈ { , . . . , p − } f and let i def = P f − j =0 i j p j . (i) We have θ i = ( − f − f − Y j =0 i j ! ! Y p − − i in F [ N /N p ] for ≤ i < q − . (ii) For f , . . . , f q − and φ as defined in [BP12, §2] we have f i = ( − f − f − Y j =0 i j ! ! Y p − − i (cid:16) (cid:17) φ for ≤ i < q − .Proof. Part (i) follows from [Mor, Lemma 0.2] after conjugation by (cid:16) p (cid:17) . Indeed,in the notation of loc.cit. we take m = n = 1 (so that A , is the group algebra of (cid:16) p O K /p O K (cid:17) ): we see that θ i corresponds (under conjugation) to F i if 0 ≤ i ≤ q − κ p − − i equals ( − f − (cid:16) Q f − j =0 i j ! (cid:17) − . Part (ii) follows immediatelyfrom (i) and the definition of θ i .As in [BP12] we write ( s , s , . . . , s f − ) ⊗ η for the Serre weightSym s F ⊗ F (Sym s F ) Fr ⊗ · · · ⊗ F (Sym s f − F ) Fr f − ⊗ F η ◦ det , where the s i are integers between 0 and p − η is a character F × q → F × and GL ( F q )acts on (Sym s i F ) Fr i via σ i : F q ֒ → F . If χ = χ ⊗ χ is a character of H = (cid:16) F × q F × q (cid:17) ,we let χ s def = χ ⊗ χ . Lemma 3.2.2.6.
Let σ def = ( s , . . . , s f − ) ⊗ η , s def = ( s , s , . . . , s f − ) ∈ { , . . . , p − } f ,and fix v ∈ σ N , v = 0 . Let χ σ denote the H -eigencharacter on σ N . (i) The F J N /N K = F J Y K -module σ N is cyclic of dimension min { s , . . . , s f − } +1 . (ii) If ≤ i ≤ s and i < p − then σ contains a unique H -eigenvector Y − i v that is sent by Y i to v . The corresponding H -eigencharacter is χ σ α − i . Also, Y j Y − i v = 0 if i j = 0 . (iii) If ≤ i ≤ min { s , . . . , s f − } and i < p − , then σ N contains a unique (cid:16) F × p F × p (cid:17) -eigenvector Y − i v that is sent by Y i to v . The corresponding eigencharacter is χ σ α − i . We have Y − i v = P i, k i k = i Y − i v . roof. (i) Note that σ N is a torsion module over F J N /N K = F J Y K as σ N is finite-dimensional. To show cyclicity it suffices to note that σ N = σ N [ X ] is 1-dimensional.Then from [Mor17, Prop. 3.3] applied with n = 1 we have an isomorphism F J Y , . . . , Y f − K / ( Y s j +1 j , ≤ j ≤ f − ∼ −→ σg ( Y ) g ( Y ) (cid:16) (cid:17) v. (130)(Restrict equation (9) in [Mor17] to (cid:16) p O K (cid:17) and conjugate by (cid:16) p (cid:17) . Note that σ isself-dual up to twist.) In particular, { Y k (cid:16) (cid:17) v, ≤ k ≤ s } is a basis of σ consistingof H -eigenvectors.Let m def = min { s , . . . , s f − } . We claim that the vectors v i def = X ≤ k ≤ s k k k = k s k− i Y k (cid:16) (cid:17) v ≤ i ≤ m (131)form a basis of σ N . If i < m and k k k = k s k − i , then k j > j . Byusing also (130) we see that v i = Y j v i +1 . Also, Y j v = 0 for all j . In particular, Y j − Y j ′ annihilates v i for all i , so v i ∈ σ N by Lemma 3.2.2.4. Moreover, Xv i +1 = v i (0 ≤ i < m ) and Xv = 0. It remains to show that v m / ∈ Xσ N . Choose j suchthat s j = m . Then Q j = j Y s j j (cid:16) (cid:17) v is the only term appearing in the sum (131) for i = m that is not divisible by Y j . Hence v m / ∈ Y j σ , and thus v m / ∈ Xσ N .(ii) Let v ′ def = Y s (cid:16) (cid:17) v , which is a scalar multiple of v . By (130), (cid:16) Y k (cid:16) (cid:17) v (cid:17) ≤ k ≤ s forms a basis of σ consisting of H -eigenvectors with eigencharacters χ sσ α k = χ σ α k − r .The eigencharacters are pairwise distinct, except if s = p − Y p − (cid:16) (cid:17) v and (cid:16) (cid:17) v have the same eigencharacter. Hence, as i < p −
1, the unique H -eigenvectorin the preimage ( Y i ) − ( v ′ ) is Y s − i (cid:16) (cid:17) v . Note also that Y j Y s − i (cid:16) (cid:17) v = 0 if i j = 0by (130).(iii) Using the notation in (ii), we have v i = P k i k = i Y − i v ′ for 0 ≤ i ≤ m and itis a (cid:16) F × p F × p (cid:17) -eigenvector with eigencharacter χ σ α − i . These characters for 0 ≤ i ≤ m are pairwise distinct, except if s = p −
1, in which case v and v p − have the sameeigencharacter. As we assume i < p − Lemma 3.2.2.7.
Suppose V is a representation of GL ( F q ) generated by some vector v ∈ V N that is an eigenvector for the action of H . If dim F V ≤ q , then the map F J Y , . . . , Y f − K Vf ( Y ) f ( Y ) (cid:16) (cid:17) v is surjective and its kernel is generated by monomials. In particular, if Y i (cid:16) (cid:17) v = Y j (cid:16) (cid:17) v = 0 , then i = j . roof. Let χ denote the eigencharacter of H on v . Then we have a GL ( F q )-equiva-riant surjection S : Ind GL ( O K ) I ( χ ) ։ V sending φ to v , where φ is the unique functionsupported on I which sends 1 to 1. Consider i : F [ Y , . . . , Y f − ] / ( Y p , . . . , Y pf − ) → Ind GL ( O K ) I ( χ ) sending f ( Y ) to f ( Y ) (cid:16) (cid:17) φ . By Lemma 3.2.2.5, f j ∈ Im( i ) for all j (even if j = q − GL ( O K ) I ( χ ) = Im( i ) ⊕ F φ (as F -vectorspaces) and i is injective.Suppose first χ = χ s . By [BP12, Lemma 2.7(i)] and as dim V ≤ q we have f r ± φ ∈ Ker( S ) for some r = P f − j =0 p j s j ∈ { , . . . , q − } and some sign ± (both dependingon χ ), so S ◦ i is surjective. If Ker( S ) is irreducible (as a GL ( F q )-representation),then by [BP12, Lemma 2.7], Ker( S ) = h f P p j d j , ≤ d j ≤ s j (not all equal) , f r ± φ i F .Intersecting with Im( i ) = h f P p j d j , ≤ d j ≤ p − i F we getKer( S ) ∩ Im( i ) = D f P p j d j , ≤ d j ≤ s j (not all equal) E F . By Lemma 3.2.2.5(ii), it follows in particular that Ker( S ◦ i ) is generated by mono-mials. If Ker( S ) is reducible, the argument is analogous using [BP12, Lemma 2.7(ii)].If χ = χ s , it is again almost identical, using [BP12, Lemma 2.6] instead. Lemma 3.2.2.8.
Suppose f > . In F [ N /N p ] we have X λ ∈ F q , Tr F q/ F p ( λ )=0 (cid:16) λ (cid:17) = ( − f − Y p − + X k i k =( p − f − ≤ i j ≤ p − Y i ! . Proof.
First we have (using x p − = 1 if x ∈ F × p ): X λ ∈ F q , Tr F q/ F p ( λ ) =0 (cid:16) λ (cid:17) = X λ ∈ F q (Tr F q / F p ( λ )) p − (cid:16) λ (cid:17) = X λ ∈ F q ( λ + λ p + · · · + λ p f − ) p − (cid:16) λ (cid:17) = X λ ∈ F q X i ∈ Z f ≥ k i k = p − ( p − Q j i j ! λ i + i p + ··· + i f − p f − (cid:16) λ (cid:17) = X i ∈ Z f ≥ k i k = p − ( p − Q j i j ! ( − f − Y j i j ! ! Y p − − i , where the last equality follows from Lemma 3.2.2.5(i), noting that P f − j =0 i j p j < q − f >
1. Letting i ′ def = p − − i we get (as ( p − − F p ): X λ ∈ F q , Tr F q/ F p ( λ ) =0 (cid:16) λ (cid:17) = ( − f X i ′ ∈ Z f ≥ k i ′ k =( p − f − Y i ′ . X λ ∈ F q (cid:16) λ (cid:17) = ( − f − Y p − . The result follows.
Proposition 3.2.2.9.
Fix j ∈ { , . . . , f − } . In F J N /N p K ∼ = F J Y , . . . , Y f − K / (cid:16) ( Y i − Y j ) p , i = j (cid:17) we have X n ∈ N /N p n = ( − f − Y j = j ( Y j − Y j ) p − modulo terms of degree ≥ f ( p − .Proof. The statement being trivial if f = 1, we can assume f >
1. We prove thefirst isomorphism. As Y i − Y j ∈ Ker (cid:16) F J N K → F J N /N K (cid:17) by Lemma 3.2.2.4, wededuce that ( Y i − Y j ) p ∈ Ker (cid:16) F J N K → F J N /N p K (cid:17) , and we thus have a surjection F J Y , . . . , Y f − K / (cid:16) ( Y i − Y j ) p , i = j (cid:17) ։ F J N /N p K . Since both terms are free modulesof rank p ( f −
1) over a power series ring in one variable over F , the surjection has tobe an isomorphism.Let A def = F J N /N p K , B def = F J N /N p K and B def = F J N /N p K , they are completelocal commutative rings of respective maximal ideals denoted by m A , m B , m B . Let Z def = P n ∈ N /N p n ∈ A . As A ∼ = F [ Z , . . . , Z f − ] / ( Z p , . . . , Z pf − ) and Z is killed by m A (as N /N p is a group) we deduce that Z ∈ m ( p − f − A . Note that m ( p − f − A = 0.Let ı : A ֒ → B denote the inclusion and denote by gr m ( ı ) the induced map m mA / m m +1 A → m mB / m m +1 B for m ≥
0. We claim that gr ( ı ) is injective with imagegenerated by all Y j − Y j ( j = j ) in m B / m B . If so, then gr ( p − f − ( ı ) has to send the1-dimensional F -vector space m ( p − f − A to a multiple of Q j = j ( Y j − Y j ) p − modulo m ( p − f − B . But (cid:16) ˜ λ
00 ˜ µ (cid:17) Z = Z (cid:16) ˜ λ
00 ˜ µ (cid:17) for λ, µ ∈ F × p , and considering the action of H ,it follows from the sentence following Lemma 3.2.2.1 that we must have ı ( Z ) = c Y j = j ( Y j − Y j ) p − + (element of m f ( p − B )for some c ∈ F (note that every element of B can be written uniquely as P i c i Y i with i j < p for all j = j and that m B is generated by the Y i , i = 0). By passing to B andusing Lemma 3.2.2.8, we deduce that we must have c = ( − f − .It remains to prove the claim. As B ∼ = B/ ( Y p , . . . , Y pf − ), we have m B / m B ∼ → m B / m B and it is equivalent to prove the claim with ı : A → B . As N /N p ∼ = F f − p N /N p ∼ = F fp , it is clear that ı is injective. Consider the natural map s : B ։ C def = F [ N /N N p ] ∼ = F [ Y ] / ( Y p ). As gr ( s ◦ ı ) = 0 and s ( Y i ) = Y by Lemma 3.2.2.4,we deduce from loc.cit. that the image of gr ( ı ) is indeed spanned by all Y j − Y j ( j = j ). F We give a crucial computation for the operator F on π N for π as at the end of §3.2.1.The main result of this section is Proposition 3.2.3.1(ii).We keep the notation of §3.2.2. For σ = ( t , . . . , t f − ) ⊗ η ∈ W ( ρ ), recall wehave t j ∈ { r j , r j + 1 , p − − r j , p − − r j } if j > ρ is reducible and t ∈{ r − , r , p − − r , p − − r } if ρ is irreducible (see e.g. [Bre11, §2]). We deducefrom (127) that t j ∈ { f − , . . . , p − − f } for all j . (132)We identify W ( ρ ) with the subsets of { , , . . . , f − } as in [Bre11, §2] and let J σ ⊆ { , . . . , f − } be the subset associated to σ . We have t j ∈ { p − − r j , p − − r j } for j ∈ J σ if j > ρ is reducible, t ∈ { p − − r , p − − r } if 0 ∈ J σ and ρ isirreducible.Let σ = ( t , . . . , t f − ) ⊗ η ∈ W ( ρ ). Denote δ ( σ ) def = δ red ( σ ) if ρ is reducible and δ ( σ ) def = δ irr ( σ ) if ρ is irreducible the Serre weights δ red ( σ ), δ irr ( σ ) defined in [Bre11,§5]. We write δ ( σ ) = ( s , . . . , s f − ) ⊗ η ′ . Let x σ ∈ σ N \ { } and let χ σ : H → F × denote the H -eigencharacter of x σ . We also identify the irreducible constituentsof Ind GL ( O K ) I ( χ sσ ) with the subsets of { , . . . , f − } as in [BP12, §2] (for instance ∅ corresponds to the socle σ of Ind GL ( O K ) I ( χ sσ )). For any J ⊆ { , . . . , f − } let Q ( χ sσ , J )denote the unique quotient of Ind GL ( O K ) I ( χ sσ ) whose GL ( O K )-socle is parametrizedby J (see [BP12, Thm.2.4(iv)]). We know that the Serre weight δ ( σ ) occurs inInd GL ( O K ) I ( χ sσ ) (see the proof of [Bre11, Prop.5.1]) and we denote by J max ( σ ) ⊆{ , . . . , f − } the associated subset. We thus havesoc GL ( O K ) Q ( χ sσ , J max ( σ )) ∼ = δ ( σ )(by definition of δ ( σ ), it is the only constituent of Q ( χ sσ , J max ( σ )) that is in W ( ρ )).We also have from [BP12, §2] (with − f − s j = p − − t j + J max ( σ ) ( j −
1) if j ∈ J max ( σ ) ,s j = t j − J max ( σ ) ( j −
1) if j / ∈ J max ( σ ) . (133)Moreover, using [BP12, Lemma 2.7] it is a combinatorial exercise (left to the reader)to prove J max ( σ ) = ( J σ ∪ J δ ( σ ) ) \ ( J σ ∩ J δ ( σ ) ) . (134)130e define m def = | J max ( σ ) | ∈ { , . . . , f } . We have m = 0 if and only if δ ( σ ) ∼ = σ , and this occurs precisely if ρ is reducible and σ is an “ordinary” Serre weight of ρ , i.e. such that J σ = ∅ or J σ = { , . . . , f − } (thisfollows, for example, from the proof of Lemma 3.2.3.2 below).We consider a GL ( K )-representation π as at the end of §3.2.1, and fix an embed-ding σ ֒ → soc GL ( O K ) ( π ) (recall there are r copies of σ inside soc GL ( O K ) ( π )). Fromthe assumption on π , we know that (cid:16) p (cid:17) x σ generates Q ( χ sσ , J max ( σ )) as GL ( O K )-subrepresentation of π | GL ( O K ) , in particular δ ( σ ) can also be seen in soc GL ( O K ) ( π )(its embedding being determined by that of σ ). Proposition 3.2.3.1. (i)
The vector x δ ( σ ) def = Y j ∈ J max ( σ ) Y s j j Y j / ∈ J max ( σ ) Y p − j (cid:16) p
00 1 (cid:17) x σ (135) spans δ ( σ ) N as F -vector space. (ii) We have in π N that Y P j ∈ J max( σ ) s j F ( Y − m x σ ) = ( − f − Y − m x δ ( σ ) if m > , Y p − F ( x σ ) = ( − f − x δ ( σ ) if m = 0 .Proof of Proposition 3.2.3.1(i). Suppose first m >
0. From [BP12, Lemma 2.7(ii)]and Lemma 3.2.2.5(ii) we see that δ ( σ ) has basis Y i (cid:16) p
00 1 (cid:17) x σ , where 0 ≤ i j ≤ s j if j ∈ J max ( σ ) and p − − s j ≤ i j ≤ p − j / ∈ J max ( σ ). Hence the only vectorsin δ ( σ ) that are killed by all Y j are the multiples of x δ ( σ ) . The statement follows byan inspection of the H -action on this basis (which is formed by H -eigenvectors), seeRemark 3.2.2.3.If m = 0, then δ ( σ ) is the socle of Ind GL ( O K ) I ( χ sσ ). By [BP12, Lemma 2.7(i)], f is the unique I -invariant element of δ ( σ ) ⊆ Ind GL ( O K ) I ( χ sσ ). The statement followsfrom Lemma 3.2.2.5(ii).In order to prove Proposition 3.2.3.1(ii), we first need several lemmas. Lemma 3.2.3.2.
We have | J max ( σ ) | = | J max ( δ ( σ )) | .Proof. If ρ is reducible, identifying { , . . . , f } with Z /f we have J δ ( σ ) = J σ − Z /f by [Bre11, §5], and the statement follows in that case by (134). If ρ is irreducible, let J ′ σ def = J σ ` ( J σ + f ) ⊆ { , . . . , f − } as in [Bre11, §5], where J σ J σ in { , . . . , f − } . It follows from (134) that | J max ( σ ) | = | ( J ′ σ ∪ J ′ δ ( σ ) ) \ ( J ′ σ ∩ J ′ δ ( σ ) ) | . Identifying { , . . . , f − } with Z / f , we again have J ′ δ ( σ ) = J ′ σ − Z / (2 f ) by [Bre11, §5], and the statement follows.The three lemmas that follow only apply to m >
0. In these three lemmas, weidentify without comment { , . . . , f − } with Z /f Z (so − f − f = 0, etc.). Lemma 3.2.3.3.
Assume m > and let i ∈ Z f ≥ with k i k ≤ m − . Then we have D GL ( O K ) (cid:16) p
00 1 (cid:17) Y − i x σ E (cid:30) X ≤ j
00 1 (cid:17) Y − j x σ E ∼ = Q (cid:16) χ sσ α i , { j ∈ J max ( σ ) , i j +1 = 0 } (cid:17) . (136) Proof.
Note first that t j ∈ { i j + 1 , . . . , p − } for all j by (132) and the assumptionon i , so that the vectors Y − i x σ and Y − j x σ are well-defined elements of σ by Lemma3.2.2.6(ii). We rewrite h GL ( O K ) (cid:16) p
00 1 (cid:17) Y − j x σ i = h GL ( O K ) (cid:16) p (cid:17) Y − j x σ i and, usingnotation from [BHH + , §§2.1,2.2], σ ∼ = F ( λ ) where λ = ( λ , . . . , λ f − ) with λ j =( λ j, , λ j, ) ∈ { , . . . , p − } . We have λ j, − λ j, = t j for all j .Let W ′ (resp. W ) be the I -subrepresentation of π generated by Y − i x σ (resp. (cid:16) p (cid:17) Y − i x σ ). We deduce from Lemma 3.2.2.6(ii) that W ′ = h N Y − i x σ i has F -basis Y − j x σ for all 0 ≤ j ≤ i , and soc I ( W ′ ) = F x σ . We moreover have W = (cid:16) p (cid:17) W ′ since I is normalized by (cid:16) p (cid:17) . In particular we see that W injects into the I -representation J χ σ of [BHH + , Cor.6.1.4] and that W has Jordan–Hölder factors χ sσ α j for 0 ≤ j ≤ i , each occurring with multiplicity 1. Let V def = Ind GL ( O K ) I ( W ). Then V isthe representation appearing in the first paragraph of the proof of [BHH + , Prop.6.2.2],with B j taken to be 2 i j + 1 for all j (and note the bounds on λ j, − λ j, which let usinvoke loc.cit. ). Hence, by [BHH + , Prop.6.2.2] and its proof in the case ε j = − B j = 2 i j + 1 for all j , we get that V is multiplicity-free, has Jordan–Hölder factors σ a def = F ( t λ ( − P a j η j )) for 0 ≤ a ≤ i + 1 with the notation of [BHH + , §2.4], andGL ( O K )-socle σ . Moreover, the unique subrepresentation of V with cosocle σ a hasconstituents σ b for 0 ≤ b ≤ a . On the other hand, Ind GL ( O K ) I ( W ) has a filtrationwith subquotients Ind GL ( O K ) I ( χ sσ α j ) for 0 ≤ j ≤ i , and by [BHH + , Lemma 6.2.1(i)]the constituents of Ind GL ( O K ) I ( χ sσ α j ) are the Serre weights σ a with 2 j ≤ a ≤ j + 1.By the proof of [BHH + , Lemma 6.2.1(i)], one easily checks that the constituent σ a ofInd GL ( O K ) I ( χ sσ α j ) corresponds to the subset { ℓ, a ℓ +1 is odd } ⊆ { , . . . , f − } in theparametrization of [BP12, §2] (note that twisting χ sσ by α j corresponds to shifting by − P j ℓ η ℓ in the extension graph).By Frobenius reciprocity V def = h GL ( O K ) (cid:16) p (cid:17) Y − i x σ i is the image of a nonzeromap Ind GL ( O K ) I ( W ) → π and any Serre weight in its GL ( O K )-socle has to be in132 ( ρ ). By [BHH + , Prop.2.4.2] if σ a ∈ W ( ρ ), then 0 ≤ a ≤
1, so σ a is a constituentof Ind GL ( O K ) I ( χ sσ ) ⊆ V . Thus by the definition of δ ( σ ) and as π K / soc GL ( O K ) π doesnot contain any Serre weight of W ( ρ ) it follows that V is the unique quotient of V with GL ( O K )-socle δ ( σ ). By the previous paragraph and the definition of J max ( σ ),we have δ ( σ ) ∼ = σ b , where b j = J max ( σ )+1 ( j ) for all j , and V has constituents σ a with J max ( σ )+1 ( j ) ≤ a j ≤ i j + 1 for all j . By construction, the left-hand side of (136) is aquotient of Ind GL ( O K ) I ( χ sσ α i ). Moreover, by what is before, it must have constituents σ a with max( J max ( σ )+1 ( j ) , i j ) ≤ a j ≤ i j + 1 for all j . It follows that its GL ( O K )-socle is irreducible and isomorphic to σ c , where c j def = max( J max ( σ )+1 ( j ) , i j ) for all j .Since 2 i j +1 is even and > i j +1 = 0, we see that c j +1 is odd if and only if i j +1 = 0. Hence the GL ( O K )-socle of this quotient of Ind GL ( O K ) I ( χ sσ α i ) correspondsto the subset { j ∈ J max ( σ ) , i j +1 = 0 } , as required. Lemma 3.2.3.4.
Assume m > and let i ∈ Z f ≥ , ℓ ∈ J max ( σ ) such that k i k ≤ m − and i ℓ +1 = 0 . Then Y p − t ℓ +2 i ℓ ℓ (cid:16) p
00 1 (cid:17) Y − i x σ = 0 . Proof.
