Rigid analytic vectors of crystalline representations arising in p -adic Langlands
aa r X i v : . [ m a t h . N T ] J un RIGID ANALYTIC VECTORS OF CRYSTALLINEREPRESENTATIONS ARISING IN p -ADIC LANGLANDS JISHNU RAY
Abstract.
Let B ( V ) be the admissible unitary GL ( Q p )-representation associatedto two dimensional crystalline Galois representation V by p -adic Langlands con-structed by Breuil. Berger and Breuil conjectured an explicit description of the locallyanalytic vectors B ( V ) la of B ( V ) which is now proved by Liu. Emerton recently studied p -adic representations from the viewpoint of rigid analytic geometry. In this article,we consider certain rigid analytic subgroups of GL (2) and give an explicit descrip-tion of the rigid analytic vectors in B ( V ) la . In particular, we show the existenceof rigid analytic vectors inside B ( V ) la and prove that its non-null. This gives us arigid analytic representation (in the sense of Emerton) lying inside the locally analyticrepresentation B ( V ) la . Introduction
Let p > L be a finite extension over Q p . Breuil initiated the con-struction of p -adic local Langlands for GL ( Q p ). To any two dimensional, crystallineGalois L -representation V he attached a p -adic admissible unitary representation of GL ( Q p ) which we denote by B ( V ). Further, the main work of Colmez was to extendthis correspondence to any two dimensional L -representation of the absolute Galoisgroup of Q p . When V is crystalline, Berger and Breuil formulated a conjecture on thelocally analytic vectors B ( V ) la of the L -Banach representation B ( V ). The representa-tion B ( V ) la is an admissible locally analytic representation in the sense of Schneider andTeitelbaum [ST03]. This conjecture was proved by Liu in [Liu12, Thm. 4.1]. Colmezand Dospinescu generalized this to all GL ( Q p )-representations appearing in the p -adicLanglands [CD14]. In [Eme17, Sec. 3.3], Emerton studies rigid analytic vectors of p -adic representations from the viewpoint of rigid analytic geometry [Bos14]. Then, hebuilds up a notion of an admissible rigid analytic representation which is completelyanalogous to the original construction of Schneider and Teitelbaum for an admissiblelocally analytic representation. This work is supported by PIMS-CNRS postdoctoral research grant from the University of BritishColumbia.
Keywords: rigid analytic geometry, p -adic Langlands correspondence. AMS subject classifications:
Let I (1) be the pro- p Iwahori subgroup of GL ( Z p ), that is matrices of the form (cid:18) Z p p Z p Z p Z p (cid:19) . Let Q be the intersection of the standard upper triangular Borelsubgroup of GL ( Z p ) with the pro- p Iwahori. The pro- p Iwahori I (1) can be consideredas Z p -points of a rigid analytic group. In [Clo18, Sec. 3.3], Laurent Clozel showed theexistence of I (1)-rigid analytic vectors inside the principal series representation, whichis the representation induced by a rigid analytic character χ of the Borel subgroup Q to I (1). This gave him an admissible rigid analytic representation, in the sense of Emerton,of the rigid analytic pro- p Iwahori subgroup. Furthermore, he found conditions on thecharacter χ such that his rigid analytic principal series is irreducible [Clo18, Thm.3.7]. This result of Clozel is later generalized by the author of this article for GL n ( Z p )[Ray18, Thms. 3.8, 3.9]. Clozel once asked the author the following question. Question 1.1 (by Clozel) . Start with a two dimensional, Galois L -representation V .By work of Fontaine, we can attach to it a rank two ´etale ( ϕ, Γ) -module D over the Robbaring R L (cf. Section 2.2). Then Colmez and others have constructed the corresponding GL ( Q p ) -representation Π( D ) which is of type Π ( D ) ∼ = ( D ⊠ P ) / ( D ♮ ⊠ P ) (cf. Section2.3) and set the p -adic Langlands as V → Π( D ) .Does there exist rigid analytic vectors inside Π( D )? If so, how to compute them andconstruct a rigid analytic representation, in the sense of Emerton?