Recall p − t ℓ + 2 i ℓ ≥ Y p − t ℓ +2 i ℓ ℓ (cid:16) p
00 1 (cid:17) Y − i x σ is well-defined.Suppose on the contrary that Y p − t ℓ +2 i ℓ ℓ (cid:16) p
00 1 (cid:17) Y − i x σ = 0 for some ℓ ∈ J max ( σ ) suchthat i ℓ +1 = 0 and k i k ≤ m −
1. By Lemma 3.2.2.1(ii) and Lemma 3.2.2.6(ii) this is aneigenvector for { (cid:16) ˜ λ
00 ˜ µ (cid:17) , λ, µ ∈ F × q } with eigencharacter χ σ α − i α ( p − t ℓ +2 i ℓ ) p ℓ . By Lemma3.2.3.3 it suffices to show that the H -eigencharacter χ σ α − i α ( p − t ℓ +2 i ℓ ) p ℓ does not occurin V i ′ def = Q ( χ sσ α i ′ , J i ′ )for any i ′ such that 0 ≤ i ′ ≤ i , where J i ′ def = { j ∈ J max ( σ ) , i ′ j +1 = 0 } .Using the notation λ = ( λ ( x ) , . . . , λ f − ( x f − )) and P ( x , . . . , x f − ) of [BP12,Thm.2.4], the irreducible constituents of V i ′ are given by the Serre weights ( λ ( t − i ′ ) , . . . , λ f − ( t f − − i ′ f − )) (up to twist) for those λ ∈ P ( x , . . . , x f − ) such that J ( λ ) ⊇ J i ′ . Recall that λ j ( x ) = p − − x + J ( λ ) ( j −
1) if j ∈ J ( λ ) and λ j ( x ) = x − J ( λ ) ( j −
1) if j / ∈ J ( λ ). By [BP12, Lemma 2.5(i)] and [BP12, Lemma 2.7], the H -eigencharacters that occur in V i ′ are χ σ α − i ′ α k , where0 ≤ k j ≤ p − − ( t j − i ′ j ) + J ( λ ) ( j −
1) if j ∈ J ( λ ) ,p − − ( t j − i ′ j − J ( λ ) ( j − ≤ k j ≤ p − j / ∈ J ( λ ) . (137)(Note that J i ′ = ∅ as k i ′ k ≤ m − ≤ f − χ σ α − i α ( p − t ℓ +2 i ℓ ) p ℓ = χ σ α − i ′ α k for some λ and k as above. Then − f − X j =0 i j p j + ( p − t ℓ + 2 i ℓ ) p ℓ ≡ − f − X j =0 i ′ j p j + f − X j =0 k j p j (mod q − p − t ℓ + 2 i ℓ ) p ℓ − f − X j =0 ( i j − i ′ j ) p j ≡ f − X j =0 k j p j (mod q − . (138)Note that ℓ ∈ J i ′ , as 0 ≤ i ′ ℓ +1 ≤ i ℓ +1 = 0 and ℓ ∈ J max ( σ ), so ℓ ∈ J ( λ ).If i ′ j = i j for all j = ℓ (for example if i ′ = i or if f = 1), then (138) gives( p − t ℓ + i ℓ + i ′ ℓ ) p ℓ ≡ P j k j p j , so k ℓ = p − t ℓ + i ℓ + i ′ ℓ as (using (132) for t ℓ and i ′ ℓ ≤ i ℓ ≤ m − ≤ f − p − t j + i j + i ′ j ∈ { i j + i ′ j , . . . , p − − ( i j − i ′ j ) } . (139)This contradicts (137) as ℓ ∈ J ( λ ) and i ′ ℓ ≤ i ℓ . Therefore f > i ′ j < i j for some j = ℓ . For m ∈ Z ≥ , let [ m ] the unique element of { , . . . , f − } which is congruentto m modulo f . In particular p m ≡ p [ m ] (mod q − h ∈ { ℓ + 1 , . . . , ℓ + f − } be minimal such that i ′ [ h ] < i [ h ] . Then modulo q − f − X j =0 ( i j − i ′ j ) p j ≡ ℓ + f X j = ℓ +1 ( i [ j ] − i ′ [ j ] ) p [ j ] = ℓ + f X j = h ( i [ j ] − i ′ [ j ] ) p [ j ] and we deduce the following congruences modulo q − p − t ℓ + 2 i ℓ ) p ℓ − f − X j =0 ( i j − i ′ j ) p j ≡ ( p − − t ℓ + 2 i ℓ ) p ℓ + p ℓ − ℓ + f X j = h ( i [ j ] − i ′ [ j ] ) p [ j ] ≡ ( p − − t ℓ + 2 i ℓ ) p ℓ + ℓ + f − X j = h +1 ( p − p j + p h +1 − ℓ + f X j = h ( i [ j ] − i ′ [ j ] ) p [ j ] ≡ ( p − − t ℓ + 2 i ℓ ) p ℓ + ℓ + f − X j = h +1 ( p − p [ j ] + p [ h ]+1 − ℓ + f X j = h ( i [ j ] − i ′ [ j ] ) p [ j ] ≡ ( p − − t ℓ + i ℓ + i ′ ℓ ) p ℓ + ℓ + f − X j = h +1 ( p − − ( i [ j ] − i ′ [ j ] )) p [ j ] + ( p − ( i [ h ] − i ′ [ h ] )) p [ h ] . (140)Note that all powers of p in (140) are distinct in { , . . . , f − } and all coefficients arein { , . . . , p − } . Moreover these coefficients cannot all equal 0 as p − ( i [ h ] − i ′ [ h ] ) = 0,nor p − k ℓ = p − − t ℓ + i ℓ + i ′ ℓ . As ℓ ∈ J ( λ ) and i ′ ℓ ≤ i ℓ , we get from (137) that i ℓ = i ′ ℓ and ℓ − ∈ J ( λ ). By (137) for j = ℓ − p − − ( i ℓ − − i ′ ℓ − ) ≤ k ℓ − ≤ p − − t ℓ − + 2 i ′ ℓ − (note that by (140) the left-hand side is an equality as soon as ℓ − = h mod f whichcan only occur if f > t ℓ − ≤ i ℓ − + i ′ ℓ − ≤ m − ≤ f −
2, whichcontradicts genericity (132). This finishes the proof.134 emma 3.2.3.5.
Assume m > and let k ∈ Z f ≥ . (i) If Y k (cid:16) p
00 1 (cid:17) Y − m x σ = 0 , then k k k ≤ ( f − p −
1) + ( m −
1) + X j ∈ J max ( σ ) s j . If moreover equality holds, then Y k (cid:16) p
00 1 (cid:17) Y − m x σ = x δ ( σ ) ( see (135)) and k j ≡ s j (mod p ) if j ∈ J max ( σ ) , k j ≡ − p ) if j / ∈ J max ( σ ) . (ii) If k k k = ( f − p − P J max ( σ ) s j then Y k (cid:16) p
00 1 (cid:17) Y − m x σ ∈ δ ( σ ) , more precisely: Y k (cid:16) p
00 1 (cid:17) Y − m x σ ∈ D Y − ℓ x δ ( σ ) , k ℓ k = m − E F . Proof.
We prove the following statements inductively on k i k ≤ m − i ∈ Z f ≥ :(a) If Y k (cid:16) p
00 1 (cid:17) Y − i x σ = 0 then k k k ≤ ( f − p −
1) + ( m −
1) + X j ∈ J max ( σ ) s j − ( m − − k i k ) p. If moreover equality holds, then Y k (cid:16) p
00 1 (cid:17) Y − i x σ = x δ ( σ ) and k j = i j +1 p + s j if j ∈ J max ( σ ), k j = i j +1 p + ( p −
1) if j / ∈ J max ( σ ).(b) If k k k = ( f − p −
1) + P j ∈ J max ( σ ) s j − ( m − − k i k ) p then Y k (cid:16) p
00 1 (cid:17) Y − i x σ = Y − ℓ x δ ( σ ) for some k ℓ k = m −
1, or it is zero.By Lemma 3.2.2.6(iii) we have Y − m x σ = X i ∈ Z f ≥ k i k = m − Y − i x σ and we see that (a) and (b) for k i k = m − Y k (cid:16) p
00 1 (cid:17) Y − i x σ = 0 and equality holds, then i is uniquely determined by k and J max ( σ )). 135e first prove by induction on k i k ≤ m − i ∈ Z f ≥ that if k k k ≥ ( f − p − P J max ( σ ) s j − ( m − −k i k ) p and Y k (cid:16) p
00 1 (cid:17) Y − i x σ = 0, then Y k (cid:16) p
00 1 (cid:17) Y − i x σ = Y k ′ (cid:16) p
00 1 (cid:17) x σ for k ′ ∈ Z f ≥ such that k ′ j = k j − i j +1 p for all j . A examination of (a) and (b) showsit will then be enough to prove them for i = 0 (replacing k by k ′ ).There is nothing to prove for i = 0, so we can assume i = 0. If k j ≥ p for some j , then using Lemma 3.2.2.1(ii): Y k (cid:16) p
00 1 (cid:17) Y − i x σ = Y k − pε j Y pj (cid:16) p
00 1 (cid:17) Y − i x σ = Y k − pε j (cid:16) p
00 1 (cid:17) Y − ( i − ε j ) x σ , where ε j def = (0 , . . . , , , , . . . ,
0) with 1 in position j and 0 elsewhere (note that Y j +1 Y − i x σ = Y − ( i − ε j ) x σ is nonzero by assumption, and hence i − ε j +1 ∈ Z f ≥ bythe last statement in Lemma 3.2.2.6(ii)). As k i − ε j +1 k = k i k − k k − pε j k = k k k − p ≥ ( f − p −
1) + P J max ( σ ) s j − ( m − − k i − ε j +1 k ) p , we can apply theinduction hypothesis and a small computation shows that k ′ is the right one, so weare done in that case.We assume k j < p for all j and derive below a contradiction (so this case can’thappen). Define J def = { j ∈ J max ( σ ) , i j +1 = 0 } , then by Lemma 3.2.3.4 (applied to ℓ = j and using Y jk j (cid:16) p
00 1 (cid:17) Y − i x σ = 0): k j ≤ p − − t j + 2 i j if j ∈ J , k j ≤ p − j / ∈ J , which implies k k k ≤ ( f − | J | )( p −
1) + P j ∈ J ( p − − t j + 2 i j ). From (133) we deduce k k k ≤ ( p − f − | J | ) + X j ∈ J ( s j + 2 i j ) + | J \ ( J max ( σ ) + 1) | . So to get a contradiction it is enough to show that( p − f − | J | ) + X j ∈ J ( s j + 2 i j ) + | J \ ( J max ( σ ) + 1) | < ( p − f − X j ∈ J max ( σ ) s j − ( m − − k i k ) p, or equivalently pm + | J \ ( J max ( σ ) + 1) | ≤ ( p − | J | + p X j J i j + ( p − X j ∈ J i j + X j ∈ J max ( σ ) \ J s j = ( p − k i k + ( p − | J | + (cid:18) X j J i j + X j ∈ J max ( σ ) \ J s j (cid:19) . (141)136ase 1: assume | J max ( σ ) \ J | > j ∈ J max ( σ ) \ J , then i j +1 >
0, so | J max ( σ ) \ J | ≤ k i k . As | J max ( σ ) \ J | = m − | J | ,this means m ≤ k i k + | J | , hence (141) is implied by2 m + | J \ ( J max ( σ ) + 1) | ≤ | J | + (cid:18) X j J i j + X j ∈ J max ( σ ) \ J s j (cid:19) . (142)Using | J \ ( J max ( σ ) + 1) | ≤ | J | , (142) is implied by2 m ≤ X j ∈ J max ( σ ) \ J s j . (143)Genericity (132) with (133) give s j ≥ f − ≥ m − j ∈ J max ( σ ), hence (143)holds if either s j ≥ m for at least one j ∈ J max ( σ ) \ J or if | J max ( σ ) \ J | ≥ m − ≥ J max ( σ ) \ J = { j } (for some j ) and moreover J \ ( J max ( σ ) + 1) = J and i j = 0 forall j J . But then i j +1 > j + 1 ∈ J ∩ ( J max ( σ ) + 1), which contradicts J ∩ ( J max ( σ ) + 1) = ∅ . Hence inequality (142) holds.Case 2: assume J max ( σ ) = J .Then using | J \ ( J max ( σ )+1) | ≤ |{ , . . . , f − }\ ( J max ( σ )+1) | = |{ , . . . , f − }\ J max ( σ ) | = f − m and | J | = m , we see that (141) is implied by ( p − m + f ≤ ( p − k i k + ( p − m which is true as k i k > f ≤ p − i = 0, which weprove now.Recall D GL ( O K ) (cid:16) p (cid:17) x σ E ∼ = Q ( χ sσ , J max ( σ )). By [BP12, Thm.2.4(iv)] the con-stituents of this GL ( O K )-representation are the Serre weights ( λ ( t ) , . . . , λ f − ( t f − ))up to twist, where λ ∈ P ( x , . . . , x f − ), J ( λ ) ⊇ J max ( σ ) and λ j ( t j ) = p − − t j + J ( λ ) ( j −
1) if j ∈ J ( λ ) (we use the notation of [BP12, §2] as in the proof of Lemma3.2.3.4). By [BP12, Lemma 2.7, Lemma 2.6] and Lemma 3.2.2.5(ii), Q ( χ sσ , J max ( σ ))has F -basis Y k (cid:16) p
00 1 (cid:17) x σ , where 0 ≤ k j ≤ λ j ( t j ) if j ∈ J ( λ ) ,p − − λ j ( t j ) ≤ k j ≤ p − j J ( λ ) (144)for some λ ∈ P ( x , . . . , x f − ) with J ( λ ) ⊇ J max ( σ ). We see that (144) implies k k k ≤ ( p − f − | J ( λ ) | ) + X j ∈ J ( λ ) ( p − − t j + J ( λ ) ( j − k j = λ j ( t j ) if j ∈ J ( λ ) and k j = p − Y k (cid:16) p
00 1 (cid:17) x σ ∈ δ ( σ ) \ { } if and only if (144) holds with J ( λ ) = J max ( σ ).137ence if Y k (cid:16) p
00 1 (cid:17) x σ δ ( σ ) we deduce that (144) holds for some λ ∈ P ( x , . . . , x f − )with J ( λ ) ) J max ( σ ).We claim that the right-hand side of (145) is smaller or equal than ( p − f −
1) + m − P J max ( σ ) s j − p ( m −
1) if J ( λ ) = J max ( σ ) and strictly smaller than( p − f −
1) + P J max ( σ ) s j − p ( m −
1) if J ( λ ) ) J max ( σ ). Recalling that s j = p − − t j + J max ( σ ) ( j −
1) for j ∈ J max ( σ ), the first case follows from ( p − f − | J max ( σ ) | ) =( p − f −
1) + m − − p ( m − p − f − − p ( m −
1) =( p − f − | J max ( σ ) | ) − ( m − p − f − | J ( λ ) | ) + X j ∈ J ( λ ) ( p − − t j ) + | J ( λ ) ∩ ( J ( λ ) + 1) | < ( p − f −| J max ( σ ) | ) + X j ∈ J max ( σ ) ( p − − t j ) + | J max ( σ ) ∩ ( J max ( σ ) + 1) |− ( m − , or equivalently (by an easy calculation):( m −
1) + | J ( λ ) ∩ ( J ( λ ) + 1) | − | J max ( σ ) ∩ ( J max ( σ ) + 1) | < X j ∈ J ( λ ) \ J max ( σ ) ( t j + 1) . This is true, as m − ≤ f − f −
1) + f ), J ( λ ) \ J max ( σ ) = ∅ and t j + 1 ≥ f for any j by genericity (132).Therefore k k k ≤ ( p − f −
1) + ( m −
1) + P J max ( σ ) s j − p ( m −
1) if Y k (cid:16) p
00 1 (cid:17) x σ = 0and Y k (cid:16) p
00 1 (cid:17) x σ ∈ δ ( σ ) if k k k ≥ ( p − f −
1) + P J max ( σ ) s j − p ( m − i = 0). If k k k ≥ ( p − f −
1) + P J max ( σ ) s j − p ( m −
1) and Y k (cid:16) p
00 1 (cid:17) x σ = 0, we know by above that J ( λ ) = J max ( σ ).By (144) we then have k j ≤ s j if j ∈ J max ( σ ) and k j ≤ p − j / ∈ J max ( σ ). By thedefinition of x δ ( σ ) in (135) and by Lemma 3.2.2.6(ii) (and Remark 3.2.2.3) we deduce Y k (cid:16) p
00 1 (cid:17) x σ = Y − ℓ x δ ( σ ) , where ℓ j = s j − k j if j ∈ J max ( σ ) and ℓ j = p − − k j if j / ∈ J max ( σ ). This implies k ℓ k = ( p − f − m ) + P J max ( σ ) s j − k k k , and in particular k ℓ k = 0 if k k k = ( p − f −
1) + ( m −
1) + P max ( σ ) s j − ( m − p and k ℓ k = m − k k k = ( p − f −
1) + P J max ( σ ) s j − p ( m − Proof of Proposition 3.2.3.1(ii).
Suppose first that m > j ∈ J max ( σ ). By138emma 3.2.2.4 and Proposition 3.2.2.9, we have Y P j ∈ J max( σ ) s j F ( Y − m x σ )= ( − f − Y j ∈ J max ( σ ) Y s j j Y j = j ( Y j − Y j ) p − + f ( Y ) (cid:16) p
00 1 (cid:17) Y − m x σ for some f ( Y ) ∈ F J Y , . . . , Y f − K of total Y -adic valuation ≥ P J max ( σ ) s j + ( p − f .As p > f ≥ m we have ( p − f > ( p − f −
1) + m − f ( Y ) (cid:16) p
00 1 (cid:17) Y − m x σ = 0, hence Y P J max( σ ) s j F ( Y − m x σ ) = ( − f − Y j ∈ J max ( σ ) Y s j j Y j = j ( Y j − Y j ) p − (cid:16) p
00 1 (cid:17) Y − m x σ . Moreover, the right-hand side is contained in h Y − ℓ x δ ( σ ) , k ℓ k = m − i F ⊆ δ ( σ )by Lemma 3.2.3.5(ii). As it is also N -invariant, it is contained in F Y − m x δ ( σ ) byLemma 3.2.2.6(iii). It is therefore enough to show that Y m − P J max( σ ) s j F ( Y − m x σ ) =( − f − x δ ( σ ) , or again by Lemma 3.2.2.4, Proposition 3.2.2.9 and Lemma 3.2.3.5(i)that Y m − j Y j ∈ J max ( σ ) Y s j j Y j = j ( Y j − Y j ) p − (cid:16) p
00 1 (cid:17) Y − m x σ = x δ ( σ ) . As (cid:16) p − i (cid:17) = ( − i for 0 ≤ i ≤ p −
1, the left-hand side equals Y m − j Y J max ( σ ) Y s j j X k k ′ k =( p − f − k ′ j ≤ p − j = j Y k ′ (cid:16) p
00 1 (cid:17) Y − m x σ . (146)By Lemma 3.2.3.5(i), as k ′ j + s j can never be congruent to s j modulo p when k ′ j ∈{ , . . . , p − } , only the terms with k ′ j = 0 for j ∈ J max ( σ ) \ { j } and k ′ j = p − j / ∈ J max ( σ ) survive. As k k ′ k = ( p − f − k ′ j = ( p − m − x δ ( σ ) , as required.Finally suppose m = 0. As Y pj (cid:16) p
00 1 (cid:17) x σ = 0 for all j , we get again by Lemma3.2.2.4, Proposition 3.2.2.9 and (135): Y p − F ( x σ ) = ( − f − Y p − Y j =0 ( Y j − Y ) p − (cid:16) p
00 1 (cid:17) x σ = ( − f − f − Y j =0 Y p − j (cid:16) p
00 1 (cid:17) x σ = ( − f − x δ ( σ ) . V GL ( π ) : proof We prove Theorem 3.2.1.1. 139e keep the notation of §§3.2.1, 3.2.2, 3.2.3. Fix σ ∈ W ( ρ ) and define σ i ∈ W ( ρ )inductively by σ def = σ and σ i def = δ ( σ i − ) for i > σ i here shouldn’t be confused withthe embedding σ i = σ ◦ ϕ i ). Let n ≥ σ n +1 ∼ = σ and write σ i = ( s ( i )0 , . . . , s ( i ) f − ) ⊗ η i . Recall that n = 1 if and only if J max ( σ ) = ∅ ifand only if ρ is reducible and σ corresponds to J σ = ∅ or J σ = S (see the beginningof §3.2.3). We set m def = | J max ( σ i ) | if n > i ∈ { , . . . , n } byLemma 3.2.3.2) and m def = 1 if n = 1, so that m ∈ { , . . . , f } . For i ∈ { , . . . , n } welet χ i denote the H -eigencharacter on σ N i = σ I i . We also define for i ∈ { , . . . , n } : s i def = X j ∈ J max ( σ i ) s ( i +1) j if n > s def = p − n = 1.The following lemma will be useful later. Lemma 3.2.4.1.
We have P ni =1 s i ≡ p − .Proof. Let s ( χ i ) ∈ { , . . . , q − } such that χ i +1 = χ i α − s ( χ i ) and denote by | s ( χ i ) | ∈{ , . . . , ( p − f } the sum of the digits of s ( χ i ) in its p -expansion. Then it followsfrom (153) below that we have α P j ∈ J max( σi ) s ( i +1) j p j + P j / ∈ J max( σi ) ( p − p j χ i = χ i +1 and so s ( χ i ) = X j ∈ J max ( σ i ) ( p − − s ( i +1) j ) p j (147)which implies | s ( χ i ) | = ( p − m − s i . As χ n +1 = χ = χ α − P ni =1 s ( χ i ) , we have P ni =1 s ( χ i ) ≡ q − P ni =1 | s ( χ i ) | ≡ p −
1) and the resultfollows.Recall π is as at the end of §3.2.1. In [Bre11, §4] there is defined an F -linearisomorphism S : (soc GL ( O K ) π ) I ∼ −→ (soc GL ( O K ) π ) I . (148)Fixing an embedding σ ֒ → soc GL ( O K ) π , for i ∈ { , . . . , n } there are unique embed-dings σ i ֒ → soc GL ( O K ) π such that the morphism S cyclically permutes the lines σ I i .In particular there exists ν ∈ F × (which depends on σ but not on the fixed embedding σ ֒ → soc GL ( O K ) π ) such that S n | σ I i is the multiplication by ν for all i ∈ { , . . . , n } .We define µ i ∈ F × for 1 ≤ i ≤ n by µ def = ν if n = 1 and if n > µ i def = (cid:18) Q ≤ i ′ ≤ n Q j ∈ J max ( σ i ′ ) ( p − − s ( i ′ +1) j )! (cid:19) − ν if i = n ,1 otherwise.140e let M σ be the F J X K [ F ]-submodule of π N , or equivalently the F J Y K [ F ]-submo-dule, generated by Y − m σ N i = Y − m σ I i for 1 ≤ i ≤ n . Recall γ ∈ Z × p acts on M σ ⊗ χ − π by the action of (cid:16) γ − (cid:17) (see the end of §3.2.1). Proposition 3.2.4.2.