This question is answered positively in this article for crystalline representations V .When V is not crystalline, this question is still open. The rigid analytic groups we willconsider are the so-called ‘good open congruence subgroups’ which have the propertythat G ( n ) ◦ ( C p ) = 1+ ̟ n ( m C p ) where ̟ n ∈ O C p is a uniformizer and m C p is the maximalideal. The group G ( n ) ◦ is a rigid analytic subgroup of GL rig2 which is a rigid analyticgroup associated to the algebraic group GL . For m a positive integer, let G ( m ) bethe rigid analytic group whose Z p -points is the m -th principal congruence subgroup of GL ( Z p ). Then G ( n ) ◦ is the rigid analytic group which is the union of the rigid analyticgroups G ( m ) for all m > n .When V is crystalline, the theme of the paper is to compute the subspace (cid:0) B ( V ) la (cid:1) G ( n ) ◦ − an of G ( n ) ◦ -analytic vectors of the locally analytic GL ( Q p )-representation B ( V ) la associ-ated to V by p -adic Langlands of Berger and Breuil. We give an explicit presentationof (cid:0) B ( V ) la (cid:1) G ( n ) ◦ − an and show that its non-null.Our main Theorem is the following. Theorem 1.2 (see Thm. 6.6) . Suppose V = V ( α, β ) is crystalline, irreducible, Frobe-nius semisimple Galois representation with distinct Hodge-Tate weights. Let B ( V ) la be the locally analytic vectors of the GL ( Q p ) -representation B ( V ) associated to V via p -adic local Langlands constructed by Berger and Breuil. Then, (cid:0) B ( V ) la (cid:1) G ( n ) ◦ − an ∼ = Coker (cid:16) π ( β ) G ( n ) ◦ − an ֒ → LA ( α ) G ( n ) ◦ − an ⊕ LA ( β ) G ( n ) ◦ − an (cid:17) is a rigid-analytic G ( n ) ◦ ( Z p ) -representation. Furthermore, IGID ANALYTIC GEOMETRY AND p -ADIC LANGLANDS 3 (i) π ( β ) G ( n ) ◦ − an ∼ = Sym k − L ⊗ lim ←− m>n C ( β ) m ,(ii) LA ( α ) G ( n ) ◦ − an ∼ = GA ( α ) and LA ( β ) G ( n ) ◦ − an ∼ = GA ( β ) .In particular, it follows that (cid:0) B ( V ) la (cid:1) G ( n ) ◦ − an = 0 . Here LA ( α ) or LA ( β ) are locally analytic principal series (cf. (3.1)) and π ( β ) is aclosed GL ( Q p )-equivariant subspace of locally algebraic functions inside LA ( β ) (cf.(3.2) and (2.3)). Here, C ( β ) m , GA ( α ) and GA ( β ) are rigid analytic functions whosedescriptions are explicitly given in Theorem 6.4 and Corollary 6.5. Our main Theoremtells us that the representation of G ( n ) ◦ -analytic vectors of B ( V ) la is a non-trivialquotient of the direct sum of two principal series representations.There are special advantages (see Section 7.1) of considering this group G ( n ) ◦ , whichis a σ -affinoid space (cf. Section 4) rather than the pro- p Iwahori considered orig-inally by Clozel in [Clo18]. One of them is the fact that its distribution algebra D an ( G ( n ) ◦ , L ) is a coherent ring (cf. Section 5). The other is the fact that the (strong)dual of (cid:0) B ( V ) la (cid:1) G ( n ) ◦ − an is a finitely presented module over the distribution algebra D an ( G ( n ) ◦ , L ) (cf. Proposition 5.1). On the contrary, nothing is know about the ringtheoretic structure of the distribution algebra of the affinoid pro- p Iwahori and so wedon’t know if there is a theory of coadmissible modules over them, analogous to thecase of Schneider and Teitelbaum for admissible locally analytic representations [ST03].(Coadmissible modules arise by taking the strong dual of admissible representations).We do have a theory of coadmissible modules over D an ( G ( n ) ◦ , L ), thanks to the work ofPatel, Schmidt and Strauch [PSS19, Section 5]. Furthermore, there are several patholo-gies concerning the distribution algebra of the pro- p Iwahori that Clozel explains in theappendix of his paper [Clo18].The organization of the article is as follows. In Section 2, we introduce the basicnotations and recall facts about trianguline representation and Colmez’s constructionof p -adic Langlands. Then, in Section 3, we give the description of B ( V ) la after recall-ing some basics of locally analytic representation theory. In Sections 4 and 5, followingEmerton, we define rigid analytic representation and show its connection with locallyanalytic representation of Schneider and Teitelbaum. Further, in these Sections, wealso collect crucial results on rigid analytic representations from the literature whichare useful for the proof of our main Theorem. The notion of rigid analytic representa-tions, due to Emerton, is relatively new and therefore we find it valuable to include ashort exposition on it which explains our main theorem. A reader familiar with p -adicLanglands and analytic representations can completely ignore reading these Sectionsand move on to Section 6. However, we chose to include them so that our article is selfcontained. Our main Theorem is proved in Section 6. Moreover, we also include someopen questions on Langlands base change of rigid analytic vectors in Section 7. JISHNU RAY Basic setup
Notations.