The module M σ ⊗ χ − π is admissible as an F J X K -module ( see§2.1.1 ) , Z × p -stable, and such that ( M σ ⊗ χ − π ) ∨ is free of rank n as F J X K -module.Moreover the étale ( ϕ, Γ) -module ( M σ ⊗ χ − π ) ∨ [1 /X ] admits a basis ( e , . . . , e n ) over F J X K [1 /X ] such that for i ∈ { , . . . , n } ( with e n +1 def = e ) : ϕ ( e i ) = µ − i X s i e i +1 , (149) γ ( e i ) ∈ χ i (cid:16)(cid:16) γ (cid:17)(cid:17) γ m (1 + X F J X K ) e i for all γ ∈ Z × p , (150) where γ is the image of γ ∈ Z × p in F . Moreover γ ( e i ) is uniquely determined by (149) and (150) . To prepare for the proof, fix x ∈ σ N \ { } and define for 1 ≤ i ≤ n − x i +1 def = ( − f − Y j ∈ J max ( σ i ) Y s ( i +1) j j Y j / ∈ J max ( σ i ) Y p − j (cid:16) p
00 1 (cid:17) x i ∈ σ N i +1 \ { } and x n +1 def = x (note that this formula is (135) multiplied by ( − f − ). Lemma 3.2.4.3.
For i ∈ { , . . . , n } we have S ( x i ) = (cid:18) Y j ∈ J max ( σ i ) ( p − − s ( i +1) j )! (cid:19) µ i x i +1 (151) and Y s i F ( Y − m x i ) = µ i Y − m x i +1 . (152) Proof. If i ∈ { , . . . , n } we have( − f − Y j ∈ J max ( σ i ) Y s ( i +1) j j Y j / ∈ J max ( σ i ) Y p − j (cid:16) p
00 1 (cid:17) x i = (cid:18) Y j ∈ J max ( σ i ) ( p − − s ( i +1) j )! (cid:19) − θ P J max( σi ) ( p − − s ( i +1) j ) p j (cid:16) p
00 1 (cid:17) x i = (cid:18) Y j ∈ J max ( σ i ) ( p − − s ( i +1) j )! (cid:19) − S ( x i ) , (153)where the first equality follows from Lemma 3.2.2.5(i) and the second from the defi-nition of the function S in [Bre11, §4]. From the definition of x i +1 , we obtain (151)for i < n . For i = n , using inductively x i +1 = (cid:18) Y j ∈ J max ( σ i ) ( p − − s ( i +1) j )! (cid:19) − S ( x i )141or i = n − i = n − i = 1 we obtain (as S is F -linear): S ( x n ) = (cid:18) Y j ∈ J max ( σ n − ) ( p − − s ( n ) j )! (cid:19) − S ( x n − )= · · · = (cid:18) Y ≤ i ≤ n − Y j ∈ J max ( σ i ) ( p − − s ( i +1) j )! (cid:19) − S n ( x ) . Since S n ( x ) = νx and from the definition of µ n , we get (151) for i = n . The lastpart follows from Proposition 3.2.3.1 combined with (153) and (151).The following lemma is stated with the variable Y , but remains the same withthe variable X . Lemma 3.2.4.4.
Suppose M is a torsion F J Y K -module. Let Σ ⊆ M be a subsetspanning M as F -vector space and set e Σ def = S v ∈ Σ F × v . If (i) Y Σ ⊆ e Σ ∪ { } ; (ii) F Y v = F Y v = 0 = ⇒ v = v for v , v ∈ Σ ; (iii) Σ ∩ M [ Y ] is a finite set of F -linearly independent vectors,then Σ is an F -basis of M and M is an admissible F J Y K -module. If moreover Y e Σ = e Σ ∪ { } , then M ∨ is a finite free F J Y K -module of rank dim F M [ Y ] .Proof. Write Σ ∩ M [ Y ] = { v , . . . , v d } (assuming Σ ∩ M [ Y ] = ∅ otherwise M =0 and there is nothing to prove). For ℓ ∈ { , . . . , d } let Σ ℓ def = { v ∈ Σ , Y j v ∈ F × v ℓ for some j ≥ } . Then M ℓ def = ⊕ v ∈ Σ ℓ F v is an F J Y K -module using (i). If v, v ′ ∈ Σ ℓ ,then using (ii) there is j ≥ F v = F Y j v ′ , or F v ′ = F Y j v , from whichone easily deduces M ℓ [ Y ] = F v ℓ , in particular M ℓ is admissible. Since Σ spans M over F and Σ = ` nℓ =1 Σ ℓ , the natural map f : L dℓ =1 M ℓ → M is surjective, and thus M is also admissible. Since L ℓ M ℓ [ Y ] = L ℓ F v ℓ ֒ → M [ Y ] (the last injection followingfrom (iii)), we deduce that Ker( f )[ Y ] = 0, hence Ker( f ) = 0 and f is an isomor-phism. This proves the first part of the statement. It follows from Y e Σ = e Σ ∪ { } that the multiplication by Y is surjective on each M ℓ , i.e. we have exact sequences0 → F v ℓ → M ℓ Y → M ℓ →
0. Dualizing, this gives 0 → M ∨ ℓ Y → M ∨ ℓ → ( F v ℓ ) ∨ → M ∨ ℓ is free of rank 1 over F J Y K . The last statement follows.Recall that M σ is the F J Y K [ F ]-submodule of π N generated by Y − m x i for 1 ≤ i ≤ n . LetΣ def = ( Y j F k ( Y − m x i ) , ≤ i ≤ n, k ≥ , ≤ j < p k − s i if k ≥ ≤ j < m if k = 0 ) . M σ and Σ satisfy all the assumptions in Lemma 3.2.4.4. Definefor ℓ ∈ Z ≥ : Σ ℓ def = n Y j F k ( Y − m x i ) ∈ Σ , k + i ≡ ℓ (mod n ) o and M ℓ,σ def = L v ∈ Σ ℓ F v . We have Σ = ` nℓ =1 Σ ℓ . Applying F k − to (152) for k ≥ F ◦ Y = Y p ◦ F on π N ): Y p k − s i F k ( Y − m x i ) ∈ F × F k − ( Y − m x i +1 ) , (154)hence Σ spans M σ and condition (i) of Lemma 3.2.4.4 holds for Σ. Using (154) we alsosee that the multiplication by Y induces an injection Σ ℓ ֒ → e Σ ℓ ∪ { } and that Y e Σ ℓ = e Σ ℓ ∪ { } , hence M ℓ,σ is an F J Y K -submodule of M σ and condition (ii) of Lemma 3.2.4.4holds for Σ ℓ and Σ. Moreover, Y e Σ = e Σ ∪ { } . Finally, Σ ∩ M σ [ Y ] = { x , . . . , x n } (and Σ ∩ M ℓ,σ [ Y ] = x ℓ ). By Lemma 3.2.4.4 and its proof, we deduce that Σ is an F -basis of M σ , that M σ = L nℓ =1 M ℓ,σ and that each M ∨ ℓ,σ is free of rank 1 over F J Y K .In fact one can visualize the “ Y -divisible line” M i +1 ,σ as follows using (152): F x i +1 Y m − ←− F Y − m x i +1 Y si ←− F F ( Y − m x i ) Y psi − ←− F F ( Y − m x i − ) Y p si − ←− F F ( Y − m x i − ) ←− · · · , where F x i +1 = M i +1 ,σ [ Y ] and the arrows mean “multiplication by the power of Y just above”. In particular we see that if d ( v ) def = min { j ≥ , Y j v = 0 } for v ∈ Σ, then v ∈ Σ i +1 is contained in F ( e Σ) if and only if d ( v ) ≡ s i + m (mod p ).Define a basis f , . . . , f n of the free F J Y K -module M ∨ σ by f i ( x i ) def = 1 and f i (Σ \ { x i } ) def = 0 , i ∈ { , . . . , n } . From what is above we then easily deduce the following formula, where F ( f )( v ) def = f ( F ( v )) for f ∈ M ∨ σ and v ∈ M σ (and using conventions as in §2.1.1): F ( Y ℓ +( s i + m − f i +1 ) = µ i Y m − f i if ℓ = 0,0 if 1 ≤ ℓ ≤ p − . (155) Lemma 3.2.4.5.
The module M σ ⊗ χ − π is Z × p -stable, hence Z × p acts on ( M σ ⊗ χ − π ) ∨ .Moreover we have for γ ∈ Z × p ( recall γ ( f )( v ) = f (cid:16)(cid:16) γ (cid:17) v (cid:17) for f ∈ ( M σ ⊗ χ − π ) ∨ , v ∈ M σ ) : γ ( f i ) ∈ χ i (cid:16)(cid:16) γ (cid:17)(cid:17) (1 + Y F J Y K ) f i for ≤ i ≤ n .Proof. As M σ = L ni =1 F J Y K [ F ] Y − m x i and Y − m x i is a Z × p -eigenvector by Lemma3.2.2.6(iii) we deduce that M σ , and hence M σ ⊗ χ − π , are Z × p -stable.143rom γ ◦ X = ((1+ X ) γ − ◦ γ and Lemma 3.2.2.2 it is easy to deduce that γ ◦ Y = f γ ( Y ) ◦ γ for some f γ ( Y ) ∈ γY + Y F J Y K , hence Z × p preserves the decomposition of F J Y K -modules M σ ⊗ χ − π = L ni =1 M i,σ ⊗ χ − π . In particular, γ ( f i ) annihilates M i ′ ,σ ⊗ χ − π for all i ′ = i . Let Y − j x i for j ≥ e Σ i such that Y j ( Y − j x i ) = x i (this is compatible with our previous notation in Lemma 3.2.2.6(iii)).Then γ ( f i ) = X j ≥ ( γ ( f i )( Y − j x i )) Y j f i ∈ χ i (cid:16)(cid:16) γ (cid:17)(cid:17) (1 + Y F J Y K ) f i . Proof of Proposition 3.2.4.2.
We have already seen above that M σ ⊗ χ − π is admissible, Z × p -stable, and that ( M σ ⊗ χ − π ) ∨ is free of rank n as F J N /N K -module. To find thebasis ( e i ) i , first note from Lemma 3.2.2.2 and (155) that (using F ◦ Y p = Y ◦ F on( M σ ⊗ χ − π ) ∨ ): F ( X s i + m − f i +1 ) = F (cid:16) X j ≥ c j Y s i + m − j f i +1 (cid:17) = µ i X j ≥ c jp Y m − j f i ∈ ( − s i µ i (1 + X F J X K ) X m − f i (156)for some c j ∈ F with c = ( − s i + m − . Similarly for ℓ ∈ { , . . . , p − } : F (cid:16) X s i + m − ℓ f i +1 (cid:17) ∈ F J X K X m f i . (157)It easily follows from (14) that p − X ℓ =0 (1 + X ) − ℓ ϕ (cid:16) F ((1 + X ) ℓ f ) (cid:17) = f (158)for all f ∈ ( M σ ⊗ χ − π ) ∨ [1 /X ]. Let f def = X s i + m − f i +1 , by (156) and (157) we have for ℓ ∈ { , . . . , p − } : F ((1 + X ) ℓ f ) ∈ ( − s i µ i (1 + X F J X K ) X m − f i , and so ϕ (cid:16) F ((1 + X ) ℓ f ) (cid:17) ∈ ( − s i µ i (1 + X p F J X K ) ϕ ( X m − f i ) . Using p − X ℓ =0 (1 + X ) − ℓ = (cid:18) X X (cid:19) p − ≡ X p − (mod X p ) , we see that (158) applied to f = X s i + m − f i +1 becomes( − s i µ i X p − ϕ ( X m − f i ) ∈ (1 + X F J X K ) X s i + m − f i +1 M σ ⊗ χ − π ) ∨ [1 /X ]: ϕ ( X m f i ) = ( − s i µ − i g i ( X ) X s i + m f i +1 (159)for some g i ( X ) ∈ X F J X K .Let e i def = ( − P i − j =1 s j h i ( X ) X m f i for some h i ( X ) ∈ X F J X K and note that thesign doesn’t change if i is replaced by i + n by Lemma 3.2.4.1. Then (149) is equivalentto h i ( X p ) ϕ ( X m f i ) = ( − s i µ − i h i +1 ( X ) X s i X m f i +1 , or equivalently h i ( X p ) g i ( X ) = h i +1 ( X ) by (159). This system has the unique solution h i ( X ) = ∞ Y j =1 g i − j ( X p j − )in 1 + X F J X K , where the indices are considered modulo n . Then (150) follows fromLemma 3.2.4.5. The final uniqueness assertion follows from γ ◦ ϕ = ϕ ◦ γ and is leftas an exercise (similar to [Bre11, Lemma 4.5]).Let O ( π ) (resp. O ( ρ )) be a set of representatives for the orbits of δ on the set ofSerre weights in soc GL ( O K ) π counted with their multiplicity r (resp. on the set W ( ρ )).We define M π def = L σ ∈O ( π ) M σ (with M σ as above). It follows from the assumptionson π that we have M π ∼ = M σ ∈O ( ρ ) M ⊕ rσ . In particular ( M π ⊗ χ − π ) ∨ [1 /X ] is an étale ( ϕ, Γ)-module over F (( X )) of rank r | W ( ρ ) | = r f . From the description of M σ [ X ], we also see that the natural map M π → π N oftorsion F J X K -modules is injective as the following composition is injective: M π [ X ] ∼ = ⊕ σ I ֒ → π I ⊆ π N [ X ] , where the direct sum is over all Serre weights σ in soc GL ( O K ) π (counting their mul-tiplicity r ). Proposition 3.2.4.6.
We have an isomorphism of representations of
Gal( Q p / Q p ) over F : V (( M π ⊗ χ − π ) ∨ [1 /X ]) ∼ = (cid:16) ind ⊗ Q p K ( ρ ) (cid:17) ⊕ r . Proof.
We are going to use a computation of [Bre11, §4]. Associated to the diagram D def = D ( ρ ) ⊕ r of §3.2.1, there is defined in loc.cit. an étale ( ϕ, Γ)-module over F (( X ))denoted there M ( D ) and which is of the form M ( D ) = ⊕ σ ∈O ( π ) M ( D ) σ , where M ( D ) σ A more consistent notation with the ones of this article would have been M ( D ) ∨ and M ( D ) ∨ σ . . . n étale ( ϕ, Γ)-module over F (( X )) associated to the orbit of σ , i.e. to thecycle σ = σ , . . . , σ n as above (so in fact one has M ( D ) = ⊕ σ ∈O ( ρ ) M ( D ) ⊕ rσ ).Let N def = F (( X )) e be the rank 1 étale ( ϕ, Γ)-module over F (( X )) defined by ϕ ( e ) = X − ( p − P j ( r j +1) e,γ ( e ) = γX (1 + X ) γ − !P j ( r j +1) e. We have V ( N ) ∼ = ω P j ( r j +1) = ind ⊗ Q p K (det ρ ) (using ind ⊗ Q p K ( ω f ) ∼ = ω ) by [Bre11,Prop.3.5] and V ( M ( D )) ∼ = (cid:16) ind ⊗ Q p K ( ρ ⊗ (det ρ ) − ) (cid:17) ⊕ r ∼ = (cid:16) ind ⊗ Q p K ( ρ ) ⊗ ind ⊗ Q p K (det ρ − ) (cid:17) ⊕ r by [Bre11, Thm.6.4]. We therefore deduce V ( M ( D ) ⊗ F (( X )) N ) ∼ = (cid:16) ind ⊗ Q p K ( ρ ) (cid:17) ⊕ r . Therefore it suffices to show that M ( D ) σ ⊗ F (( X )) N ∼ = M ∨ σ [1 /X ] for each σ ∈ O ( π ), orequivalently each σ ∈ O ( ρ ).Let x ∨ , . . . , x ∨ n ∈ ( L ni =1 σ I i ) ∨ be the dual basis of the F -basis ( x i ) i of L ni =1 σ I i ,it follows from its definition in [Bre11, §4] and from (147) that M ( D ) σ has basis x ∨ , . . . , x ∨ n as F (( X ))-module with ϕ ( x ∨ i ) = X s i +( p − f − m ) Y j ∈ J max ( σ i ) ( p − − s ( i +1) j )! ! ( x ∨ i ◦ S | − ⊕ σ I i ) , where S − is the inverse of the bijection S of (148) (which preserves L ni =1 σ I i ). By(151) we have x ∨ i ◦ S | − ⊕ σ I i = Y J max ( σ i ) ( p − − s ( i +1) j )! ! − µ − i x ∨ i +1 , so we obtain ϕ ( x ∨ i ) = µ − i X s i +( p − f − m ) x ∨ i +1 . Also we have for γ ∈ Z × p (using the hypothesis on the central character of π ): x ∨ i ◦ (cid:16) γ −
00 1 (cid:17) = γ − P j r j (cid:16) x ∨ i ◦ (cid:16) γ (cid:17)(cid:17) = γ − P j r j χ i (cid:16)(cid:16) γ (cid:17)(cid:17) x ∨ i , hence with the definition of γ ( x ∨ i ) given in [Bre11, Lemma 4.5]: γ ( x ∨ i ) ∈ χ i (cid:16)(cid:16) γ (cid:17)(cid:17) γ − P j r j (1 + X F J X K ) x ∨ i . M ( D ) σ ⊗ F (( X )) N ∼ = L ni =1 F J X K ( x ∨ i ⊗ e ) with ϕ ( x ∨ i ⊗ e ) = µ − i X s i − ( p − m + P j r j ) ( x ∨ i +1 ⊗ e ) ,γ ( x ∨ i ⊗ e ) ∈ χ i (cid:16)(cid:16) γ (cid:17)(cid:17) γ − P j r j (1 + X F J X K )( x ∨ i ⊗ e ) . Now, let e ′ i def = X m + P j r j ( x ∨ i ⊗ e ) for all i . Then e ′ , . . . , e ′ n is a basis of M ( D ) σ ⊗ F (( X )) N and we have for i ∈ { , . . . , n } (with e ′ n +1 def = e ′ ): ϕ ( e ′ i ) = µ − i X s i e ′ i +1 ,γ ( e ′ i ) ∈ χ i (cid:16)(cid:16) γ (cid:17)(cid:17) γ m (1 + X F J X K ) e ′ i . From Proposition 3.2.4.2 we see that M ( D ) σ ⊗ F (( X )) N ∼ = M ∨ σ [1 /X ].By Lemma 3.2.1.2 this completes the proof of Theorem 3.2.1.1 when the constants ν i are as in [Bre11, Thm.6.4]. When they are arbitrary, the proof of Proposition3.2.4.6 gives V (( M π ⊗ χ − π ) ∨ [1 /X ]) | I Q p ∼ = (cid:16) ind ⊗ Q p K ( ρ ) (cid:17) | ⊕ rI Q p using [Bre11, Cor.5.4], whichfinishes the proof of Theorem 3.2.1.1. GL ( K ) We prove results on the structure of an admissible smooth representation π of GL ( K )over F associated to a semisimple sufficiently generic representation ρ of Gal( Q p /K )as in [BP12] when π satisfies a further multiplicity one assumption as in [BHH + ] anda self-duality property. In particular we prove that such a π is irreducible if and onlyif ρ is, and is semisimple when f = 2 (Corollary 3.3.5.8 and Corollary 3.3.5.6).We keep the notation at the beginning of §§3, 3.1, and set Λ def = F J I /Z K . Werecall that the graded ring gr(Λ) is isomorphic to ⊗ f − i =0 F [ y i , z i , h i ] with h i lying in thecenter (see (118)). We set R def = gr(Λ) / ( h , . . . , h f − ) , which is commutative and isomorphic to F [ y i , z i , ≤ i ≤ f − R = R/ ( y i z i , ≤ i ≤ f −
1) = gr(Λ) /J (see (123)). Moreover the finite torus H naturally acts on Λ by the conjugation on I (via its Teichmüller lift) and we see (using(100)) that the induced action on gr(Λ) is trivial on h i and is the multiplication by thecharacter α i (resp. α − i ) on y i (resp. z i ), where α i (cid:16)(cid:16) λ µ (cid:17)(cid:17) def = σ i ( λµ − ) for (cid:16) λ µ (cid:17) ∈ H .Notice that gr(Λ) is an Auslander regular ring (see [LvO96, Def.III.2.1.7], [LvO96,Def.III.2.1.3]) by the first statement in [BHH + , Thm.5.3.4] and so is Λ itself by[LvO96, Thm.III.2.2.5]. This allows us to apply (many) results of [LvO96, §III.2].147 .3.1 Combinatorial results We define some explicit ideals a ( λ ) of R and study some of their properties.We fix a continuous representation ρ : Gal( Q p /K ) → GL ( F ) which is generic inthe sense of [BP12, §11] and let D ( ρ ) be the representation of GL ( F q ) over F definedin [BP12, §13] (see also §3.2.1 when ρ is semisimple). Recall from [BP12, Cor.13.6]that D ( ρ ) I is multiplicity-free as a representation of H ∼ = I/I . By [Bre14, §4],there is a bijection between the characters of H appearing in D ( ρ ) I and a certainset of f -tuples, denoted by PI D ( x , . . . , x f − ) , resp . PRD ( x , . . . , x f − ) , resp . PD ( x , . . . , x f − ) , if ρ is irreducible, resp. reducible split, resp. reducible nonsplit. We refer to [Bre14,§4] for the precise definition of these sets and we simply write P for the set associatedto ρ . We write χ λ for the character of H associated to λ ∈ P (more precisely, in loc.cit. one rather associates a Serre weight σ λ to λ , and χ λ is the action of H = I/I on the 1-dimensional subspace σ I λ , different σ λ giving different χ λ ).On the other hand, the set W ( ρ ) is in bijection with another set of f -tuples,denoted by (see [BP12, §11]) I D ( x , . . . , x f − ) , resp . RD ( x , . . . , x f − ) , resp . D ( x , . . . , x f − ) , depending on ρ as above. We simply write D for the set associated to ρ . Since thesocle of D ( ρ ) is ⊕ σ ∈ W ( ρ ) σ , we may view D as a subset of P . For example, if ρ isreducible split, then D is the subset of P consisting of λ such that λ j ( x j ) ∈ { x j , x j + 1 , p − − x j , p − − x j } , while if ρ is nonsplit, then we require moreover that λ j ( x j ) ∈ { x j + 1 , p − − x j } implies j ∈ J ρ , where J ρ is a certain subset of { , . . . , f − } uniquely determined bythe Fontaine–Laffaille module of ρ (cf. [Bre14, (17)]). Definition 3.3.1.1.
We associate to λ ∈ P an ideal a ( λ ) of R as follows.• If ρ is irreducible, then a ( λ ) = ( t , . . . , t f − ), where t def = z if λ ( x ) ∈ { x − , p − − x } y if λ ( x ) ∈ { x + 1 , p − x } y z if λ ( x ) ∈ { x , p − − x } , and if j = 0 t j def = z j if λ j ( x j ) ∈ { x j , p − − x j } y j if λ j ( x j ) ∈ { x j + 2 , p − − x j } y j z j if λ j ( x j ) ∈ { x j + 1 , p − − x j } .
148 If ρ is reducible nonsplit, then a ( λ ) = ( t , . . . , t f − ), where t j def = z j if λ j ( x j ) ∈ { x j , p − − x j } and j ∈ J ρ y j if λ j ( x j ) ∈ { x j + 2 , p − − x j } and j ∈ J ρ y j z j if λ j ( x j ) ∈ { x j , p − − x j } and j / ∈ J ρ y j z j if λ j ( x j ) ∈ { x j + 1 , p − − x j } . • If ρ is reducible split, then a ( λ ) = ( t , . . . , t f − ) is defined as in the nonsplitcase by letting J ρ = { , . . . , f − } , namely t j def = z j if λ j ( x j ) ∈ { x j , p − − x j } y j if λ j ( x j ) ∈ { x j + 2 , p − − x j } y j z j if λ j ( x j ) ∈ { x j + 1 , p − − x j } . In particular, if ρ is reducible nonsplit and J ρ = ∅ , then a ( λ ) = ( y z , . . . , y f − z f − )for any λ ∈ P . Note that R/ a ( λ ) is always a quotient of R . Remark 3.3.1.2.
An equivalent form of Definition 3.3.1.1 is as follows (comparethe proof of Theorem 3.3.2.1). Given λ ∈ P , t j = y j (resp. t j = z j ) if and only ifthe character χ λ α − j (resp. χ λ α j ) occurs in D ( ρ ) I (i.e. has the form χ λ ′ for some λ ′ ∈ P ), and t j = y j z j if and only if neither of χ λ α ± j occurs in D ( ρ ) I . Lemma 3.3.1.3.
Let λ ∈ P . (i) Assume ρ is semisimple. Then λ ∈ D if and only if y j / ∈ a ( λ ) for any j ∈{ , . . . , f − } . (ii) Assume ρ is reducible nonsplit and let ρ ss be the semisimplification of ρ . Thenthere is a bijection between D ( ρ ss ) ( defined as the set D associated to ρ ss ) andthe set of λ ∈ P such that y j / ∈ a ( λ ) for any j ∈ { , . . . , f − } .Proof. (i) It is clear by definition.(ii) Let λ ∈ P such that y j / ∈ a ( λ ) for any j ∈ { , . . . , f − } . By definition, we have(for ρ reducible nonsplit) λ j ( x j ) ∈ { x j , x j + 1 , p − − x j , p − − x j , p − − x j } and from the definition of a ( λ ) if λ j ( x j ) = p − − x j then j / ∈ J ρ (note that if λ j ( x j ) = p − − x j then it is automatic that j ∈ J ρ ). We define an f -tuple µ by µ j ( x j ) def = ( p − − x j if λ j ( x j ) = p − − x j λ j ( x j ) otherwise . It is then easy to see that µ is an element of D ( ρ ss ) and that any element of D ( ρ ss )arises (uniquely) in this way. 149 orollary 3.3.1.4. The set { λ ∈ P , y j / ∈ a ( λ ) ∀ j ∈ { , . . . , f − }} has cardinality f .Proof. This is a direct consequence of Lemma 3.3.1.3 and of | W ( ρ ss ) | = 2 f .Given λ ∈ P , write a ( λ ) = ( t , . . . , t f − ) as in Definition 3.3.1.1 and define A ( λ ) def = { j ∈ { , . . . , f − } , t j = y j z j } ⊆ { , . . . , f − } . (160)The following proposition will only be used in Corollary 3.3.2.5 below. Proposition 3.3.1.5.
We have P λ ∈ P |A ( λ ) | = 4 f .Proof. We will only give the proof in the case ρ is reducible (split or not), the irre-ducible case can be treated similarly.First assume that ρ is split. Given λ ∈ P , we define an element λ ∈ D as follows: λ j ( x j ) def = x j if λ j ( x j ) ∈ { x j , x j + 2 } p − − x j if λ j ( x j ) ∈ { p − − x j , p − − x j } λ j ( x j ) otherwise . It is easy to see that λ ∈ D . By definition of P (see [Bre14, §4] and recall P = PRD ( x , . . . , x f − )), for each λ ∈ D , there are exactly 2 |{ ,...,f − }\A ( λ ) | elements λ in P giving rise to λ under the above rule. Moreover, it is direct from the definitionsthat A ( λ ) = A ( λ ). Hence X λ ∈ P |A ( λ ) | = X λ ∈ D (2 f −|A ( λ ) | |A ( λ ) | ) = 2 f | D | = 2 f f = 4 f . Now assume that ρ is nonsplit. Let P be the subset of P considered in the proofof Lemma 3.3.1.3(ii), namely λ ∈ P if and only if λ j ( x j ) ∈ { x j , x j + 1 , p − − x j , p − − x j , p − − x j } and λ j ( x j ) = p − − x j implies j / ∈ J ρ . By the proof of loc.cit. , we have | P | = | D ( ρ ss ) | = 2 f . Given λ ∈ P , we define an element λ ∈ P as follows: λ j ( x j ) def = x j if λ j ( x j ) ∈ { x j , x j + 2 } p − − x j if λ j ( x j ) = p − − x j or ( λ j ( x j ) = p − − x j and j ∈ J ρ ) λ j ( x j ) otherwise . As in the split case it is easy to see that A ( λ ) = A ( λ ) and that given λ ∈ P , thereexist exactly 2 |{ ,...,f − }\A ( λ ) | elements λ in P giving rise to λ . The result follows asin the split case. 150 efinition 3.3.1.6. Given λ ∈ P , we define another f -tuple λ ∗ as follows: λ ∗ j ( x j ) def = p − − λ j ( x j ) if t j = z j p + 1 − λ j ( x j ) if t j = y j p − − λ j ( x j ) if t j = y j z j . If λ ∈ D , we define its “length” ℓ ( λ ) to be (see [BP12, §4]): ℓ ( λ ) def = |{ j ∈ { , . . . , f − } , λ j ( x j ) ∈ { p − − x j ± , x j ± }}| . (161) Lemma 3.3.1.7.
Let λ ∈ P . (i) We have λ ∗ ∈ P and a ( λ ) = a ( λ ∗ ) . (ii) Assume that ρ is semisimple. Then λ ∈ D if and only if λ ∗ ∈ D , and in thiscase ℓ ( λ ∗ ) = f − ℓ ( λ ) .Proof. (i) The first statement can be checked directly using the definition of P andthe second one is obvious from the definitions.(ii) The first statement follows from (i) and Lemma 3.3.1.3(i). By definition of D (see [BP12, §11]), ℓ ( λ ) can be computed as the cardinality of the following set: n j ∈ { , . . . , f − } , λ j ( x j ) ∈ { p − − x j , p − − x j , p − − x j } o . For example, when ρ is reducible split, we have (cf. the beginning of [BP12, §11]) λ j ( x j ) ∈ { p − − x j , p − − x j } ⇐⇒ λ j +1 ( x j +1 ) ∈ { p − − x j +1 , x j +1 + 1 } . The second statement of (ii) follows from this and Definition 3.3.1.6.
Lemma 3.3.1.8.
Let λ ∈ P , χ λ the character of H associated to λ , ( t , . . . , t f − ) the ideal a ( λ ) in Definition 3.3.1.1 and η λ be the character of H acting on Q f − j =0 t j .Then we have χ λ χ λ ∗ = η λ ( η ◦ det) , where λ ∗ is as in Definition 3.3.1.6 and η ( a ) def = χ λ (cid:16)(cid:16) a a (cid:17)(cid:17) for a ∈ F × q ( η does notdepend on λ ∈ P ) .Proof. This is an easy computation, but we give some details. Note that λ j ( x j ) + λ ∗ j ( x j ) = ( p −
1) + 2 ε j , where ε j equals 1, 0 or − t j equals y j , y j z j or z j respectively.Moreover, in the notation of [Bre14, §4], we have e ( λ ) + e ( λ ∗ ) = 12 (cid:18) p f − f − X j =0 p j ( x j − λ j ( x j ) + x j − λ ∗ j ( x j )) (cid:19) = f − X j =0 p j ( x j − ε j ) . (cid:16) a b (cid:17) ∈ Hχ λ (cid:16)(cid:16) a b (cid:17)(cid:17) = σ ( a ) (cid:16) P f − j =0 p j λ j ( r j ) (cid:17) + e ( λ )( r ,...,r f − ) σ ( b ) e ( λ )( r ,...,r f − ) (see [Bre14, §4]) and that H acts on y i (resp. z i ) via α i (resp. α − i ).Note that H acts on I /Z by conjugation and hence on Λ and gr(Λ). This induces H -actions also on R , R , and R/ a ( λ ) for any λ ∈ P . We say that M is a gr(Λ)-modulewith compatible H -action if H acts on M such that h ( rm ) = h ( r ) h ( m ) for h ∈ H , r ∈ R , and m ∈ M . In this case E i gr(Λ) ( M ) is again a gr(Λ)-module with compatible H -action for any i ≥ Lemma 3.3.1.9. If M is a gr(Λ) -module with compatible H -action that is annihilatedby ( h , . . . , h f − ) , then we have isomorphisms of gr(Λ) -modules with compatible H -action for i ≥ : E i + f gr(Λ) ( M ) ∼ = E iR ( M ) . (162) If moreover M is annihilated by J , then we have isomorphisms of gr(Λ) -modules withcompatible H -action for i ≥ : E i +2 f gr(Λ) ( M ) ∼ = E i + fR ( M ) ∼ = E iR ( M ) . (163) Proof.
Since ( h , . . . , h f − ) is a regular sequence of central elements in gr(Λ) and( y z , . . . , y f − z f − ) is a regular sequence in R (which is commutative), the isomor-phisms (162) and (163) as gr(Λ)-modules are proved as in the proof of [BHH + , Lemma5.1.3]. Moreover, H acts trivially on h j and y j z j (for 0 ≤ j ≤ f − H -equivariant, from which the results follow.We don’t use the following proposition in the sequel, but it is consistent withRemark 3.3.2.6(i) and the essential self-duality assumption (iii) in §3.3.5 below (seeProposition 3.3.4.6). Proposition 3.3.1.10.
For λ ∈ P there is an isomorphism of gr(Λ) -modules withcompatible H -action: E f gr(Λ) (cid:16) χ − λ ⊗ R/ a ( λ ) (cid:17) ∼ = (cid:16) χ − λ ∗ ⊗ R/ a ( λ ) (cid:17) ⊗ η ◦ det . Proof.
Applying (163) with i = 0 and M = χ − λ ⊗ R/ a ( λ ), we are left to proveHom R ( χ − λ ⊗ R/ a ( λ ) , R ) ∼ = (cid:16) χ − λ ∗ ⊗ R/ a ( λ ) (cid:17) ⊗ η ◦ det . Using Lemma 3.3.1.8, it suffices to construct an isomorphism of gr(Λ)-modules withcompatible H -action Hom R ( R/ a ( λ ) , R ) ∼ = η − λ ⊗ R/ a ( λ ) , (164)152here η λ is the character of H acting on Q f − j =0 t j if we write a ( λ ) = ( t , . . . , t f − ) with t j ∈ { y j , z j , y j z j } . Put t ′ def = Q f − j =0 ( y j z j /t j ). One easily checks that t ′ R = R [ a ( λ )] andthere is an isomorphism of R -modules θ : η − λ ⊗ R/ a ( λ ) ∼ −→ t ′ R, where the first map sends 1 to t ′ . As H acts on t ′ via η − λ , θ is also H -equivariant.The isomorphism (164) is then obtained by sending r ∈ η − λ ⊗ R/ a ( λ ) to φ ∈ Hom R ( R/ a ( λ ) , R ) such that φ (1) def = θ ( r ). gr( π ∨ )We give a partial result on the structure of gr( π ∨ ) for certain admissible smoothrepresentations π of GL ( K ) over F associated to ρ when gr( π ∨ ) comes from the m I /Z -adic filtration on π ∨ .We let ρ be as in §3.3.1 (in particular ρ is not necessarily semisimple) and keepthe notation of loc.cit. As in §3.2.1 when ρ is semisimple, we consider D ( ρ ) as arepresentation of GL ( O K ) K × , where GL ( O K ) acts via its quotient GL ( F q ) andthe center K × acts by the character det( ρ ) ω − . We now write m for m I /Z .We consider an admissible smooth representation π of GL ( K ) over F satisfyingthe following two conditions:(i) there is r ≥ π K ∼ = D ( ρ ) ⊕ r as a representation of GL ( O K ) K × (inparticular π has a central character);(ii) for any λ ∈ P , we have an equality of multiplicities[ π [ m ] : χ λ ] = [ π [ m ] : χ λ ] . Note that (ii) implies that the gr(Λ)-module gr( π ∨ ) (defined with the m -adic filtrationon π ∨ ) is annihilated by the ideal J in (119) by the proof of [BHH + , Cor.5.3.5], andin particular is an R -module. Theorem 3.3.2.1.
For π as above, there is a surjection of gr(Λ) -modules with com-patible H -action (cid:16) M λ ∈ P χ − λ ⊗ R/ a ( λ ) (cid:17) ⊕ r ։ gr( π ∨ ) , (165) where a ( λ ) is as in Definition 3.3.1.1.Proof. Consider the gr(Λ)-module with compatible H -action: M def = (cid:16) M λ ∈ P χ − λ ⊗ R/ a ( λ ) (cid:17) ⊕ r . λ χ λ between P and the characters of H on D ( ρ ) I (see§3.3.1), we can choose a basis of π I over F , say { v λ,k , λ ∈ P , ≤ k ≤ r } , such thateach v λ,k is an eigenvector for I of character χ λ . We denote by { e λ,k , λ ∈ P , ≤ k ≤ r } the basis of gr ( π ∨ ) over F which is the dual basis of { v λ,k } , and note that { e λ,k } generates the gr(Λ)-module gr( π ∨ ). To prove that there exists a surjective morphism M ։ gr( π ∨ ) it suffices to prove that, for any λ ∈ P and any k ∈ { , . . . , r } , thevector e λ,k is annihilated by the ideal a ( λ ) of R = gr(Λ) / ( h , . . . , h f − ). Writing a ( λ ) = ( t , . . . , t f − ) as in Definition 3.3.1.1, we already see that if t j = y j z j , then t j kills all the e λ,k since gr( π ∨ ) is annihilated by J .Let j ∈ { , . . . , f − } such that t j ∈ { y j , z j } and define χ ′ def = χ λ α − j if t j = y j , χ ′ def = χ λ α j if t j = z j . By Definition 3.3.1.1 one checks that χ ′ = χ λ ′ , where λ ′ ∈ P is defined by λ ′ i ( x i ) def = λ i ( x i ) if i = j , and λ ′ j ( x j ) def = λ j ( x j ) + ε j , where ε j equalseither − t j equals either y j or z j respectively. Note that χ ′− is equalto the character of I acting on t j e λ,k ∈ gr ( π ∨ ). Thus, if t j e λ,k = 0, then dually the χ ′ -isotypic subspace of π [ m ] /π [ m ] would be nonzero. But this contradicts condition(ii) above. Hence e λ,k is annihilated by the whole ideal a ( λ ) and we are done. Corollary 3.3.2.2.
Let π ′ be a subrepresentation of π and P ′ ⊆ P be the subsetcorresponding to the characters ( without multiplicities ) of H appearing in π ′ I . Then gr( π ′∨ ) ( with the m -adic filtration on π ′∨ ) is a quotient of (cid:16) L λ ∈ P ′ χ − λ ⊗ R/ a ( λ ) (cid:17) ⊕ r .Proof. We have a natural quotient map π ∨ ։ π ′∨ which induces a quotient mapgr( π ∨ ) ։ gr( π ′∨ ). It is enough to prove that the composition (cid:16) M λ ∈ P ′ χ − λ ⊗ R/ a ( λ ) (cid:17) ⊕ r ֒ → (cid:16) M λ ∈ P χ − λ ⊗ R/ a ( λ ) (cid:17) ⊕ r ։ gr( π ∨ ) ։ gr( π ′∨ )is surjective (where the second map is the surjection of Theorem 3.3.2.1). The as-sumption implies that it is surjective on gr ( − ), and we conclude using that gr( π ′∨ )is generated by gr ( π ′∨ ) as a gr(Λ)-module.If N is a finitely generated R -module and q a minimal prime ideal of R , recallthat m q ( N ) ∈ Z ≥ denotes the multiplicity of N at q , see (124). Theorem 3.3.2.3.
We have dim F V GL ( π ) = dim F (( X )) D ∨ ξ ( π ) ≤ m p (gr( π ∨ )) ≤ f r ,where the minimal ideal p is as in §3.1.4.Proof. This is a direct consequence of (16), of Corollary 3.1.4.5, of Theorem 3.3.2.1and of Corollary 3.3.1.4, noting that, if y j ∈ a ( λ ) for some j ∈ { , . . . , f − } ,then m p ( R/ a ( λ )) = 0 (as y j / ∈ p ), and if y j / ∈ a ( λ ) ∀ j ∈ { , . . . , f − } , then m p ( R/ a ( λ )) = 1 (as ( R/ a ( λ ))[( y · · · y f − ) − ] ∼ = F [ y , . . . , y f − ][( y · · · y f − ) − ] ∼ =gr( A )). 154ombined with the results of §3.2, we can deduce the following important corol-lary. Corollary 3.3.2.4.
Assume moreover that ρ is semisimple, satisfies the genericitycondition (127) and that condition (i) above can be enhanced into an isomorphismof diagrams ( π I ֒ → π K ) ∼ = D ( ρ ) ⊕ r , where D ( ρ ) is as in (128) . Then we have anisomorphism of representations of I Q p : V GL ( π ) | I Q p ∼ = (cid:16) ind ⊗ Q p K ( ρ ) (cid:17) | ⊕ rI Q p . In particular we have dim F V GL ( π ) = dim F (( X )) D ∨ ξ ( π ) = m p ( π ∨ ) = 2 f r . If moreoverthe constants ν i associated to D ( ρ ⊗ χ ) ( χ as in §3.2.1 ) at the beginning of [Bre11,§6] are as in [Bre11, Thm.6.4] , then we have an isomorphism of representations of Gal( Q p / Q p ) : V GL ( π ) ∼ = (cid:16) ind ⊗ Q p K ( ρ ) (cid:17) ⊕ r . Proof.
It follows from Theorem 3.2.1.1 and Theorem 3.3.2.3 as dim F (cid:16) ind ⊗ Q p K ( ρ ) (cid:17) ⊕ r =2 f r .It is also worth mentioning the following corollary of Theorem 3.3.2.1. Corollary 3.3.2.5.
We have P q m q (gr( π ∨ )) ≤ f r , where the sum is taken over allminimal prime ideals q of R .Proof. By an easy computation, we have P q m q ( R/ a ( λ )) = 2 |A ( λ ) | (see (160) for A ( λ )).Thus the result follows from Proposition 3.3.1.5 and Theorem 3.3.2.1. Remark 3.3.2.6. (i) It seems possible to us that the surjection in Theorem 3.3.2.1could actually be an isomorphism, as least for π coming from the global theory as in§3.4.1 below. Note that such an isomorphism implies in particular E i gr(Λ) (gr( π ∨ )) = 0if and only if i = 2 f (i.e. the gr(Λ)-module gr( π ∨ ) is Cohen–Macaulay of grade 2 f ),which in turns implies E i Λ ( π ∨ ) = 0 if and only if i = 2 f (use [Ven02, Cor.6.3] and thesimilar result with gr(Λ) instead of Λ, the first statement in [Ven02, Thm.3.21(ii)]and [LvO96, Thm.I.7.2.11(1)]).(ii) It is worth recalling here the following implications that we have seen. Considerthe following conditions on an admissible smooth representation π of GL ( K ) over F with a central character:(a) [ π [ m ] : χ ] = [ π [ m ] : χ ] for every character χ : I → F × ;(b) gr( π ∨ ) is killed by J , where gr( π ∨ ) is computed with the m -adic filtration on π ∨ ; 155c) gr( π ∨ ) is killed by some power of J , where gr( π ∨ ) is computed with any goodfiltration on the Λ-module π ∨ ;(d) π is in the category C of §3.1.2.Then we have (a) ⇒ (b) ⇒ (c) ⇒ (d). We suspect that every implication is strict. We completely compute the gr( F J I/Z K )-module gr( V ∨ ) for certain irreducible admis-sible smooth representations V of GL ( K ) over F (with V ∨ endowed with the m -adicfiltration). We assume p ≥ V is a smooth representation of I /Z over F ,we write gr( V ∨ ) for the graded module associated to the m -adic filtration on V ∨ . Lemma 3.3.3.1.
Let V be a smooth representation of I /Z over F such that V | N is admissible as a representation of N and such that the natural map gr m N ( V ∨ ) → gr( V ∨ ) ( induced by the inclusions m nN V ∨ ⊆ m n V ∨ for n ≥ is surjective. Then thismap is an isomorphism.Proof. Since V | ∨ N is a finite type F J N K -module by assumption, it is a completefiltered F J N K -module for the m N -adic filtration. As all the maps m nN V ∨ / m n +1 N V ∨ → m n V ∨ / m n +1 V ∨ are surjective, any element in v ∈ m n V ∨ can be written v = v + w ,where v ∈ P m ≥ n m mN V ∨ = m nN V ∨ (as V | ∨ N is complete) and w ∈ ∩ m ≥ n m m V ∨ = 0(as the m -adic filtration is separated since V is smooth). Thus the inclusion m nN V ∨ ⊆ m n V ∨ is an equality for n ≥
0, and we are done.The following two lemmas are motivated by [Paš10, Prop.7.1, Prop.7.2]. Weconsider the finite group H as subgroup of I via the Teichmüller lift. Lemma 3.3.3.2.
Let V be an admissible smooth representation of I/Z over F . As-sume that dim F V N = 1 and that V | HN is isomorphic to an injective envelope ofsome character χ in the category of smooth representations of HN over F . Then Ext I/Z ( χα − j , V ) = 0 for any ≤ j ≤ f − .Proof. Consider an extension class in Ext I/Z ( χα − j , V ) represented by 0 → V → V ′ → χα − j →
0. By assumption on V , this extension splits when restricted to HN ,hence we may find v ′ ∈ V ′ \ V on which HN acts via χα − j (in particular v ′ ∈ V ′ N ).Notice that ( g − v ′ ∈ V for any g ∈ I . Let v ∈ V N be a nonzero vector so that V N = F v by assumption. 156irst take g ∈ (cid:16) p O K
00 1+ p O K (cid:17) . It is easy to see that ( g − v ′ is again fixed by N and H acts on it via χα − j . But, by assumption V N is 1-dimensional on which H actsvia χ , thus we must have ( g − v ′ = 0. We deduce that v ′ is fixed by I ∩ B ( O K ).We claim that v ′ is fixed by N − def = (cid:16) p O K (cid:17) . This will imply that v ′ is fixed by I by the Iwahori decomposition, and consequently V ′ splits as I -representation. Let k ≥ v ′ is fixed by N − k def = (cid:16) p k O K (cid:17) ; such an integeralways exists, as V is a smooth representation of I . Suppose k ≥ g ∈ N − k − .Using the matrix identity (see [Paš10, Eq.(14)]) (cid:16) b (cid:17)(cid:16) c (cid:17) = (cid:16) c (1+ bc ) − (cid:17)(cid:16) bc b bc ) − (cid:17) and the fact that v ′ is fixed by (cid:16) p O K O K p k O K p O K (cid:17) , one checks that ( g − v ′ ∈ V N . Con-sequently, F v ⊕ F v ′ gives rise to an extension in Ext HN − k − ( χα − j , χ ) which is nonsplitby the choice of k . But, as in [Paš10, Lemma 5.6], one shows that Ext HN − k − ( χ ′ , χ ) = 0if and only if χ ′ = χα i for some 0 ≤ i ≤ f −
1. Indeed, after conjugating by (cid:16) p k −
00 1 (cid:17) ,we are reduced to the case k = 2, in which case the result is proved by determining the H -action on Hom( N − , F ) as in [Paš10, Lemma 5.3] (see the proof of [BP12, Prop.5.1]for the computation). This finishes the proof as χα − j = χα i for any 0 ≤ i, j ≤ f − p ≥ Lemma 3.3.3.3.