Let G Q p be the absolute Galois group of Q p and W Q p be the Weilgroup which is dense in G Q p . Let χ : G Q p → Z × p be the cyclotomic character. Let F ∞ = Q p ( µ p ∞ ) and H Q p = ker χ = Gal( Q p /F ∞ ) . We can view χ as an isomorphismfrom Γ = G Q p /H Q p = Gal( F ∞ / Q p ) to Z × p . Let val p ( − ) denote the p -adic valuation on Q p normalized by val p ( p ) = 1. Let c T ( L ) be the set of continuous characters δ : Q × p → L × .Following Colmez [Col08], we remark that this notation c T ( L ) is justified by the factthat it is the L -rational point of a character variety c T . We define the weight w ( δ ) of δ by the formula w ( δ ) = log δ ( u )log u where u ∈ Z × p is not a root of unity. By class fieldtheory, we can view an element of c T ( L ) as a continuous character of the Weil group W Q p . Furthermore, if δ is a unitary character (having values in O × L ), then one canuniquely extend δ to a continuous character of W Q p . Then, one can reinterpret w ( δ ) asgeneralized Hodge-Tate weight of δ . We denote x ∈ c T ( L ) to be the character inducedby the inclusion of Q p in L , and | x | the character sending x ∈ Q × p to p − val p ( x ) . Then x | x | is unitary corresponding to the cyclotomic character χ .2.2. Trianguline representations.
Let R L denote the Robba ring over L with theFrobenius ϕ -action and Γ-action given by ϕ ( T ) = (1+ T ) p − γ ( T ) = (1+ T ) χ ( γ ) − γ ∈ Γ. For δ ∈ c T ( L ), let R L ( δ ) be the Robba twisted by the character δ . Ithas R L -basis e such that ϕ, Γ-actions are given by ϕ ( xe ) = δ ( p ) ϕ ( x ) e and γ ( xe ) = δ ( χ ( γ )) γ ( x ) e for any x ∈ R L . Colmez shows that if M is a rank one ( ϕ, Γ)-moduleover R L , then M ∼ = R L ( δ ) for some δ [Col08, Prop. 3.1]. Recall that a ( ϕ, Γ)-moduleover R L is called triangulable if it can be expressed as successive extensions of rankone ( ϕ, Γ)-modules over R L . An L -representation V of G Q p is called trianguline if itsassociated ( ϕ, Γ)-module, denoted by D † rig ( V ), over R L is triangulable, i.e. it fits intoa short exact sequence 0 → R L ( δ ) → D † rig ( V ) → R L ( δ ) → . The representation V is then determined by a triple ( δ , δ , h ) where h ∈ Ext ( R L ( δ ) , R L ( δ )) , the extension corresponding to D † rig ( V ). The generalized Hodge-Tate weights of V willthen be w ( δ ) and w ( δ ) and it follows that val p ( δ ( p )) + val p ( δ ( p )) = 0. Colmezstudied Ext ( R L ( δ ) , R L ( δ )) and showed that for any two characters δ , δ ∈ c T ( L ),Ext ( R L ( δ ) , R L ( δ )) is an L -vector space of dimension 1 unless δ δ − is of the form x − i ( i ≥ | x | x i ( i ≥ ( R L ( δ ) , R L ( δ )) isof dimension 2 and the associated projective space is naturally isomorphic to P ( L )[Col10a, Prop. 8.2]. Let S be the set of ( δ , δ , L ), where ( δ , δ ) is an element of c T ( L ) × c T ( L ) and L ∈ Proj(Ext ( R L ( δ ) , R L ( δ ))), the later space being identifiedwith P ( L ) = {∞} (resp. P ( L )) if Ext ( R L ( δ ) , R L ( δ )) is of dimension 1 (resp. 2).Take s = ( δ , δ , L ) which determines an extension D ( s ) of twisted Robba rings upto IGID ANALYTIC