Let V be an admissible smooth representation of I/Z over F . As-sume that dim F V N = 1 and that V | HN is isomorphic to an injective envelope ofsome character χ in the category of smooth representations of HN over F . Then wehave an isomorphism of gr( F J I/Z K ) -modules: gr( V ∨ ) ∼ = χ − ⊗ R/ ( z , . . . , z f − ) . Proof.
By assumption, V [ m ] = V [ m N ] is one-dimensional and isomorphic to χ , hencewe may view gr( V ∨ ) as a cyclic module over gr(Λ) generated by e χ ∈ gr ( V ∨ ) = V [ m ] ∨ , where H acts on e χ by χ − . Let a ⊆ gr(Λ) be the annihilator of e χ .We first prove that z j ∈ a for 0 ≤ j ≤ f −
1. Since H acts on z j via α − j (see justabove §3.3.1), to prove z j e χ = 0 in gr ( V ∨ ) it is equivalent to prove thatHom H ( χα j , V [ m ] /V [ m ]) = 0 ∀ j ∈ { , . . . , f − } . If not, then V would admit a subrepresentation isomorphic to E χ,χα j (for some j ),where E χ,χα j denotes the unique I/Z -representation which is a nonsplit extension of χα j by χ . But by [BHH + , Lemma 6.1.1(ii)] (after conjugating by the element (cid:16) p (cid:17) ), N acts trivially on E χ,χα j , which implies dim F V [ m N ] ≥
2, a contradiction to theassumption on V . Using [BHH + , Lemma 6.1.1], we then deduce an embedding V [ m ] /V [ m ] ֒ → ⊕ f − j =0 χα − j . (166)157n the other hand, since Hom I ( χα − j , V ) = 0, we deduce from Lemma 3.3.3.2 thatHom I ( χα − j , V [ m ] /V [ m ]) = Hom I ( χα − j , V /V [ m ]) ∼ −→ Ext I/Z ( χα − j , χ )which have dimension 1 over F by [BHH + , Lemma 6.1.1] again. Combining this with(166), we obtain 0 → χ → V [ m ] → ⊕ f − j =0 χα − j → . (167)and that V [ m ] = V [ m N ].Next, we prove that Ext I/Z ( χ, E χ,χα − j ) has dimension 1 over F for any 0 ≤ j ≤ f −
1. A straightforward dévissage using Ext I/Z ( χ, χ ) = 0 and dim F Ext I/Z ( χ, χα − j ) = 1(see [BHH + , Lemma 6.1.1]) yields dim F Ext I/Z ( χ, E χ,χα − j ) ≤
1. So it suffices toexplicitly construct a nonzero element in this space, as follows. Let E j def = F v ⊕ F v ⊕ F v equipped with the action of I/Z determined by:• H acts on v , v , v by χ, χα − j , χ respectively;• if g = (cid:16) pa bpc pd (cid:17) ∈ I , then gv = v , gv = v + σ j ( b ) v ,gv = v + σ j ( c ) v + 12 (cid:16) σ j ( a ) − σ j ( d ) + σ j ( bc ) (cid:17) v . One easily checks that E j is well defined and yields the desired nonsplit extension ofExt I/Z ( χ, E χ,χα − j ). Moreover one also checks that E N j = F v ⊕ F v .We prove that h j ∈ a for 0 ≤ j ≤ f −
1. Since Ext I/Z ( χ, χ ) = 0, the sequence(167) induces an embeddingExt I/Z ( χ, V [ m ]) ֒ → Ext I/Z ( χ, ⊕ f − j =0 χα − j ) . Note that the right-hand side has dimension f over F . Since E j /χ is nonzero inExt I/Z ( χ, χα − j ) for 0 ≤ j ≤ f −
1, we easily see that the above embedding isactually an isomorphism and that Ext I/Z ( χ, V [ m ]) is spanned by the E j ’s. By thelast statement of the previous paragraph, if an extension E ∈
Ext I/Z ( χ, V [ m ]) isnonzero then dim F E N ≥
2. Since dim F V N = 1 by assumption, we see that thereexists no embedding E ֒ → V . From (167) we then easily deduceHom H ( χ, V [ m ] /V [ m ]) = 0 . Since H acts trivially on h j and h j e χ ∈ gr ( V ∨ ) ∼ = ( V [ m ] /V [ m ]) ∨ , we thus musthave h j e χ = 0, i.e. h j ∈ a for 0 ≤ j ≤ f −
1. This proves the claim.158e deduce a surjection gr(Λ) / ( z j , h j , ≤ j ≤ f − ։ gr( V ∨ ). As the left-handside is F [ y , . . . , y f − ] ∼ = gr( F J N K ) and ( V | N ) ∨ ∼ = F J N K from the assumption, weobtain a surjection gr m N ( V ∨ ) ։ gr( V ∨ ). By Lemma 3.3.3.1 this surjection is anisomorphism (and hence a = ( z j , h j , ≤ j ≤ f − χ = χ ⊗ χ is a character of H or of T ( K ), recall χ s = χ ⊗ χ . Proposition 3.3.3.4.
Let V be an irreducible smooth F -representation of GL ( K ) with a central character. (i) If V ∼ = ψ ◦ det for some smooth character ψ : K × → F × , then gr( V ∨ ) ∼ =( ψ ⊗ ψ ) − ⊗ F , where ψ ⊗ ψ is viewed as a character of H . (ii) If V ∼ = Ind GL ( K ) B ( K ) χ for some smooth character χ : T ( K ) → F × , then gr( V ∨ ) ∼ = (cid:16) ( χ s | H ) − ⊗ R/ ( z , . . . , z f − ) (cid:17) ⊕ (cid:16) ( χ | H ) − ⊗ R/ ( y , . . . , y f − ) (cid:17) . (iii) If V ∼ = (Ind GL ( K ) B ( K ) / is the special series, then gr( V ∨ ) ∼ = R/ ( y i z j , ≤ i, j ≤ f − . (iv) Assume K = Q p . If V is supersingular, i.e. isomorphic to (c-Ind GL ( Q p )GL ( Z p ) Q × p σ ) /T for some Serre weight σ ( recall that c-Ind here means compact induction andthat End GL ( Q p ) (c-Ind GL ( Q p )GL ( Z p ) Q × p σ ) ∼ = F [ T ]) , then gr( V ∨ ) ∼ = (cid:16) χ − σ ⊗ R/ ( y z ) (cid:17) ⊕ (cid:16) ( χ sσ ) − ⊗ R/ ( y z ) (cid:17) , where χ σ is the action of H on σ I .Proof. (i) It is trivial.(ii) The restriction of V to I admits a decomposition V | I ∼ = Ind II ∩ B ( K ) χ ⊕ Ind II ∩ B − ( K ) χ s , (168)(cf. the proof of [Paš10, Prop.11.1]). By loc.cit. , when restricted to HN , Ind II ∩ B − ( K ) χ s is an injective envelope of χ s in the category of smooth representations of HN over F , hence gr((Ind II ∩ B − ( K ) χ s ) ∨ ) ∼ = ( χ s | H ) − ⊗ R/ ( z , . . . , z f − )by Lemma 3.3.3.3. One handles the other direct summand by taking conjugation bythe element (cid:16) p (cid:17) .(iii) By assumption we have a short exact sequence 0 → → Ind GL ( K ) B ( K ) → V → W = (Ind GL ( K ) B ( K ) | I and decompose W = W ⊕ W as in (168). The image159f 1 ֒ → W is equal to the subspace of constant functions, hence the composition1 ֒ → W ։ W i is nonzero for i ∈ { , } . Consequently, the dual morphism gr( W ∨ i ) → gr(1 ∨ ) is also nonzero, and using (ii) (applied to W ) we obtain an exact sequence ofgr( F J I/Z K )-modules0 → R/ ( y i z j , ≤ i, j ≤ f − → gr( W ∨ ) ⊕ gr( W ∨ ) → gr(1 ∨ ) → . (169)Denote by F the induced filtration on V ∨ from the m -adic filtration on W ∨ . By (169)we have an isomorphism gr F ( V ∨ ) ∼ = R/ ( y i z j , ≤ i, j ≤ f − F coincides with the m -adic filtration on V ∨ , or equivalently theinclusion m n V ∨ ⊆ m n W ∨ ∩ V ∨ (for n ≥
0) is an equality. As in the proof of Lemma3.3.3.1 it suffices to prove that the induced graded morphism gr m ( V ∨ ) → gr F ( V ∨ )is surjective. But, gr F ( V ∨ ) is generated by gr F ( V ∨ ), so it suffices to show thatgr m ( V ∨ ) → gr F ( V ∨ ) is surjective, which follows from (169) and the exact sequencegr m ( V ∨ ) → gr m ( W ∨ ) → gr m (1 ∨ ) → → I → W I → V I (this sequence is actually right exact but we don’tneed this fact).(iv) The proof is analogous to (iii), using [Paš10, Thm.1.2] together with [Paš10,Prop.4.7].By the classification of irreducible admissible smooth representations of GL ( Q p )over F , we deduce from Proposition 3.3.3.4 and the results of §3.1.2: Corollary 3.3.3.5.
Let V be an admissible smooth representation of GL ( Q p ) over F which has a central character and is of finite length. Then there is an integer n ≥ such that gr( V ∨ ) is annihilated by J n . In particular V is in the category C of §3.1.2. We define the characteristic cycle of a finitely generated filtered Λ-module M suchthat gr( M ) is annihilated by a power of J and prove an important property (Theorem3.3.4.5).Recall from §3.1.4 that the minimal prime ideals of R = R/ ( y j z j , ≤ j ≤ f − y i , z j , i ∈ J , j / ∈ J ) with J a subset of { , . . . , f − } . Definition 3.3.4.1.
Let N be a finitely generated module over gr(Λ) which is an-nihilated by some power of J . We define the characteristic cycle of N , denoted by Z ( N ) as follows: Z ( N ) def = X q m q ( N ) q ∈ ⊕ q Z ≥ q , where q runs over all minimal prime ideals of R . A more standard notation is Z f ( N ), where f indicates the dimension of the cycles. emma 3.3.4.2. Let n ≥ . If → N → N → N → is a short exact sequence offinitely generated gr(Λ) /J n -modules, then Z ( N ) = Z ( N ) + Z ( N ) in ⊕ q Z ≥ q .Proof. It is a direct consequence of Lemma 3.1.4.3.Let M be a finitely generated Λ-module which is equipped with a good filtration F def = { F n M, n ∈ Z } (in the sense of [LvO96, §I.5]) such that gr F ( M ) is annihilated bysome power of J . Recall that this condition doesn’t depend on the choice of the goodfiltration F (see just before Proposition 3.1.2.11) and that gr F ( M ) is also finitelygenerated over gr(Λ) ([LvO96, Lemma I.5.4]). Lemma 3.3.4.3. If F, F ′ are two such good filtrations on M , then Z (gr F ( M )) = Z (gr F ′ ( M )) . Proof.
The proof is (almost) the same as in [Bjö89, §4]. We recall it for the conve-nience of the reader. Since F and F ′ are equivalent by [LvO96, Lemma I.5.3], wemay find c ∈ Z ≥ such that F n − c M ⊆ F ′ n M ⊆ F n + c M, ∀ n ∈ Z . For i ∈ {− c, − c + 1 , . . . , c } define a sequence of filtrations F ( i ) = { F ( i ) n M, n ∈ Z } on M by F ( i ) n M def = F n + i M ∩ F ′ n M. It is clear that F ( − c ) = F [ − c ] and F ( c ) = F ′ , where F [ − c ] denotes the shifted filtration F [ − c ] n def = F n − c , n ∈ Z . Hence it suffices to show that each F ( i ) is a good filtration on M such that Z (gr F ( i ) ( M )) = Z (gr F ( i +1) ( M )) . (170)Put for − c ≤ i ≤ c : T i def = M n ∈ Z ( F n + i M ∩ F ′ n M ) / ( F n + i M ∩ F ′ n − M ) ,S i def = M n ∈ Z ( F n + i +1 M ∩ F ′ n M ) / ( F n + i M ∩ F ′ n M ) . Since T i is a gr(Λ)-submodule of gr F ′ ( M ) and S i is a gr(Λ)-submodule of gr F ( M )[ i +1],both T i and S i are finitely generated gr(Λ)-modules and are annihilated by some powerof J . Moreover, one checks that there are short exact sequences of gr(Λ)-modules(annihilated by some power of J ):0 → T i → gr F ( i +1) ( M ) → S i → , → S i [ − → gr F ( i ) ( M ) → T i → . Hence, gr F ( i ) ( M ) is also finitely generated over gr(Λ) and annihilated by a power of J .Consequently, F ( i ) is a good filtration on M by [LvO96, Thm.I.5.7] and (170) followsfrom Lemma 3.3.4.2. 161hanks to Lemma 3.3.4.3, we can define m q ( M ) to be m q (gr F ( M )) and Z ( M ) tobe Z (gr F ( M )) for any minimal prime ideal q of R and any good filtration F on M . Lemma 3.3.4.4.
Let M be as above and let → M → M → M → be an exactsequence of Λ -modules. Then we have in ⊕ q Z ≥ q : Z ( M ) = Z ( M ) + Z ( M ) . Proof.
We may equip M (resp. M ) with the induced filtration (resp. quotient filtra-tion) from the one of M , which are automatically good by [LvO96, Cor.I.5.5(1)] and[LvO96, Rem.I.5.2(2)]. Moreover the sequence 0 → gr( M ) → gr( M ) → gr( M ) → M ) and gr( M ) are finitely generatedgr(Λ)-modules annihilated by some power of J , and the result follows from Lemma3.3.4.2.If M is a finitely generated Λ-module, recall from [LvO96, Def.III.2.1.1] that thegrade of M is by definition the smallest integer j Λ ( M ) ≥ j Λ ( M )Λ ( M ) = 0(with j Λ ( M ) def = + ∞ if E j Λ ( M ) = 0 for all j ≥ F on M , wedefine similarly the grade j gr(Λ) (gr F ( M )) of the gr(Λ)-module gr F ( M ). By [LvO96,Thm.III.2.5.2] we have j gr(Λ) (gr F ( M )) = j Λ ( M ) (note that Λ is a left and right Zariskiring by [LvO96, Prop.II.2.2.1]), in particular j gr(Λ) (gr F ( M )) doesn’t depend on thegood filtration F .Recall that the Krull dimension dim R ( N ) of a finitely generated module N over R (which is commutative) is the Krull dimension of R/ Ann R ( N ). For such a module N , by the argument in the proof of [BHH + , Lemma 5.1.3] applied to A = gr(Λ), I = ( h , . . . , h f − ) and with N instead of gr m M there, we have j gr(Λ) ( N ) = dim( I /Z ) − dim R ( N ) . (171)Now, for M as above, assume that gr F ( M ) is annihilated by a power of J . Thenapplying (171) to the R -modules N = J i gr F ( M ) /J i +1 gr F ( M ) for i ≥ j Λ ( M ) ≥ dim( I /Z ) − dim( R ) = 3 f − f = 2 f. (172)Moreover, by the same dévissage using [LvO96, Cor.III.2.1.6] (note that all assump-tions are satisfied since gr(Λ) is Auslander regular) and (171), we deduce that if j Λ ( M ) = j gr(Λ) (gr F ( M )) > f , then we have dim R ( J i gr F ( M ) /J i +1 gr F ( M )) < f forall i , hence Z ( J i gr F ( M ) /J i +1 gr F ( M )) = 0 for all i ≥ Z ( M ) = 0 (see (124)). Theorem 3.3.4.5.
Let M be a finitely generated Λ -module such that gr( M ) is an-nihilated by a power of J for one ( equivalently every ) good filtration on M . Then Z (E f Λ ( M )) is well-defined and we have Z ( M ) = Z (E f Λ ( M )) . roof. If j Λ ( M ) > f , then the result is trivial since both terms are 0 by the sentencejust before the proposition. So from (172) we may assume j Λ ( M ) = 2 f in the rest ofthe proof.Choose a good filtration F of M so that Z ( M ) = Z (gr F ( M )). We first show thatthe gr(Λ)-module E f gr(Λ) (gr F ( M )) is also annihilated by some power of J . Indeed,gr F ( M ) has a finite filtration whose graded pieces are annihilated by J , hence bydévissage it suffices to show that E f gr(Λ) ( N ) is annihilated by J if N is a finitelygenerated R -module. As in the proof of Proposition 3.3.1.10 it is equivalent to provethe same property for E fR ( N ), which is obvious as R is commutative.As a consequence, by the first statement in Proposition 3.3.4.6 below the gradedmodule associated to the filtration on E f Λ ( M ) in loc.cit. is again finitely generatedover gr(Λ) and annihilated by some power of J . Hence Z (E f Λ ( M )) can be defined.By Proposition 3.3.4.6 the cokernel of the injection gr(E f Λ ( M )) ֒ → E f gr(Λ) (gr F ( M ))has grade > f , hence its associated characteristic cycle is 0, as explained above.From Lemma 3.3.4.2 we deduce an equality of cycles Z (cid:16) gr(E f Λ ( M )) (cid:17) = Z (cid:16) E f gr(Λ) (gr F ( M )) (cid:17) . Hence, we are left to show that Z (gr F ( M )) = Z (cid:16) E f gr(Λ) (gr F ( M )) (cid:17) . As gr(Λ) is an Auslander regular ring, any subquotient N of gr F ( M ) has grade ≥ f (by [LvO96, Prop.III.2.1.6]) and is such that E j gr(Λ) ( N ) has grade ≥ j for any j ≥ j gr(Λ) ( N ) and all its subquotients have zero cycle if j < f or if j > f (byLemma 3.3.4.2 and the discussion before the proposition for the latter). Hence, for n large enough so that J n annihilates gr F ( M ), we deduce using again Lemma 3.3.4.2: Z (cid:16) E f gr(Λ) (gr F ( M )) (cid:17) = n − X i =0 Z (cid:16) E f gr(Λ) ( J i gr F ( M ) /J i +1 gr F ( M )) (cid:17) . By the definition of Z and of m q ( N ), see (124), it thus suffices to show Z ( N ) = Z (E f gr(Λ) ( N ))for any finitely generated R -module N . Using Lemma 3.3.1.9 it suffices to show Z ( N ) = Z (Hom R ( N, R )) , which is equivalent to show that for any minimal prime ideal q of R ,lg R q ( N q ) = lg R q (Hom R ( N, R ) q ) . Using the isomorphism Hom R ( N, R ) q ∼ = Hom R q ( N q , R q ) and noting that R q is a field(being artinian, and reduced as R is), the result is clear.163he following general result was used in the proof of Theorem 3.3.4.5. Recall thata finitely generated gr(Λ)-module of grade j is Cohen–Macaulay if all its E i gr(Λ) are 0when i = j . Proposition 3.3.4.6.
Let M be a finitely generated Λ -module of grade j with a goodfiltration. Then there exists a good filtration on E j Λ ( M ) such that gr(E j Λ ( M )) is asubmodule of E j gr(Λ) (gr( M )) and the corresponding cokernel has grade ( over gr(Λ)) ≥ j + 1 . If gr( M ) is moreover Cohen–Macaulay, then gr(E j Λ ( M )) ∼ → E j gr(Λ) (gr( M )) . Proof.
See [Bjö89, Prop.3.1] and the remark following it. We explain the proof fol-lowing the presentation of [LvO96, §III.2.2].As in [LvO96, §III.2.2], we may construct a filtered free resolution of M · · · → L j → L j − → · · · → L → M → ( − ) = Hom Λ ( − , Λ) obtain a filtered complex of finitely generated Λ-modules 0 → E ( L ) → E ( L ) → · · · , (173)where each E ( L j ) is endowed with a good filtration as in loc.cit. . Taking the associ-ated graded complex of (173), we obtain a complex of gr(Λ)-modules (denoted G ( ∗ )in loc.cit. ): 0 → gr(E ( L )) → gr(E ( L )) → · · · and by [LvO96, Lemma III.2.2.2(2)] we have isomorphisms E (gr( L j )) ∼ =gr(E ( L j )) for j ≥
0. Next, as in [LvO96, §III.1] we may associate a spectral sequence { E rj , r ≥ , j ≥ } to the filtered complex (173) and define a good filtration onE j Λ ( M ) for j ≥ E j = gr(E ( L j )) and E j = E j gr(Λ) (gr( M )) for any j ;(b) for any fixed r ≥
1, there is a complex0 → E r → · · · → E rj → E rj +1 → · · · whose homology gives E r +1 j ;(c) for r large enough (depending on j ), E ∞ j = E rj ∼ = gr(E j Λ ( M )).Since j Λ ( M ) = j by assumption, we also have j gr(Λ) (gr( M )) = j by [LvO96,Thm.III.2.5.2] and so E j = 0 for j < j . By (b), this implies short exact sequences0 → E r +1 j → E rj → E rj +1 , ∀ r ≥ . r large enough, gr(E j Λ ( M )) = E ∞ j is a submodule of E j .Moreover, since E rj +1 has grade ≥ j + 1 for all r and so do its subquotients, thecokernel of E ∞ j ֒ → E j also has grade ≥ j + 1.If moreover gr( M ) is Cohen–Macaulay, then E j = 0 except for j = j , hence E ∞ j = E j which implies the last claim. π in the semisimple case For ρ as in §3.3.1 assumed moreover semisimple and strongly generic, and π as in§3.3.2 with moreover r = 1 and satisfying one more hypothesis, we prove that π isgenerated over GL ( K ) by its GL ( O K )-socle, is irreducible if ρ is, and is semisimpleof length 3 if ρ is reducible split and f = 2.We keep the notation in §3.3.2 and we assume moreover that ρ is semisimple andsatisfies the strong genericity condition (127) (we will use the results of §3.2). Wefix an admissible smooth representation π of GL ( K ) over F satisfying the conditions(i), (ii) in loc.cit. with r = 1 in (i), i.e. π K ∼ = D ( ρ ). Recall this implies that gr( π ∨ )is annihilated by J , where gr( π ∨ ) is computed with the m -adic filtration. We assumemoreover:(iii) π ∨ is essentially self-dual of grade 2 f , i.e. there is a GL ( K )-equivariant iso-morphism of Λ-modules E f Λ ( π ∨ ) ∼ = π ∨ ⊗ (det( ρ ) ω − ) (174)(recall det( ρ ) ω − is the central character of π ). Here E j Λ ( π ∨ ) is endowed with theaction of GL ( K ) (compatible with the Λ-module structure) defined in [Koh17,Prop.3.2]. Remark 3.3.5.1.
Conditions (i) to (iii), with r = 1 in (i), will be satisfied for π coming from the global theory in the so-called minimal case (see §3.4.4). The reasonto impose the extra assumption r = 1 in (i) is that although for general r we have anequality of diagrams ( π I ֒ → π K ) = ( D ( ρ ) I ֒ → D ( ρ )) ⊕ r for the representations π coming from cohomology (see Theorem 3.4.1.1 below), wedo not know if this implies that π has the form π ′⊕ r for some representation π ′ ofGL ( K ).Given σ ∈ W ( ρ ), we define the length of σ as follows: if λ ∈ D corresponds to σ (see §3.3.1), then ℓ ( σ ) def = ℓ ( λ ), see (161). For 0 ≤ ℓ ≤ f , let W ℓ ( ρ ) def = { σ ∈ W ( ρ ) , ℓ ( σ ) = ℓ } τ ℓ ( ρ ) def = ⊕ σ ∈ W ℓ ( ρ ) σ . We call W ℓ ( ρ ), or by abuse of notation τ ℓ ( ρ ), an orbit in W ( ρ ). Note that this is different from an orbit of δ in W ( ρ ) as defined in §3.2.4(see §3.2.3 for δ ), i.e. in general τ ℓ ( ρ ) contains several orbits of δ . Lemma 3.3.5.2. If π ′ is a nonzero subrepresentation of π , then soc GL ( O K ) ( π ′ ) is adirect sum of orbits in W ( ρ ) .Proof. It is clear that ( π ′ I ֒ → π ′ K ) is a subdiagram of ( π I ֒ → π K ). The result fol-lows from this using [BP12, Thm.15.4] together with the proof of [BP12, Thm.19.10].Actually, when ρ is irreducible, we even have soc GL ( O K ) ( π ′ ) = soc GL ( O K ) ( π ) by (theproof of) [BP12, Thm.19.10].We use without comment the notation and definitions in §3.1.4 and denote bylg( τ ) the length of a finite-dimensional representation τ of GL ( O K ) over F . Proposition 3.3.5.3.
Let π ′ be a subquotient of π . (i) We have dim F (( X )) D ∨ ξ ( π ′ ) = m p ( π ′∨ ) . (ii) Assume that π ′ is a subrepresentation of π . Then dim F (( X )) D ∨ ξ ( π ′ ) = m p ( π ′∨ ) = lg(soc GL ( O K ) ( π ′ )) . In particular, if π ′ = 0 , then D ∨ ξ ( π ′ ) = 0 . (iii) Assume that π ′ is a nonzero quotient of π . Then D ∨ ξ ( π ′ ) = 0 .Proof. (i) First, for any subquotient π ′ of π , we equip the Λ-module π ′∨ with a goodfiltration F by choosing two submodules π ∨ ⊆ π ∨ of π ∨ (with filtrations inducedfrom the m -adic one on π ∨ ) such that π ′∨ ∼ = π ∨ /π ∨ and taking the induced filtration. Then gr F ( π ′∨ ) is again an R -module, and dim F (( X )) D ∨ ξ ( π ′ ) ≤ m p ( π ′∨ ) by Corollary3.1.4.5. Since dim F (( X )) D ∨ ξ ( π ) = m p ( π ∨ ) by Corollary 3.3.2.4, since D ∨ ξ ( − ) is an exactfunctor by Theorem 3.1.3.3 and since Z ( − ), and in particular m p ( − ), are additiveby Lemma 3.3.4.4, the result follows.(ii) By assumption π ′ is a subrepresentation of π . Using that soc GL ( O K ) ( π ′ ) is aunion of orbits of δ , or equivalently of S as in (148), by Lemma 3.3.5.2, it followsfrom Proposition 3.2.4.2 thatdim F (( X )) D ∨ ξ ( π ′ ) ≥ lg(soc GL ( O K ) ( π ′ )) . The filtrations on π ∨ and π ∨ might not be the m -adic ones, and the resulting filtration on π ′∨ might depend on the choice of π ∨ and π ∨ . m p ( π ′∨ ) ≤ lg(soc GL ( O K ) ( π ′ )) (see the proof of Theorem 3.3.2.3). Hence all the three quantitiesare equal by (i).(iii) Let π ′′ be the kernel of the quotient map π ։ π ′ so that we have an exactsequence of Λ-modules: 0 → π ′∨ → π ∨ → π ′′∨ → . Since π ∨ is essentially self-dual of grade 2 f by assumption, π ′∨ also has grade 2 f by[LvO96, Prop.III.4.2.8(1)] and [LvO96, Prop.III.4.2.9]. Taking E i Λ ( − ), we obtain along exact sequence of Λ-modules0 → E f Λ ( π ′′∨ ) → E f Λ ( π ∨ ) → E f Λ ( π ′∨ ) → E f +1Λ ( π ′′∨ ) (175)which gives rise by Pontryagin duality to an exact sequence of admissible smoothrepresentations of GL ( K ) with central character (see [Koh17, Cor.1.8]). Define e π tobe the admissible smooth representation of GL ( K ) such that e π ∨ ⊗ (det( ρ ) ω − ) ∼ = Im (cid:16) E f Λ ( π ∨ ) → E f Λ ( π ′∨ ) (cid:17) . (176)Since π ∨ is essentially self-dual by assumption (see (174)), e π ∨ is a quotient of π ∨ anddually e π is a subrepresentation of π . Since E f +1Λ ( π ′′∨ ) has grade ≥ f + 1 as Λ isAuslander regular, we have by (175) and the discussion before Theorem 3.3.4.5: Z (E f Λ ( π ′∨ )) = Z ( e π ∨ ⊗ (det( ρ ) ω − )) , hence Z ( π ′∨ ) = Z ( e π ∨ ) by Theorem 3.3.4.5 which implies in particular by (i):dim F (( X )) D ∨ ξ ( π ′ ) = dim F (( X )) D ∨ ξ ( e π ) . (177)Since j Λ ( π ′∨ ) = 2 f , Z ( π ′∨ ) is non-zero (using e.g. (171)), hence e π is non-zero, thus D ∨ ξ ( e π ) = 0 by (ii), and finally D ∨ ξ ( π ′ ) = 0 by (177). Remark 3.3.5.4. (i) The construction of e π in the proof of Proposition 3.3.5.3(iii)does not use the assumption that ρ is semisimple.(ii) It follows from Proposition 3.3.5.3(ii), from Corollary 3.1.4.5, from Lemma 3.1.4.4,from Lemma 3.1.4.1 and from (110) that for π ′ ⊆ π as in Proposition 3.3.5.3(ii) wehave rk A ( D A ( π ′ ) ´et ) = dim F (( X )) D ∨ ξ ( π ′ ) = m p (gr( π ′∨ )) = rk A ( D A ( π ′ )) . (178)As the A -submodule D A ( π ′ ) tor ⊆ D A ( π ′ ) of torsion elements is 0 (apply Corollary3.1.1.7 to the ideal of A which is the annihilator of the finitely generated A -module D A ( π ′ ) tor , recalling that A is a domain), it easily follows from (178) that the surjectionof A -modules D A ( π ′ ) ։ D A ( π ′ ) ´et is here an isomorphism. Theorem 3.3.5.5.
As a GL ( K ) -representation, π is generated by its GL ( O K ) -socle. roof. Let τ def = soc GL ( O K ) ( π ), let π ′ def = h GL ( K ) .τ i be the subrepresentation of π generated by τ and let π ′′ def = π/π ′ . Since D ∨ ξ ( − ) is exact by Theorem 3.1.3.6, we havedim F (( X )) D ∨ ξ ( π ) = dim F (( X )) D ∨ ξ ( π ′ ) + dim F (( X )) D ∨ ξ ( π ′′ ) . However, since π and π ′ have the same GL ( O K )-socle, we havedim F (( X )) D ∨ ξ ( π ) = dim F (( X )) D ∨ ξ ( π ′ )by Proposition 3.3.5.3(ii), thus D ∨ ξ ( π ′′ ) = 0. If π ′′ is non-zero this contradicts Propo-sition 3.3.5.3(iii). Corollary 3.3.5.6.
Assume that ρ is irreducible. Then π is irreducible and is asupersingular representation.Proof. This follows from Theorem 3.3.5.5 and [BP12, Thm.19.10(i)].
Remark 3.3.5.7. (i) A result analogous to Theorem 3.3.5.5 when ρ is not semisimpleis proved in [HW, Thm.1.6].(ii) While we believe that Proposition 3.3.5.3 and Theorem 3.3.5.5 should be truewithout assuming r = 1, we don’t know how to prove a generalization of Corollary3.3.5.6 (i.e. π is semisimple and has length r in general), as mentioned in Remark3.3.5.1. Corollary 3.3.5.8.
Assume that ρ is reducible split. Then π has the form π = π ⊕ π f ⊕ π ′ , (179) where • π and π f are irreducible principal series such that E f Λ ( π ∨ i ) ∼ = π ∨ f − i ⊗ (det( ρ ) ω − ) , i ∈ { , f } ; • π ′ is generated by its GL ( O K ) -socle and π ′∨ is essentially self-dual ( as in (174)) .Moreover, π ′ is irreducible and supersingular when f = 2 .Proof. By the definition of W ( ρ ) (see [BP12, §11]), there exists a unique Serre weight σ ∈ W ( ρ ) such that ℓ ( σ ) = 0. Let χ σ be the character of I acting on σ I . It is easyto check that JH (cid:16) Ind GL ( O K ) I χ σ (cid:17) ∩ W ( ρ ) = { σ } . Let π def = h GL ( K ) .σ i , a subrepresentation of π . We claim that π is an irreducibleprincipal series. Indeed, by [HW, Lemma 5.14] and its proof, the morphism (inducedfrom σ ֒ → π by Frobenius reciprocity)c-Ind GL ( K )GL ( O K ) K × σ → π GL ( K )GL ( O K ) K × σ / ( T − µ )for some µ ∈ F × (as soc GL ( O K ) ( π ) is multiplicity-free). Note that the genericity of ρ implies that dim F σ ≥
2, hence the representation c-Ind GL ( K )GL ( O K ) K × σ / ( T − µ ) isirreducible and isomorphic to some principal series by [BL94, Thm.30]. This provesthe claim. Moreover, the GL ( O K )-socle of π is exactly σ , and if π ∼ = Ind GL ( K ) B ( K ) χ for some smooth character χ : T ( K ) → F × then χ s | H = χ σ . Similarly, there existsa unique Serre weight σ f ∈ W ( ρ ) such that ℓ ( σ f ) = f . It satisfies againJH (cid:16) Ind GL ( O K ) I χ σ f (cid:17) ∩ W ( ρ ) = { σ f } and by the same argument as above the subrepresentation π f def = h GL ( K ) .σ f i of π is an irreducible principal series with GL ( O K )-socle equal to σ f , and if π f ∼ =Ind GL ( K ) B ( K ) χ f then χ sf | H = χ σ f . The map π ⊕ π f → π is injective since it is injectiveon the GL ( O K )-socles.Letting π ′ def = π/ ( π ⊕ π f ), we have an exact sequence of Λ-modules:0 → π ′∨ → π ∨ → π ∨ ⊕ π ∨ f → . As Λ is Auslander regular and π ∨ is of grade 2 f , it follows from [LvO96, Cor.III.2.1.6]that π ′∨ is of grade ≥ f , hence E f − ( π ′∨ ) = 0 and there is an exact sequence of(finitely generated) Λ-modules0 → E f Λ ( π ∨ ) ⊕ E f Λ ( π ∨ f ) → E f Λ ( π ∨ ) → E f Λ ( π ′∨ ) . Since π ∨ is essentially self-dual by assumption (see (174)) and since E f Λ ( π ∨ ) ∨ andE f Λ ( π ∨ f ) ∨ are also irreducible principal series by [Koh17, Prop.5.4], we see that π admits a quotient isomorphic to π ′ ⊕ π ′ f , where π ′ i (for i ∈ { , f } ) is the (irreducible)principal series such that π ′∨ i ⊗ (det( ρ ) ω − ) = E f Λ ( π ∨ f − i ) . (180)Explicitly, if π ′ i ∼ = Ind GL ( K ) B ( K ) χ ′ i for some smooth characters χ ′ i : T ( K ) → F × , and if welet α B def = ω ⊗ ω − : T ( K ) → F × and η def = det( ρ ) ω − (for short), then by [HW, Lemma10.7] (which is based on [Koh17, Prop.5.4]): χ ′ f = χ − α B ( η ⊗ η ) , χ ′ = χ − f α B ( η ⊗ η ) . (181)Let us compute the GL ( O K )-socle of π ′ f (the case of π ′ is similar). Since η is equalto the central character of π , we have χ − ( η ⊗ η ) = χ s , so that (181) becomes χ ′ f = χ s α B . Since χ s | H = χ σ as seen in the first paragraph, we deduce( χ ′ f ) s | H = χ sσ α − B = χ σ f , (182)169here the last equality holds by an easy check using the definition of σ and σ f (see[BP12, §11]). In particular, our genericity assumption on ρ implies that χ ′ f = χ ′ sf when restricted to T ( O K ). Using [BL94, Thm.34(2)], this implies that the GL ( O K )-socle of π ′ f is irreducible and actually isomorphic to σ f by (182). Similarly, theGL ( O K )-socle of π ′ is isomorphic to σ .We claim that the composite morphism π ⊕ π f ֒ → π ։ π ′ ⊕ π ′ f is an isomorphism. Since π is generated by its GL ( O K )-socle, namely L σ ∈ W ( ρ ) σ ,the composite morphism ι : M σ ∈ W ( ρ ) σ ֒ → π ։ π ′ is non-zero. Since the image is contained in soc GL ( O K ) ( π ′ ), which is equal to σ asseen in the last paragraph, ι is non-zero when restricted to σ . But, by constructionwe have h GL ( K ) .σ i = π inside π , hence the composite morphism π ֒ → π ։ π ′ is non-zero, hence an isomorphism as both π and π ′ are irreducible. In the sameway the composite morphism π f ֒ → π ։ π ′ f is also an isomorphism. This proves theclaim, from which the decomposition (179) immediately follows. From (180) we alsodeduce the isomorphism E f Λ ( π ∨ i ) ∼ = π ∨ f − i ⊗ η for i ∈ { , f } .We now finish the proof. First, π ′ is generated by its GL ( O K )-socle by Theorem3.3.5.5. Explicitly, we have soc GL ( O K ) ( π ′ ) = M σ ∈ W ( ρ ) <ℓ ( σ )
7. We makethe following extra assumptions on the field F and the unitary group H :(i) F/F + is unramified at all finite places;(ii) p is unramified in F + ;(iii) H is defined over O F + and H × O F + F + is quasi-split at all finite places of F + .Condition (i) (together with the fact that any p -adic place of F + splits in F ) implies[ F + : Q ] is even (see [GK14, §3.1]). By [GK14, §3.1.1] such groups H always exist.We denote by R (cid:3) r ˜ w the universal framed deformation ring of r ˜ w over W ( F ) ( ˜ w is anyfinite place of F ). We set K def = F + v and f def = [ K : Q p ].We let r : Gal( F /F ) → GL ( F ) as in §2.1.3 and make the following extra assump-tions on r (recall that S p is the set of places of F + dividing p ):(iv) r | Gal(
F /F ( p √ is adequate ([Tho17, Def.2.20]);(v) r ˜ w is unramified if ˜ w | F + is inert in F ;(vi) R (cid:3) r ˜ w is formally smooth over W ( F ) if r ˜ w is ramified and ˜ w | F + / ∈ S p ;(vii) r ˜ w is generic in the sense of [BP12, Def.11.7] if ˜ w | F + ∈ S p \{ v } ;(viii) r ˜ v is, up to twist, of one of the following forms for ˜ v | F + = v :• r ˜ v | I K ∼ = ω ( r +1)+ ··· + p f − ( r f − +1) f
00 1 ! ≤ r i ≤ p − r ˜ v | I K ∼ = ω ( r +1)+ ··· + p f − ( r f − +1)2 f ω p f (same)2 f ≤ r ≤ p −
5, 3 ≤ r i ≤ p − i > w | F + and ˜ v | F + using condition (i) in§2.1.3 (the genericity conditions in (viii) are satisfied in [DL, §3.3] and don’t dependon the choices of σ , σ ′ ). We denote by S r the finite set of finite places of F + suchthat ˜ w | F + ∈ S r if and only if r ˜ w is ramified. Thus S p ⊆ S r and by (ii) any place in S r splits in F + . We fix a finite place v of F + which is not in S r and satisfies theassumptions in [EGS15, §6.6], and we choose f v | v in F .171e choose S a finite set of finite places of F + that split in F containing S r butnot v , and a compact open subgroup U = Q w U w ⊆ H ( A ∞ F + ) such that(ix) U w ⊆ H ( O F + w ) if w splits in F ;(x) U w is maximal hyperspecial in H ( F + w ) if w is inert in F ;(xi) U w = H ( O F + w ) if w / ∈ S ∪ { v } and w splits in F or if w ∈ S p ;(xii) ι e v ( U v ) is contained in the upper-triangular unipotent matrices mod f v .We also define V def = U p Q w ∈ S p V w , where U p def = Q w / ∈ S p U w and V w is a pro- p normalsubgroup of U w if w ∈ S p (hence V is normal in U ). We set Σ def = S ∪ { v } and assume S ( V, F )[ m Σ ] = 0 (see §2.1.2). Note that S ( V, F )[ m Σ ] doesn’t depend on S as above bythe proof of [BDJ10, Lemma 4.6(a)]. For each place w ∈ S p we choose a place ˜ w | w in F . For w ∈ S p recall from §3.2.1 that W ( r ˜ w (1)) is the set of Serre weights associatedto r ˜ w (1) def = r ˜ w ⊗ ω defined as in [BDJ10, §3]. Then it follows from [GLS14, Thm.A]and [BLGG13, Def.2.9] that we haveHom U (cid:16) ⊗ w ∈ S p σ ˜ w , S ( V, F )[ m Σ ] (cid:17) = 0 ⇐⇒ σ ˜ w ∈ W ( r ˜ w (1)) ∀ w ∈ S p , (183)where we consider ⊗ w ∈ S p σ ˜ w as a representation of U via U ։ U/V ∼ → Q w ∈ S p U w /V w and the isomorphisms ι ˜ w . Note that the left-hand side of (183) is also isomorphic toHom U ( ⊗ w ∈ S p σ ˜ w , S ( U p , F )[ m Σ ]), where S ( U p , F )[ m Σ ] is defined as in §2.1.2, replacing U v by U p .We freely use the previous local notation ( I is the pro- p Iwahori subgroup inGL ( O K ) = GL ( O F ˜ v ) etc.) and set ρ def = r ˜ v (1). Theorem 3.4.1.1.
Choose Serre weights σ ˜ w ∈ W ( r ˜ w (1)) for w ∈ S p \{ v } and set π def = Hom U v ( ⊗ w ∈ S p \{ v } σ ˜ w , S ( V v , F )[ m Σ ]) . Then there exist an integer r ≥ only depending on v , U v , V v , ⊗ w ∈ S p \{ v } σ ˜ w and r and a diagram D ( ρ ) = ( D ( ρ ) ֒ → D ( ρ )) as in (3.2.1) only depending on ρ = r ˜ v (1)( and satisfying the assumptions in loc.cit. on the constants ν i ) such that there is anisomorphism of diagrams D ( ρ ) ⊕ r ∼ = ( π I ֒ → π K ) . The case r = 1 of Theorem 3.4.1.1 is known and due to Dotto and Le ([DL,Thm.1.3]). We generalize below their proof to the case r ≥ + , §8.2]. Moreover the diagram D ( ρ ) in Theorem 3.4.1.1 is in fact the same asthe diagram D ( π glob ( ρ )) of [DL, Thm.1.3].172 .4.2 Review of patching functors We recall the patching functors of [EGS15, §6.6] and some results of [BHH + , §8.2].We keep the notation of §3.4.1. We choose Serre weights σ ˜ w ∈ W ( r ˜ w (1)) for w ∈ S p \{ v } and set σ v def = O w ∈ S p \{ v } σ ˜ w . For each w ∈ S p \{ v } we fix a tame inertial type τ ˜ w at the place ˜ w such that, denotingby σ ( τ ˜ w ) the irreducible smooth representation of GL ( O F ˜ w ) over E associated byHenniart to τ ˜ w in the appendix to [BM02], JH( σ ( τ ˜ w )) contains exactly one Serreweight in W ( r ˜ w (1)) (where ( − ) means the mod p semisimplification). The existenceof such τ ˜ w follows from [EGS15, Prop.3.5.1], and the fact σ ( τ ˜ w ) can be realized over E = W ( F )[1 /p ] follows from [EGS15, Lemma 3.1.1]. For each w ∈ S p \{ v } we also fixa GL ( O F ˜ w )-invariant W ( F )-lattice σ ( τ ˜ w ) in σ ( τ ˜ w ).We define σ ,v def = O w ∈ S p \{ v } σ ( τ ˜ w ) , and for any continuous representation σ ˜ v of GL ( O F ˜ v ) on a finite type W ( F )-module,we consider σ ,v ⊗ W ( F ) σ ˜ v as a representation of U via U ։ Q w ∈ S p U w and the iso-morphisms ι ˜ w . We define S ( U p , W ( F )) m Σ exactly as in §2.1.2 replacing F by W ( F )and U v by U p . Then, as in [EGS15, §§6.2,6.6], by “patching” Hom U ( σ ,v ⊗ W ( F ) σ ˜ v , S ( U p , W ( F )) m Σ ) ∗ for various U (where ( − ) ∗ def = Hom W ( F ) (( − ) , E/W ( F )) as in loc.cit. ), we obtain a patching functor M ∞ : σ ˜ v M ∞ ( σ ,v ⊗ W ( F ) σ ˜ v )which is an exact functor from the category of continuous representations σ ˜ v ofGL ( O F ˜ v ) on finite type W ( F )-modules to the category of finite type R ∞ -modules(though this patching functor depends on σ ,v , we just write M ∞ ( σ ˜ v ) in the sequel).The local ring R ∞ is (see [GK14, §4.3] or [DL, §6.2]): R ∞ def = R loc J X , . . . , X q − [ F + : Q ] K , where q is an integer ≥ [ F + : Q ] and R loc def = (cid:18) b ⊗ w ∈ S \ S p R (cid:3) r ˜ w (1) (cid:19) b ⊗ W ( F ) (cid:18) b ⊗ w ∈ S p \{ v } R (cid:3) , (1 , ,τ ˜ w r ˜ w (1) (cid:19) b ⊗ W ( F ) R (cid:3) r v (1) . Recall R (cid:3) , (1 , ,τ ˜ w r ˜ w (1) is the reduced p -torsion free quotient of R (cid:3) r ˜ w (1) parametrizing framedpotentially Barsotti–Tate deformations with inertial type τ ˜ w (by local-global compat-ibility and the inertial Langlands correspondence, for w ∈ S p \{ v } the action of R (cid:3) r ˜ w (1) on M ∞ ( σ ,v ⊗ W ( F ) σ v ) factors through this quotient, see [EGS15, §6.6]). As in [BHH + ,1738.1] (see the discussion before [BHH + , Rem.8.1.3] but note that we do not need tofix the determinant here) we have isomorphisms R (cid:3) r ˜ w (1) ∼ = W ( F ) J X , X , X , X K for w ∈ S \ S p , and, by genericity of r v , R (cid:3) r v (1) ∼ = W ( F ) J X , . . . , X F ˜ v : Q p ] K . By [EGS15,Thm.7.2.1(2)] (and [GK14, Rk.5.2.2]) we have R (cid:3) , (1 , ,τ ˜ w r ˜ w (1) ∼ = W ( F ) J X , . . . , X F ˜ w : Q p ] K ,so that we finally get R ∞ ∼ = R (cid:3) r v (1) J X , . . . , X | S |− q − [ F + v : Q p ] K ∼ = W ( F ) J X , . . . , X | S | + q +3[ F + v : Q p ] K . (184)Moreover, if σ ˜ v is free of finite type over W ( F ), then M ∞ ( σ ˜ v ) is free of finite typeover a subring S ∞ of R ∞ , where S ∞ ∼ = W ( F ) J x , . . . , x | S | + q K . Finally, denoting by m ∞ the maximal ideal of R ∞ , we have M ∞ ( σ ˜ v ) / m ∞ ∼ = Hom U (cid:16) ( ⊗ w ∈ S p \ v σ ˜ w ) ⊗ F σ ˜ v , S ( U p , F )[ m Σ ] (cid:17) ∨ ∼ = Hom U v ( σ ˜ v , π ) ∨ , (185)where π is as in Theorem 3.4.1.1.Since everything is now at the place ˜ v , we drop the index ˜ v . If τ is a tameinertial type, we set R (1 , ,τ ∞ def = R ∞ ⊗ R (cid:3) ρ R (cid:3) , (1 , ,τρ . If σ ∈ W ( ρ ), we denote by P σ the projective F [GL ( F q )]-envelope of σ and by e P σ the projective W ( F )[GL ( F q )]-module lifting P σ . We recall that the scheme theoretic support of an R ∞ -module M is R ∞ / Ann R ∞ ( M ). The following theorem then follows by exactly the same proof asfor [BHH + , Prop.8.2.3] and [BHH + , Prop.8.2.5]. Theorem 3.4.2.1.