GEOMETRY AND p -ADIC LANGLANDS 5 isomorphism by L . We have a short exact sequence0 → R L ( δ ) → D ( s ) → R L ( δ ) → . We let S ∗ to be the subset of triples S consisting of s such that val p ( δ ( p ))+val p ( δ ( p )) =0 and val p ( δ ( p )) > u ( s ) = val p ( δ ( p )) = − val p ( δ ( p )) and w ( s ) = w ( δ ) − w ( δ ). In the case when D ( s ) is ´etale, we denote by V ( s ), the L -linear rep-resentation of G Q p such that D † rig ( V ( s )) = D ( s ). In [Col08], Colmez partitions S ∗ into S ∗ = S ng ∗ ⊔ S cris ∗ ⊔ S st ∗ ⊔ S ord ∗ ⊔ S ncl ∗ . Here the exponents ‘ng’, ‘cris’, ‘st’, ‘ord’, ‘ncl’ refer to ‘non geometric’, ‘crystalline’,‘semistable’, ‘ordinary’, ‘non classical’ respectively. From these Colmez classified theirreducible ones S irr = S ng ∗ ⊔ S cris ∗ ⊔ S st ∗ , and showed that if s ∈ S irr , then D ( s ) is ´etale and V ( s ) is irreducible. Conversely,if V is irreducible and trianguline, then V ∼ = V ( s ) for some s ∈ S irr [Col08, Thm.0.5(i),(ii)]. Throughout this paper we will be interested with the crystalline ones S cris ∗ . S cris ∗ is the set of all s ∈ S irr such that w ( s ) is an integer greater or equal to 1, u ( s ) < w ( s ) and L = ∞ .We have that V ∈ S cris ∗ if and only if V is twist of an irreducible and crystabelian(becomes crystalline after going to a finite abelian extension of Q p ) representation.2.3. p -adic Langlands for crystalline representations. Let us first briefly recallColmez’s construction of p -adic local Langlands for any two-dimensional irreducible L -linear representation of G Q p . Then, we will provide an explicit description of the GL ( Q p )-representation for V crystalline which is due to Berger and Breuil [BB10]. Let D = D ( s ) be a rank two, irreducible and ´etale ( ϕ, Γ)-module over R L corresponding tothe Galois representation V = V ( s ). Then Colmez constructs a GL ( Q p )-equivariantsheaf on P = P ( Q p ) whose global sections D ⊠ P give a description of the associated GL ( Q p )-representation Π ( D ). More precisely, he shows that we have an exact sequence0 → Π ( D ) ∗ ⊗ δ D → D ⊠ P → Π ( D ) → , where δ D is the character χ − det( V ) which is also the central character of Π ( D ).Here Π ( D ) ∗ is the topological dual of Π ( D ) and Π ( D ) ∗ ⊗ δ D ∼ = D ♮ ⊠ P ; that is Π ( D ) ∼ = ( D ⊠ P ) / ( D ♮ ⊠ P ) [Col10b, Thm. 0.17]. Colmez showed that Π ( D ) isan admissible unitary representation of GL ( Q p ) and he sets the p -adic local Lang-lands correspondence as V Π ( V ) := Π ( D ( V )). Let us now suppose that V iscrystalline. In the following, we give an explicit construction of the corresponding GL ( Q p )-representation due to Berger and Breuil [BB10], [BB04]. We can twist V bea suitable power of the cyclotomic character so that its Hodge-Tate weights are 0 and k − k ≥