There exists an integer r ≥ such that (i) for any σ ∈ W ( ρ ) the module M ∞ ( σ ) is free of rank r over its scheme-theoreticsupport which is a domain; (ii) for any σ ∈ W ( ρ ) the modules M ∞ ( e P σ ) and M ∞ ( P σ ) are free of rank r overtheir respective scheme-theoretic support; (iii) for any tame inertial type τ such that JH( σ ( τ )) ∩ W ( ρ ) = ∅ and any GL ( O K ) -invariant W ( F ) -lattice σ ( τ ) in σ ( τ ) with irreducible cosocle, the module M ∞ ( σ ( τ )) is free of rank r over its scheme-theoretic support, which is thedomain R (1 , ,τ ∞ . Corollary 3.4.2.2.
Let π as in Theorem 3.4.1.1 and r as in Theorem 3.4.2.1. Wehave an isomorphism of GL ( O K ) K × -representations D ( ρ ) ⊕ r ∼ = π K .Proof. The action of the center K × being by definition the same on both sides, wecan focus on the action of GL ( O K ). It follows from Theorem 3.4.2.1(i) and (ii)and from (185) that the surjection P σ ։ σ induces an isomorphism of r -dimensional F -vector spaces Hom GL ( O K ) ( σ, π K ) ∼ → Hom GL ( O K ) ( P σ , π K ). In particular the mul-tiplicity of each σ ∈ W ( ρ ) in π K is r . It follows from M ∞ ( D ,σ ( ρ ) /σ ) = 0 (recall174 ( ρ ) = ⊕ σ ∈ W ( ρ ) D ,σ ( ρ )) and from (185) that the injection σ ֒ → D ,σ ( ρ ) inducesan isomorphism Hom GL ( O K ) ( D ,σ ( ρ ) , π K ) ∼ → Hom GL ( O K ) ( σ, π K ). This gives an in-clusion D ( ρ ) ⊕ r ֒ → π K . If this inclusion is strict, then by maximality of D ( ρ ) ⊕ r (an obvious generalization of [BP12, Prop.13.1]) this implies there exists σ ∈ W ( ρ )which appears in π K /D ( ρ ) ⊕ r , and hence has multiplicity > r in π K , which is acontradiction. Remark 3.4.2.3.
In the proof of Theorem 3.4.2.1, and hence also in Corollary 3.4.2.2,one only needs the slightly weaker bounds 1 ≤ r i ≤ p − ≤ r ≤ p − r ˜ v is irreducible) in the genericity conditions (viii) on r ˜ v (or equivalently ρ ) in §3.4.1(these bound are used in [LMS, §4] which is used in the proof of [BHH + , Prop.8.2.5]). We prove Theorem 3.4.1.1 using the method of [DL, §4].We keep the notation in §§3.4.1, 3.4.2. Everything in this section being at theplace ˜ v , we drop it from the notation. Recall we identify the set of embeddings F q ֒ → F with { , . . . , f − } via σ ◦ ϕ i i . We denote by P the set of subsets of { , . . . , f − } and by J c ∈ P the complement of a subset J ∈ P .We start by fixing a tame inertial type τ such that JH( σ ( τ )) ∩ W ( ρ ) = ∅ anda GL ( O K )-invariant W ( F )-lattice θ in σ ( τ ) with irreducible cosocle. With thenotation of [EGS15, §5.1] there is I ∈ P such that this cosocle is σ I ( τ ) and θ = σ o I ( τ ).As in [EGS15, p.77] we can reindex the irreducible constituents of θ /p by elements J ′ in P as follows: σ J ′ def = σ ( J ′ ∪ I c ) \ ( J ′ ∩ I c ) ( τ ) , so that (by [EGS15, Thm.5.1.1]) the j -th layer of the cosocle filtration of θ /p consistsof the σ J ′ for | J ′ | = f − j , 0 ≤ j ≤ f . By the beginning of the proof of [EGS15,Thm.10.1.1] (see loc.cit. p.77), there is J ′ min ⊆ J ′ max in P such that JH( θ /p ) ∩ W ( ρ ) = { σ J ′ , J ′ min ⊆ J ′ ⊆ J ′ max } . By [EGS15, Thm.7.2.1] we have R (1 , ,τ ∞ ∼ = (cid:16) W ( F ) J ( X ′ j , Y ′ j ) j ∈ J ′ max \ J ′ min K / ( X ′ j Y ′ j − p ) j ∈ J ′ max \ J ′ min (cid:17) J U , . . . , U d K for some integer d ≥
0. Up to renumbering the variables we can assume that theirreducible component of R (1 , ,τ ∞ /p corresponding to σ J ′ , J ′ min ⊆ J ′ ⊆ J ′ max , in [EGS15,p.77] (which is the support of M ∞ ( σ J ′ ) by Theorem 3.4.2.1(i)) is given by the ideal(( X ′ j ) j ∈ J ′ \ J ′ min , ( Y ′ j ) j ∈ J ′ max \ J ′ ).We first fix J ∈ P such that | J | = f −
1, so that J c = { j } for some j ∈ { , . . . , f − } . We let θ be the unique (up to homothety) GL ( O K )-invariant W ( F )-lattice in σ ( τ ) with irreducible cosocle σ J ([EGS15, Lemma 4.1.1]). Up to multiplication byan element in W ( F ) × , there is a unique GL ( O K )-equivariant saturated inclusion175 : θ ֒ → θ , i.e. such that the induced morphism ι : θ/p → θ /p is nonzero. Recallthat by Theorem 3.4.2.1(iii) both M ∞ ( θ ) and M ∞ ( θ ) are free of rank r over R (1 , ,τ ∞ . Lemma 3.4.3.1.
The image of M ∞ ( ι ) : M ∞ ( θ ) ֒ → M ∞ ( θ ) is xM ∞ ( θ ) , where x = p if j ∈ J ′ min , x = X ′ j if j ∈ J ′ max \ J ′ min and x = 1 if j / ∈ J ′ max .Proof. It follows from [EGS15, Thm.5.2.4(4)] (up to a reindexation as above) that p ( θ /ι ( θ )) = 0 and that the irreducible constituents of θ /ι ( θ ) are the σ J ′ for J ′ containing j . In particular θ /ι ( θ ) is of the form σ J for a capped interval J as in[EGS15, p.81] (namely J = { J ′ , j ∈ J ′ } ). By the proof of [BHH + , Prop.8.2.3] themodule M ∞ ( θ /ι ( θ )) = M ∞ ( σ J ) is free of rank r over its schematic support, which isthe unique reduced quotient of R (1 , ,τ ∞ /p with irreducible components correspondingto the σ J ′ such that j ∈ J ′ and J ′ min ⊆ J ′ ⊆ J ′ max . If j / ∈ J ′ max , there are no such J ′ , sothis quotient is 0 (i.e. M ∞ ( θ /ι ( θ )) = 0). If j ∈ J ′ max \ J ′ min , then this quotient is clearly( R (1 , ,τ ∞ /p ) / ( X ′ j ) = R (1 , ,τ ∞ / ( X ′ j ). Finally, if j ∈ J ′ min , all irreducible componentsremain, i.e. this quotient is R (1 , ,τ ∞ /p . The lemma follows by exactness of M ∞ .We now consider an arbitrary J ∈ P and let θ be the unique invariant W ( F )-lattice in σ ( τ ) with irreducible cosocle σ J . If J c = ∅ we set J c = { j , . . . , j h } and J i def = J ∐ { j , . . . , j h − i } for i ∈ { , . . . , h } (so J = { , . . . , f − } and J h = J ). Asabove we then denote by θ i for i ∈ { , . . . , h } the unique (up to homothety) invariant W ( F )-lattice in σ ( τ ) with irreducible cosocle σ J i and ι i : θ i ֒ → θ i − the correspondingsaturated inclusion for i ∈ { , . . . , h } (so θ is the same as before and θ h = θ ). Thecomposition ι ◦ · · · ◦ ι i : θ i ι i ֒ → θ i − ι i − ֒ → · · · θ ι ֒ → θ is still saturated since one can check using [EGS15, Thm.5.1.1] that the cosocle σ J h of θ h /p remains in the image of θ i /p → θ i − /p for all i ∈ { h, h − , . . . , } (indeed, by loc. cit. the Serre weights σ J i − σ J i − in θ /p form a nonsplit extension as J i ⊆ J i − and | J i − \ J i | = 1). In particular ι def = ι ◦ · · · ◦ ι h is the unique (up to scalar) saturatedinclusion θ ֒ → θ . Proposition 3.4.3.2.
There is x ∈ R (1 , ,τ ∞ such that the image of M ∞ ( ι ) : M ∞ ( θ ) ֒ → M ∞ ( θ ) is xM ∞ ( θ ) . Moreover the principal ideal xR (1 , ,τ ∞ only depends on ( thesemisimplification of ) θ /ι ( θ ) .Proof. The statement being trivial if J c = ∅ (equivalently if θ = θ ) we can assume J c = ∅ . For i ∈ { , . . . , h } we can apply Lemma 3.4.3.1 to ι i : θ i ֒ → θ i − instead of ι : θ ֒ → θ . Hence there is x i ∈ R (1 , ,τ ∞ such that the image of M ∞ ( ι i ) is x i M ∞ ( θ i − ).The image of M ∞ ( ι ) is thus ( Q hi =1 x i ) M ∞ ( θ ), i.e. we can take x = Q hi =1 x i . It followsthat M ∞ ( θ /ι ( θ )) ∼ = ( R (1 , ,τ ∞ / ( x )) ⊕ r . Hence the irreducible components of R (1 , ,τ ∞ / ( x )are the ones corresponding to the σ J ′ such that J ′ min ⊆ J ′ ⊆ J ′ max and σ J ′ appearsin θ /ι ( θ ), and their multiplicities are the multiplicities of the σ J ′ in θ /ι ( θ ). The176econd assertion then follows by the same argument as at the end of the proof of [DL,Prop.4.17] (it also follows from an explicit computation of x via Lemma 3.4.3.1).Till the end of this section, we now extensively use notation and results from [DL,§4] to which we refer the reader for more details.Recall that D ( ρ ) = ⊕ σ ∈ W ( ρ ) D ,σ ( ρ ). If χ : I → F × is a character appearing in D ( ρ ) I and F v χ ⊆ D ( ρ ) is the corresponding eigenspace (which is 1-dimensional), wedefine as in [DL, Def.4.1] Rχ as the character of I on (soc GL ( O K ) h F GL ( O K ) v χ i ) I ,which is also 1-dimensional as it is σ I for the unique σ ∈ W ( ρ ) such that χ appears in D ,σ ( ρ ) I . As in [BP12, p.8] we denote by χ s the character of I on (cid:16) p (cid:17) v χ ∈ D ( ρ ) I and by σ ( χ ) the Serre weight which is the cosocle of Ind GL ( O K ) I χ .We define as in [DL, Prop.4.14] an isomorphism h χ : M ∞ ( σ ( Rχ s )) / m ∞ ∼ −→ M ∞ ( σ ( Rχ )) / m ∞ (the “one-dimensional by Theorem 4.6” in the proof of loc.cit. can just be replacedby “of the same dimension by Theorem 3.4.2.1”; also note that h χ is an isomorphism,as it is dual to the isomorphism g χ in loc.cit. ). Proposition 3.4.3.3.
Let k ≥ and χ , . . . , χ k − arbitrary characters of I whichoccur on π I ( equivalently on D ( ρ ) I ) such that Rχ si = Rχ i +1 for i ∈ { , . . . , k − } and Rχ sk − = Rχ . Then the isomorphism h χ ◦ h χ ◦ · · · ◦ h χ k − ◦ h χ : M ∞ ( σ ( Rχ s )) / m ∞ ∼ −→ M ∞ ( σ ( Rχ s )) / m ∞ is the multiplication by a scalar in F × which depends neither on r nor on M ∞ . Inparticular this scalar is the same as in [DL, (34)] .Proof. We just indicate the steps in the proofs of [DL, §§4.4, 4.5], where the assump-tion r = 1 is used, and how one can extend the argument there to r ≥
1. We usewithout comment the notation of loc.cit. • The definition of the isomorphism e h χ : M ∞ ( θ Rχ s ) ∼ → M ∞ ( θ Rχ ) in [DL, (28)] holdsbecause one only needs to know that M ∞ ( θ Rχ s ) and M ∞ ( θ Rχ ) are free of the samefinite rank over R ∞ ( τ ). • By Proposition 3.4.3.2 there exists e U p ( χ ) ∈ R ∞ ( τ ) such that M ∞ ( ι )( M ∞ ( θ Rχ )) = e U p ( χ ) M ∞ ( θ Rχ s ), where ι : θ Rχ ֒ → θ Rχ s is as in the unlabelled commutative diagrambelow [DL, (27)]. Since R ∞ ( τ ) is a domain by [EGS15, Thm.7.2.1(2)] and M ∞ ( θ Rχ ), M ∞ ( θ Rχ s ) are free of rank r over R ∞ ( τ ) by Theorem 3.4.2.1(iii), there is a unique R ∞ ( τ )-linear isomorphism e ι χ : M ∞ ( θ Rχ ) ∼ → M ∞ ( θ Rχ s ) such that M ∞ ( ι ) = e ι χ ◦ e U p ( χ ),where e U p ( χ ) here means multiplication by e U p ( χ ) on M ∞ ( θ Rχ ). Then we have a com-mutative diagram analogous to [DL, (29)] replacing the multiplication by e U p ( χ ) inthe diagonal map by the map e h χ ◦ e ι χ ◦ e U p ( χ ) = e U p ( χ )( e h χ ◦ e ι χ ).177 By the commutativity of the right-hand side of (the analog of) [DL, (28)] and bythe isomorphism M ∞ ( Q ( χ s ) Rχ ) ∼ = M ∞ ( θ ( χ s ) Rχ ) /p , we deduce that the map h χ ◦ ι Q : M ∞ ( Q ( χ s ) Rχ ) −→ M ∞ ( Q ( χ s ) Rχ )is the multiplication by the image of p − e ( χ ) U p ( χ ) in R ∞ ( τ ( χ s )) /p . As the image of h χ ◦ ι Q is e U p ( χ ) M ∞ ( Q ( χ s ) Rχ ) by the commutativity of the left-hand side of (the analogof) [DL, (28)] and the definition of e U p ( χ ), we deduce that e U p ( χ )( R ∞ ( τ ( χ s )) /p ) = ( p − e ( χ ) U p ( χ ))( R ∞ ( τ ( χ s )) /p ) . In particular, multiplying e U p ( χ ) by a unit in R ∞ ( τ ) we can assume that e U p ( χ ) and p − e ( χ ) U p ( χ ) have the same image in the quotient R ∞ ( τ ( χ s )) /p of R ∞ ( τ ). As a conse-quence the analogue of [DL, Prop.4.17] holds. • Since by definition p − e ( χ ) U p ( χ ) ∈ R ∞ ( τ ( χ s )) \ pR ∞ ( τ ( χ s )), we haveAnn R ∞ ( τ ( χ s )) /p (cid:16) p − e ( χ ) U p ( χ ) (cid:17) ⊆ m ∞ ( R ∞ ( τ ( χ s )) /p ) . (186)As e U p ( χ ) p − e ( χ ) U p ( χ ) ∈ R ∞ ( τ ( χ s )) /p (previous point), we deduce e U p ( χ )( e h χ ◦ e ι χ − Id) R ∞ ( τ ( χ s )) /p ( M ∞ ( Q ( χ s ) Rχ )) by the analog of [DL, (28)]. As M ∞ ( Q ( χ s ) Rχ ) ∼ = M ∞ ( θ ( χ s ) Rχ ) /p is free of rank r over R ∞ ( τ ( χ s )) /p (by Theorem3.4.2.1(iii)), (186) implies the image of e h χ ◦ e ι χ − Id in End R ∞ ( τ ( χ s )) /p ( M ∞ ( Q ( χ s ) Rχ ))lands in m ∞ End R ∞ ( τ ( χ s )) /p ( M ∞ ( Q ( χ s ) Rχ )). Since Ker( R ∞ ( τ ) ։ R ∞ ( τ ( χ s )) /p ) ⊆ m ∞ R ∞ ( τ ), we also have e h χ ◦ e ι χ − Id ∈ m ∞ End R ∞ ( τ ) ( M ∞ ( θ Rχ )) . (187) • The big unlabelled diagram before [DL, (33)] still holds but the diagonal maps arenot simply multiplication by some e U p ( χ i ). For instance in the case k = 3 (the generalcase being similar) one has to replace the left diagonal maps in loc.cit. by successively(from top to bottom) e U p ( χ )(( e ι χ ◦ e ι χ ) − ◦ ( e h χ ◦ e ι χ ) ◦ e ι χ ◦ e ι χ ), e U p ( χ )( e ι − χ ◦ ( e h χ ◦ e ι χ ) ◦ e ι χ ), and e U p ( χ )( e h χ ◦ e ι χ ). By (187) and the R ∞ ( τ )-linearity of the isomorphisms e ι χ i , all these diagonal maps are in e U p ( χ i )(Id + m ∞ End R ∞ ( τ ) ( M ∞ ( θ Rχ s ))), and theircomposition is thus in (cid:16) k − Y i =0 e U p ( χ i ) (cid:17) (Id + m ∞ End R ∞ ( τ ) ( M ∞ ( θ Rχ s ))) . (188) • For ν ≥ e ι χ i : (cid:16) k − Y i =0 e U p ( χ i ) (cid:17) ( e ι χ ◦ e ι χ k − ◦ · · · ◦ e ι χ ) = p ν Id (189)which implies p − ν ( Q k − i =0 e U p ( χ i )) ∈ R ∞ ( τ ) × since the e ι χ i are isomorphisms. By thecommutativity in the (analog of) the big unlabelled diagram before [DL, (33)] (seethe previous point) together with (188) and (189) we finally obtain e h χ ◦ · · · ◦ e h χ k − ◦ e h χ ∈ (cid:16) p − ν k − Y i =0 e U p ( χ i ) (cid:17) (Id + m ∞ End R ∞ ( τ ) ( M ∞ ( θ Rχ s )))178hich is our analog of [DL, (33)]. Then [DL, (34)] follows by the same argument.The rest of the proof in [DL, §5] is unchanged.We can now prove Theorem 3.4.1.1. Proof of Theorem 3.4.1.1.
We let D ( ρ ) = ( D ( ρ ) ֒ → D ( ρ )) be the diagram denotedby D ( π glob ( ρ )) in [DL], which only depends on ρ . Let D ( π ) = ( D ( π ) ֒ → D ( π )) def =( π I ֒ → π K ) be the diagram defined by π . We will show that D ( ρ ) ⊕ r ∼ = D ( π ) asdiagrams.Define first R : π I → (soc GL ( O K ) π ) I as in [DL, Def.4.1], i.e. Rv = S i ( χ ) v with S i ( χ ) as in [DL, Rem.4.2] if v ∈ π I is an I -eigenvector with eigencharacter χ . Notethat the eigencharacter of Rv is Rχ .Starting from D ( ρ ) we define a groupoid G with objects x ξ , where ξ is any characterof I such that (soc GL ( O K ) D ( ρ )) I [ ξ ] = 0, and morphisms freely generated by g χ : x Rχ ∼ −→ x Rχ s , where χ is any character of I such that D ( ρ )[ χ ] = 0, as in [DL,Def.4.3].The diagram D ( π ) defines an r -dimensional representation of G , sending x ξ to thevector space (soc GL ( O K ) D ( π )) I [ ξ ] and g χ to the linear map g πχ : (soc GL ( O K ) D ( π )) I [ Rχ ] ∼ −→ (soc GL ( O K ) D ( π )) I [ Rχ s ]as in [DL, §4]. Similarly, we have an r -dimensional representation of G defined by thediagram D ( ρ ) ⊕ r ; we denote the linear maps by g ρχ .To check that the two r -dimensional representations of G are isomorphic it sufficesto check that for each object x the restrictions of the two representations to theautomorphism group G x are isomorphic (see [DL, Prop.4.5]), which is the case byProposition 3.4.3.3, remembering that g πχ is the dual of h χ by (the analog of) [DL,Prop.4.14].Therefore there exists an isomorphism λ : (soc GL ( O K ) D ( π )) I ∼ −→ (soc GL ( O K ) D ( ρ ) ⊕ r ) I of I -representations such that λ ◦ g πχ = g ρχ ◦ λ on (soc GL ( O K ) D ( π )) I [ Rχ ] for all χ .As π K ∼ = D ( ρ ) ⊕ r as K -representations we can extend λ uniquely to an isomorphism λ : D ( π ) ∼ −→ D ( ρ ) ⊕ r of K -representations (extending to the GL ( O K )-socle first).As in the proof of [DL, Prop.4.4] we deduce that λ restricts to an isomorphism λ : D ( π ) ∼ −→ D ( ρ ) ⊕ r commuting with (cid:16) p (cid:17) and I , which completes the proof.179 .4.4 Local-global compatibility results We collect our previous results to deduce (together with the results of [HW]) specialcases of Conjecture 2.1.3.1 and Conjecture 2.5.1 when n = 2 and K is unramified.We keep all the previous notation. We also keep the assumptions (i) to (xii) of§3.4.1 (in particular r ˜ v is semisimple), except that we replace the bounds on the r i in (viii) by the stronger bounds (which are those of [BHH + , §1]):9 ≤ r j ≤ p −
12 if j > ρ is reducible;10 ≤ r ≤ p −
11 if ρ is irreducible.Recall that we choose Serre weights σ ˜ w ∈ W ( r ˜ w (1)) for w ∈ S p \{ v } and consider π = Hom U v ( ⊗ w ∈ S p \{ v } σ ˜ w , S ( V v , F )[ m Σ ]) (see Theorem 3.4.1.1). Theorem 3.4.4.1.