1. Suppose that V is irreducible with distinct Hodge-Tate weights. By JISHNU RAY a result of Colmez [Col08, Prop. 4.14], we know that V is uniquely determined by apair of smooth characters of Q × p . Furthermore, suppose that V is Frobenius semisimple,i.e. the Frobenius ϕ on D cris ( V ) = ( B cris ⊗ Q p V ) G Q p is semisimple. Here B cris is oneof Fontaine’s period rings and V is considered as a Q p -linear representation by restric-tion of scalars. One can check that D cris ( V ) is an L -vector space and the Frobenius ϕ : D cris ( V ) → D cris ( V ) is L -linear. Then, there are two elements (coming from thevalues at p of the pair of smooth characters associated to V ), α, β ∈ O L (the ring ofintegers of L ) such that α = β, < val p ( β ) ≤ val p ( α ) , val p ( α ) + val p ( β ) = k − D cris ( V ) = D ( α, β ) = Le α ⊕ Le β with ϕ ( e α ) = α − e α and ϕ ( e β ) = β − e β . The module D ( α, β ) is a filtered ϕ -module over L where the filtration is given byFil i D ( α, β ) = D ( α, β ) if i ≤ − ( k − L ( e α + e β ) if − ( k − ≤ i ≤
00 if i > . V D cris ( V ) is an equivalence of categories from the category of L -linear crystalline repre-sentations to the category of (admissible) filtered ϕ -modules over L (cf. [BB10, Prop.2.4.5], see also [BB04, Thm. 2.3.1]). Let B ( Q p ) be the upper triangular Borel subgroupof GL ( Q p ). Following the notations of Berger and Breuil in [BB04], we define thelocally algebraic representations π ( α ) and π ( β ) as π ( α ) := (cid:0) Ind GL ( Q p ) B ( Q p ) unr( α − ) ⊗ x k − unr( pβ − ) (cid:1) lalg (2.1) ∼ = Sym k − L ⊗ L (cid:0) Ind GL ( Q p ) B ( Q p ) unr( α − ) ⊗ unr( pβ − ) (cid:1) sm (2.2) π ( β ) := (cid:0) Ind GL ( Q p ) B ( Q p ) unr( β − ) ⊗ x k − unr( pα − ) (cid:1) lalg (2.3) ∼ = Sym k − L ⊗ L (cid:0) Ind GL ( Q p ) B ( Q p ) unr( β − ) ⊗ unr( pα − ) (cid:1) sm , (2.4)where unr( − ) is the unramified character from Q × p to L × given by unr( λ ) : y λ val p ( y ) for λ = α − or pβ − . The subscript ‘sm’ denotes the smooth vectors in the inducedprincipal series. (Note that, to be consistent with notations of Emerton, we have used‘sm’ as a subscript whereas Berger and Breuil usually writes it as a superscript). Weequip π ( α ) (resp. π ( β )) with the unique locally convex topology such that the open setsare lattices of π ( α ) (resp. π ( β )). (A lattice of an L -vector space U is an O L -submodulewhich generates U over L ). In [BB10, Thm. 4.3.1], Berger and Breuil constructedthe universal unitary completion B ( α ) /L ( α ) of π ( α ) and a GL ( Q p )-equivariant map π ( α ) → B ( α ) /L ( α ) where(2.5) B ( α ) = (cid:0) Ind GL ( Q p ) B ( Q p ) unr( α − ) ⊗ x k − unr( pβ − ) (cid:1) C val p ( α ) , and L ( α ) is a certain closed subspace of B ( α ) (see section 7.1 of [BB04]). Similarlyinterchanging α with β , we can construct B ( β ) /L ( β ) and realize it as the universal IGID ANALYTIC GEOMETRY AND p -ADIC LANGLANDS 7 unitary completion of π ( β ). Recall that, there exists up to multiplication by a non-zeroscalar, a unique non-zero GL ( Q p )-intertwining operator [BB04, Sec. 9.1] I sm : (cid:0) Ind GL ( Q p ) B ( Q p ) unr( α − ) ⊗ unr( pβ − ) (cid:1) sm → (cid:0) Ind GL ( Q p ) B ( Q p ) unr( β − ) ⊗ unr( pα − ) (cid:1) sm . Tensoring with the identity map on Sym k − L , we get a non-zero GL ( Q p )-equivariantmorphism I : π ( α ) → π ( β ). Berger and Breuil showed the following fact [BB04,Corollary 7.2.3]. Lemma 2.1.