We have [ π [ m I /Z ] : χ ] = [ π [ m I /Z ] : χ ] for all smooth characters χ : I → F × appearing in π [ m I /Z ] .Proof. The statement of [BHH + , Thm.8.3.10] applies verbatim with the same proof to π as above using Theorem 3.4.2.1 and (185). Combining this with Corollary 3.4.2.2,we see that π satisfies all the assumptions of [BHH + , Thm.1.3], whence the result. Remark 3.4.4.2.
By a similar argument as in (ii) of the proof of [BHH + , Thm.8.4.1](which uses [GN, App.A]), we also have dim GL ( K ) ( π ) = f , where dim GL ( K ) ( π ) is theGelfand–Kirillov dimension of π as defined in [BHH + , §5.1].The following theorem is one of the main results of this paper. Theorem 3.4.4.3.
Keep all the previous assumptions and assume that the r i in r ˜ v satisfy the following stronger bounds: max { , f − } ≤ r j ≤ p − max { , f + 2 } if j > or ρ is reducible; max { , f } ≤ r ≤ p − max { , f + 1 } if ρ is irreducible. (190) Let σ v def = ⊗ w ∈ S p \{ v } σ ˜ w , where the σ ˜ w are Serre weights in W ( r ˜ w (1)) for w ∈ S p \{ v } .Then Conjecture 2.1.3.1 holds for Hom U v ( σ v , S ( V v , F )[ m Σ ]) .Proof. This follows from Corollary 3.3.2.4 applied to π = Hom U v ( σ v , S ( V v , F )[ m Σ ]),which satisfies all the assumptions there by Theorem 3.4.1.1 and Theorem 3.4.4.1,and by Remark 2.1.1.4(ii).We now give some evidence for Conjecture 2.5.1, still assuming (190). As we alsoneed r = 1, and to make things as simple as possible, we replace assumptions (v) and(vii) in §3.4.1 by 180 is unramified at all finite places outside S p and we then take S def = S p (hence Σ = S p ∪ { v } ). We also replace assumption (xii) in§3.4.1 by ι e v ( U v ) is equal to the upper-triangular unipotent matrices mod f v .We take V v = U p Q w ∈ S p \{ v } V w with ι ˜ w ( V w ) = 1 + pM ( O F ˜ w ) ⊆ GL ( O F ˜ w ) = ι ˜ w ( U w ).We let T e v be the Hecke operator acting on S ( V v , F ) by the double coset ι − e v " ι e v ( U v ) ̟ e v ! ι e v ( U v ) , where ̟ e v is a uniformizer in O F e v . Increasing F if necessary, we fix a choice ofeigenvalues α e v ∈ F of ρ (Frob e v ) (the image of a geometric Frobenius at f v ) andconsider the ideal m S def = ( m Σ , T e v − α e v ) ⊆ T Σ [ T e v ] , where α e v is any element in W ( F ) lifting α e v (see §2.1.2 for T Σ ). Then, replacing m Σ by m S everywhere in §§3.4.1, 3.4.2, 3.4.3, by a multiplicity 1 result analogous to theone in [BD14, Prop.3.5.1] (see for instance the argument in the proof of [Enn, Lemma3.1.4]) all the previous global results hold with r being 1. Proposition 3.4.4.4.
Choose Serre weights σ ˜ w ∈ W ( r ˜ w (1)) for w ∈ S p \{ v } and let π def = Hom U v ( ⊗ w ∈ S p \{ v } σ ˜ w , S ( V v , F )[ m S ]) . The representation π satisfies all the assumptions of §3.3.5 ( with ρ = r ˜ v (1)) .Proof. The only missing assumption is the essential self-duality (174). But it holdsby the same proof as for the definite case of [HW, Thm.8.2] using Remark 3.4.4.2.From the results of §3.3.5, we thus deduce the following theorems.
Theorem 3.4.4.5.
The GL ( F ˜ v ) -representation π is generated by its GL ( O F ˜ v ) -socle,in particular is of finite type. Theorem 3.4.4.6. (i)
Assume that r ˜ v is irreducible. Then π is irreducible and is a supersingularrepresentation. Assume that r ˜ v is reducible ( split ) and write ρ = r ˜ v (1) = χ χ ! . Then onehas π = Ind GL ( F ˜ v ) B − ( F ˜ v ) ( χ ω − ⊗ χ ) ⊕ π ′ ⊕ Ind GL ( F ˜ v ) B − ( F ˜ v ) ( χ ω − ⊗ χ ) , where π ′ is generated by its GL ( O F ˜ v ) -socle and π ′∨ is essentially self-dual, i.e.satisfies (174) . Moreover, when f = 2 , π ′ is irreducible and supersingular ( andhence π is semisimple ) .Proof. Everything is in Corollary 3.3.5.6 and Corollary 3.3.5.8, except the preciseform of the irreducible principal series π , π f in loc.cit. , but this easily follows from(181) and Theorem 3.4.1.1 (which is [DL, §5] since r = 1).Combining Theorem 3.4.4.6 with Theorem 3.4.4.3, we obtain: Corollary 3.4.4.7.
Keep the same assumptions as just before Proposition 3.4.4.4.If r ˜ v is irreducible or if f = 2 , then π is compatible with ρ ( Definition 2.4.2.7 ) . Inparticular in these cases Conjecture 2.5.1 holds for Hom U v ( σ v , S ( V v , F )[ m S ]) . Remark 3.4.4.8.
When r ˜ v is reducible nonsplit, a similar proof as for[HW, Thm.1.6] (with the hypothesis of loc.cit. on r ˜ v ) implies that π is generated overGL ( F ˜ v ) by π K . When moreover f = 2, a similar proof as for [HW, Thm.10.37]implies that π is at least compatible with e P ρ = P ρ = B (Definition 2.4.1.5).182 eferences [AW09] Konstantin Ardakov and Simon J. Wadsley, Γ -invariant ideals in Iwasawaalgebras , J. Pure Appl. Algebra (2009), no. 9, 1852–1864. MR 2518183[AWZ08] Konstantin Ardakov, Feng Wei, and James J. Zhang, Reflexive ideals inIwasawa algebras , Adv. Math. (2008), no. 3, 865–901. MR 2414324[BD14] Christophe Breuil and Fred Diamond,
Formes modulaires de Hilbert mod-ulo p et valeurs d’extensions entre caractères galoisiens , Ann. Sci. Éc.Norm. Supér. (4) (2014), no. 5, 905–974. MR 3294620[BD20] Christophe Breuil and Yiwen Ding, Higher L -invariants for GL ( Q p ) andlocal-global compatibility , Cambridge J. of Math. (2020), no. 4, 775–951.[BDJ10] Kevin Buzzard, Fred Diamond, and Frazer Jarvis, On Serre’s conjecturefor mod ℓ Galois representations over totally real fields , Duke Math. J. (2010), no. 1, 105–161. MR 2730374[Ber10] Laurent Berger,
Représentations modulaires de GL ( Q p ) et représentationsgaloisiennes de dimension 2 , Astérisque (2010), no. 330, 263–279. MR2642408[BG14] Kevin Buzzard and Toby Gee, The conjectural connections between auto-morphic representations and Galois representations , Automorphic formsand Galois representations. Vol. 1, London Math. Soc. Lecture Note Ser.,vol. 414, Cambridge Univ. Press, Cambridge, 2014, pp. 135–187. MR3444225[BH15] Christophe Breuil and Florian Herzig,
Ordinary representations of G ( Q p ) and fundamental algebraic representations , Duke Math. J. (2015),no. 7, 1271–1352. MR 3347316[BHH + ] Christophe Breuil, Florian Herzig, Yongquan Hu, Stefano Morra, and Ben-jamin Schraen, Gelfand–Kirillov dimension and mod p cohomology for GL , https://arxiv.org/pdf/2009.03127.pdf , preprint (2020).[Bjö89] Jan-Erik Björk, The Auslander condition on Noetherian rings , Séminaired’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris,1987/1988), Lecture Notes in Math., vol. 1404, Springer, Berlin, 1989,pp. 137–173. MR 1035224[BL94] Laure Barthel and Ron Livné,
Irreducible modular representations of GL of a local field , Duke Math. J. (1994), no. 2, 261–292. MR 1290194183BL95] , Modular representations of GL of a local field: the ordinary,unramified case , J. Number Theory (1995), no. 1, 1–27. MR 1361556[BLGG13] Thomas Barnet-Lamb, Toby Gee, and David Geraghty, Serre weights forrank two unitary groups , Math. Ann. (2013), no. 4, 1551–1598. MR3072811[BM02] Christophe Breuil and Ariane Mézard,
Multiplicités modulaires etreprésentations de GL ( Z p ) et de Gal( Q p / Q p ) en l = p , Duke Math. J. (2002), no. 2, 205–310, With an appendix by Guy Henniart. MR1944572[Bor79] Armand Borel, Automorphic L -functions , Automorphic forms, represen-tations and L -functions (Proc. Sympos. Pure Math., Oregon State Univ.,Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer.Math. Soc., Providence, R.I., 1979, pp. 27–61. MR 546608[Bou07] Nicolas Bourbaki, Éléments de mathématique. Algèbre. Chapitre 10. Al-gèbre homologique , Springer-Verlag, Berlin, 2007, Reprint of the 1980 orig-inal [Masson, Paris; MR0610795]. MR 2327161[BP12] Christophe Breuil and Vytautas Pašk¯unas,
Towards a modulo p Langlandscorrespondence for GL , Mem. Amer. Math. Soc. (2012), no. 1016,vi+114. MR 2931521[Bre03a] Christophe Breuil, Sur quelques représentations modulaires et p -adiques de GL ( Q p ) . I , Compositio Math. (2003), no. 2, 165–188. MR 2018825[Bre03b] , Sur quelques représentations modulaires et p -adiques de GL ( Q p ) .II , J. Inst. Math. Jussieu (2003), no. 1, 23–58. MR 1955206[Bre10] , Série spéciale p -adique et cohomologie étale complétée , Astérisque(2010), no. 331, 65–115. MR 2667887[Bre11] , Diagrammes de Diamond et ( φ, Γ) -modules , Israel J. Math. (2011), 349–382. MR 2783977[Bre14] , Sur un problème de compatibilité local-global modulo p pour GL ,J. Reine Angew. Math. (2014), 1–76. MR 3274546[Bre15] , Induction parabolique et ( ϕ, Γ) -modules , Algebra & Number The-ory (2015), no. 10, 2241–2291. MR 3437761[CDP14] Pierre Colmez, Gabriel Dospinescu, and Vytautas Pašk¯unas, The p -adiclocal Langlands correspondence for GL ( Q p ), Camb. J. Math. (2014),no. 1, 1–47. MR 3272011 184CEG +
18] Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, VytautasPašk¯unas, and Sug Woo Shin,
Patching and the p -adic Langlands programfor GL ( Q p ), Compos. Math. (2018), no. 3, 503–548. MR 3732208[CHT08] Laurent Clozel, Michael Harris, and Richard Taylor, Automorphy for some l -adic lifts of automorphic mod l Galois representations , Publ. Math. Inst.Hautes Études Sci. (2008), no. 108, 1–181, With Appendix A, summariz-ing unpublished work of Russ Mann, and Appendix B by Marie-FranceVignéras. MR 2470687[Col89] Michael J. Collins,
Tensor induction and transfer , Quart. J. Math. OxfordSer. (2) (1989), no. 159, 275–279. MR 1010818[Col10] Pierre Colmez, Représentations de GL ( Q p ) et ( φ, Γ) -modules , Astérisque(2010), no. 330, 281–509. MR 2642409[Con14] Brian Conrad, Reductive group schemes , Autour des schémas en groupes.Vol. I, Panor. Synthèses, vol. 42/43, Soc. Math. France, Paris, 2014,pp. 93–444. MR 3362641[CR81] Charles W. Curtis and Irving Reiner,
Methods of representation theory.Vol. I , John Wiley & Sons, Inc., New York, 1981, With applicationsto finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication. MR 632548[CS17a] Przemyslaw Chojecki and Claus Sorensen,
Strong local-global compatibil-ity in the p -adic Langlands program for U (2), Rend. Semin. Mat. Univ.Padova (2017), 135–153. MR 3652872[CS17b] , Weak local-global compatibility in the p -adic Langlands programfor U (2), Rend. Semin. Mat. Univ. Padova (2017), 101–133. MR3652871[DL] Andrea Dotto and Daniel Le, Diagrams in the mod p cohomologyof Shimura curves , https://arxiv.org/pdf/1909.12219.pdf , preprint(2019).[DM91] François Digne and Jean Michel, Representations of finite groups of Lietype , London Mathematical Society Student Texts, vol. 21, CambridgeUniversity Press, Cambridge, 1991. MR 1118841[DPS] Gabriel Dospinescu, Vytautas Pašk¯unas, and BenjaminSchraen,
Infinitesimal characters in arithmetic families , https://arxiv.org/pdf/2012.01041.pdf , preprint (2020).185EGH13] Matthew Emerton, Toby Gee, and Florian Herzig, Weight cycling andSerre-type conjectures for unitary groups , Duke Math. J. (2013), no. 9,1649–1722. MR 3079258[EGS15] Matthew Emerton, Toby Gee, and David Savitt,
Lattices in the cohomologyof Shimura curves , Invent. Math. (2015), no. 1, 1–96. MR 3323575[Emea] Matthew Emerton,
Local-global compatibilityin the p -adic Langlands program for GL / Q , , preprint(2011).[Emeb] , On a class of coherent rings, with applications to thesmooth representation theory of GL ( Q p ) in characteristic p , ,preprint (2008).[Eme06] , On the interpolation of systems of eigenvalues attached to auto-morphic Hecke eigenforms , Invent. Math. (2006), no. 1, 1–84. MR2207783[Eme10a] ,
Ordinary parts of admissible representations of p -adic reductivegroups I. Definition and first properties , Astérisque (2010), no. 331, 355–402. MR 2667882[Eme10b] , Ordinary parts of admissible representations of p -adic reductivegroups II. Derived functors , Astérisque (2010), no. 331, 403–459. MR2667883[Enn] John Enns, Multiplicities in the ordinary part of mod p cohomology for GL n ( Q p ), https://arxiv.org/pdf/1809.00278.pdf , preprint (2018).[Fon90] Jean-Marc Fontaine, Représentations p -adiques des corps locaux. I , TheGrothendieck Festschrift, Vol. II, Progr. Math., vol. 87, BirkhäuserBoston, Boston, MA, 1990, pp. 249–309. MR 1106901[GHS18] Toby Gee, Florian Herzig, and David Savitt, General Serre weight con-jectures , J. Eur. Math. Soc. (JEMS) (2018), no. 12, 2859–2949. MR3871496[GK14] Toby Gee and Mark Kisin, The Breuil-Mézard conjecture for potentiallyBarsotti-Tate representations , Forum Math. Pi (2014), e1, 56. MR3292675[GLS14] Toby Gee, Tong Liu, and David Savitt, The Buzzard-Diamond-Jarvis con-jecture for unitary groups , J. Amer. Math. Soc. (2014), no. 2, 389–435.MR 3164985 186GN] Toby Gee and James Newton, Patching and the completed homology oflocally symmetric spaces , J. Inst. Math. Jussieu, to appear.[Gro98] Benedict H. Gross,
On the Satake isomorphism , Galois representationsin arithmetic algebraic geometry (Durham, 1996), London Math. Soc.Lecture Note Ser., vol. 254, Cambridge Univ. Press, Cambridge, 1998,pp. 223–237. MR 1696481[Gro05] Alexander Grothendieck,
Cohomologie locale des faisceaux cohérents etthéorèmes de Lefschetz locaux et globaux (SGA 2) , Documents Mathéma-tiques (Paris) [Mathematical Documents (Paris)], vol. 4, Société Mathé-matique de France, Paris, 2005, Séminaire de Géométrie Algébrique duBois Marie, 1962, Augmenté d’un exposé de Michèle Raynaud. [With anexposé by Michèle Raynaud], With a preface and edited by Yves Laszlo,Revised reprint of the 1968 French original. MR 2171939[Hau16] Julien Hauseux,
Extensions entre séries principales p -adiques et modulo p de G ( F ), J. Inst. Math. Jussieu (2016), no. 2, 225–270. MR 3480966[Hau18] , Parabolic induction and extensions , Algebra & Number Theory (2018), no. 4, 779–831. MR 3830204[Hau19] , Sur une conjecture de Breuil-Herzig , J. Reine Angew. Math. (2019), 91–119. MR 3956692[Her11] Florian Herzig,
The classification of irreducible admissible mod p repre-sentations of a p -adic GL n , Invent. Math. (2011), no. 2, 373–434. MR2845621[Hu12] Yongquan Hu, Diagrammes canoniques et représentations modulo p de GL ( F ), J. Inst. Math. Jussieu (2012), no. 1, 67–118. MR 2862375[Hum78] James E. Humphreys, Introduction to Lie algebras and representation the-ory , Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978, Second printing, revised. MR 499562[HW] Yongquan Hu and Haoran Wang,
On the mod p cohomology for GL :the non-semisimple case , https://arxiv.org/pdf/2009.09640.pdf ,preprint (2020).[HW18] , Multiplicity one for the mod p cohomology of Shimura curves: thetame case , Math. Res. Lett. (2018), no. 3, 843–873. MR 3847337[Jan03] Jens C. Jantzen, Representations of algebraic groups , second ed., Mathe-matical Surveys and Monographs, vol. 107, American Mathematical Soci-ety, Providence, RI, 2003. MR 2015057187Kis10] Mark Kisin,
Deformations of G Q p and GL ( Q p ) representations ,Astérisque (2010), no. 330, 511–528. MR 2642410[Koh17] Jan Kohlhaase, Smooth duality in natural characteristic , Adv. Math. (2017), 1–49. MR 3682662[Le19] Daniel Le,
Multiplicity one for wildly ramified representations , AlgebraNumber Theory (2019), no. 8, 1807–1827. MR 4017535[LMS] Daniel Le, Stefano Morra, and Benjamin Schraen, Multiplicity one at fullcongruence level , J. Inst. Math. Jussieu, to appear.[LvO96] Huishi Li and Freddy van Oystaeyen,
Zariskian filtrations , K -Monographsin Mathematics, vol. 2, Kluwer Academic Publishers, Dordrecht, 1996. MR1420862[Lyu97] Gennady Lyubeznik, F -modules: applications to local cohomology and D -modules in characteristic p >
0, J. Reine Angew. Math. (1997), 65–130. MR 1476089[Mor] Stefano Morra,
Corrigendum to Iwasawa mod-ules and p -modular representations of GL , ,preprint (2020).[Mor17] , Iwasawa modules and p -modular representations of GL , Israel J.Math. (2017), no. 1, 1–70. MR 3642015[Paš04] Vytautas Pašk¯unas, Coefficient systems and supersingular representationsof GL ( F ), Mém. Soc. Math. Fr. (N.S.) (2004), no. 99, vi+84. MR 2128381[Paš10] , Extensions for supersingular representations of GL ( Q p ),Astérisque (2010), no. 331, 317–353. MR 2667891[Paš13] , The image of Colmez’s Montreal functor , Publ. Math. Inst.Hautes Études Sci. (2013), 1–191. MR 3150248[PQ] Chol Park and Zicheng Qian,
On mod p local-global compatibility for GL n ( Q p ) in the ordinary case , https://arxiv.org/pdf/1712.03799.pdf , preprint (2019).[Sch15] Benjamin Schraen, Sur la présentation des représentations supersingulièresde GL ( F ), J. Reine Angew. Math. (2015), 187–208. MR 3365778[Ser00] Jean-Pierre Serre, Local algebra , Springer Monographs in Mathematics,Springer-Verlag, Berlin, 2000, Translated from the French by CheeWhyeChin and revised by the author. MR 1771925188Sti10] Jakob Stix,
Trading degree for dimension in the section conjecture: thenon-abelian Shapiro lemma , Math. J. Okayama Univ. (2010), 29–43.MR 2589844[SV11] Peter Schneider and Marie-France Vigneras, A functor from smooth o -torsion representations to ( φ, Γ) -modules , On certain L -functions, ClayMath. Proc., vol. 13, Amer. Math. Soc., Providence, RI, 2011, pp. 525–601. MR 2767527[Tho12] Jack Thorne, On the automorphy of l -adic Galois representations withsmall residual image , J. Inst. Math. Jussieu (2012), no. 4, 855–920,With an appendix by Robert Guralnick, Florian Herzig, Richard Taylorand Thorne. MR 2979825[Tho17] , A 2-adic automorphy lifting theorem for unitary groups over CMfields , Math. Z. (2017), no. 1-2, 1–38. MR 3598803[Ven02] Otmar Venjakob,
On the structure theory of the Iwasawa algebra of a p -adic Lie group , J. Eur. Math. Soc. (JEMS) (2002), no. 3, 271–311. MR1924402[Wu] Zhixiang Wu, A note on presentations of supersingular representations of GL ( F ), Manuscripta Math., to appear.[Záb18a] Gergely Zábrádi, Multivariable ( ϕ, Γ) -modules and products of Galoisgroups , Math. Res. Lett. (2018), no. 2, 687–721. MR 3826842[Záb18b] , Multivariable ( ϕ, Γ) -modules and smooth o -torsion representa-tions , Selecta Math. (N.S.) (2018), no. 2, 935–995. MR 3782415[Zhu] Xinwen Zhu, A note on integral Satake isomorphisms , https://arxiv.org/pdf/2005.13056.pdfhttps://arxiv.org/pdf/2005.13056.pdf