Assume < val p ( α ) < k − , then B ( α ) /L ( α ) and B ( β ) /L ( β ) are unitary GL ( Q p ) -Banach spaces and we have a commutative GL ( Q p ) -equivariant diagram: B ( α ) /L ( α ) b I −−−→ B ( β ) /L ( β ) x x π ( α ) I −−−→ π ( β ) where the horizontal maps are isomorphisms and the vertical maps are closed GL ( Q p ) -equivariant embeddings. Here b I is the continuous GL ( Q p ) -morphism induced from I. The p -adic Langlands of Berger and Breuil associates a non-zero GL ( Q p )-Banachrepresentation B ( V ) := B ( α ) /L ( α ) to V . The relation of B ( V ) with Colmez’s construc-tion is the following. The Banach dual of B ( V ) is naturally isomorphic to D ( ˇ V ) ♮ ⊠ P as a L -Banach space representation of GL ( Q p ). Here ˇ V is the contragredient of V .Berger and Breuil conjectured that the locally analytic vectors B ( V ) la of B ( V ) havethe form B ( V ) la ∼ = LA ( α ) ⊕ π ( β ) LA ( β ) , where LA ( α ) and LA ( β ) are two locally analytic principal series (cf. Section 3.1). Thisis now proved by Liu [Liu12].3. Locally analytic automorphic representations
In the automorphic side of the p -adic Langlands, the locally analytic representationsof GL ( Q p ) were initially studied by Schneider and Teitelbaum in [ST02b], [ST02a]and [ST03]. In the following, we are going to first recall some basic general factsabout locally analytic representations. Then, we will state results of Berger, Breuil andColmez on locally analytic vectors of Π ( V ) or B ( V ).Let L be an extension of K over Q p , G be a locally K -analytic group, U be a L -Banachspace representation of G . Let C la ( G, U ) be the space of locally analytic U -valuedfunctions on G . The space of locally analytic vectors of U is the subspace of U consistingof those vectors u ∈ U for which the orbit map g gu is in C la ( G, U ). We denote thisspace of locally analytic vectors by U la (unlike the notation of Schneider and Teitelbaum,here we have followed the notation of Emerton [Eme17] and used ‘la’ as a subscript todenote locally analytic vectors). We say that U is a locally analytic G -representation JISHNU RAY if the natural map U la → U is a bijection. (This definition can be generalized tobarrelled locally convex Hausdorff spaces [Eme17] but we will mostly be interested inlocally analytic vectors of Banach spaces B ( V )). Let D la ( G, L ) be the locally analyticdistribution algebra on G (strong dual of C la ( G, L ), i.e. Hom L ( C la ( G, L ) , L ) b where thesubscript ‘b’ denotes strong topology [Sch13]). The crucial property of D la ( G, L ) is thatit is a Fr´echet-Stein algebra when G is compact [ST03, Thm. 5.1]. That is, D la ( G, L ) = lim ←− r D r ( G, L )where D r ( G, L ) are L -Banach noetherian algebras with flat transition maps. Thisallowed Schneider and Teitelbaum to describe a category of coadmissible modules over D la ( G, L ). A coadmissible module M will have the form M ∼ = lim ←− r M r where each M r is a finitely generated D r ( G, L ) module (carrying a Banach topology induced from D r ( G, L ) n → M ) together with compatible isomorphisms D r ( G, L ) ⊗ D r ′ ( G,L ) M r ′ ∼ = M r for r ′ < r. We equip M with the projective limit topology which makes it a L -Fr´echet space. Inthe case when G is compact, an admissible locally analytic G -representation over L is alocally analytic G -representation on an L -vector space of compact type U such that thestrong dual U ′ b = Hom L ( U, L ) b is a coadmissible D la ( G, L )-module equipped with itscanonical topology. For general G , a locally analytic G -representation over L is calledadmissible if it is admissible as an H representation for one (equivalently any) opencompact subgroup H of G . If G is compact, U is a locally analytic G -representation,then U is strongly admissible if U ′ b is finitely generated as a D la ( G, L )-module (as U ′ b is quotient of some D la ( G, L ) n by some closed submodule, by Lemma 3 . U admits a closed G -equivariant embedding into C la ( G, L ) n for some natural number n (which comes bydualizing the surjection D la ( G, L ) n ։ U ′ b ). Here is an important theorem of Schneiderand Teitelbaum [ST03, Thm. 7.1] which finds locally analytic vectors of a L -Banachrepresentation of G . Theorem 3.1.
Suppose G is compact and U is a L -Banach representation of G suchthat the dual U ′ b is finitely generated module over the Iwasawa algebra L [[ G ]] of G (thealgebra of Z p -valued measures on G , base changed to L ). Then(1) U la is dense in U .(2) U la is strongly admissible G -representation.(3) ( U la ) ′ b ∼ = D la ( G, L ) ⊗ L [[ G ]] U ′ b . Locally analytic vectors of crystalline representations.
With notations asin Section 2.3, we know that for a rank-two, irreducible, ´etale ( ϕ, Γ)-module D over R L , we have an exact sequence0 → Π ( D ) ∗ ⊗ δ D → D ⊠ P → Π ( D ) → . IGID ANALYTIC GEOMETRY AND p -ADIC LANGLANDS 9 Then Colmez found the locally analytic vectors of Π( D ) and showed that the followingsequence is short exact.0 → ( Π ( D ) la ) ∗ ⊗ δ D → D rig ⊠ P → Π ( D ) la → , and ( ˇΠ( D ) la ) ∗ ∼ = D ♮ rig ⊠ P ∼ = ( Π ( D ) la ) ∗ ⊗ δ D (see. Theorem 0.17 and Theorem V.2.20 of[Col10b]). Now suppose V is irreducible, crystalline, Frobenius semisimple with distinctHodge-Tate weights 0 and k −
1. Following notation as in Section 2.3, Berger and Breuildefines a locally analytic principal series(3.1) LA ( α ) = (cid:0) Ind GL ( Q p ) B ( Q p ) unr( α − ) ⊗ x k − unr( pβ − ) (cid:1) la . We set LA ( β ) by replacing α with β . There are GL ( Q p )-equivariant injections LA ( α ) ֒ → B ( α ) and LA ( β ) ֒ → B ( β ). Obviously the locally algebraic representations π ( α ) and π ( β ) have closed GL ( Q p )-equivariant inclusions(3.2) π ( α ) ֒ → LA ( α ) and π ( β ) ֒ → LA ( β ) . Then Berger and Breuil (see [BB10], [BB04, Sec. 7.2]) construct a natural GL ( Q p )-equivariant map from the amalgamated direct sum LA ( α ) ⊕ π ( β ) LA ( β ) → B ( V ) la and conjecture that it is an isomorphism if val p ( α ) < k −
1. This conjecture is nowproved by Liu [Liu12, Thm. 4.1]. More precisely Liu constructs a map(3.3) F : ( LA ( α ) ⊕ π ( β ) LA ( β )) ∗ → D ♮ rig ( ˇ V ) ⊠ P and shows that this is an isomorphism. This proves that(3.4) LA ( α ) ⊕ π ( β ) LA ( β ) ∼ = B ( V ) la . Further by Liu, we have a GL ( Q p )-equivariant commutative diagram( B ( α ) /L ( α ) = B ( V )) ∗ ∼ = −−−→ D ♮ ( ˇ V ) ⊠ P y y ( LA ( α ) ⊕ π ( β ) LA ( β )) ∗ ∼ = −−−→ D ♮ rig ( ˇ V ) ⊠ P Remark . Note that the locally analytic principal series LA ( α ) and LA ( β ) are ad-missible locally analytic representations (cf. Example 1 .
18 of [Liu12]).4.
Rigid analytic vectors of automorphic representations
In this section, we recall the basic notions in the theory of rigid (globally) analytic p -adic representations. In the following, we will follow the exposition of [Eme17]. Weassume basic notions of rigid analytic geometry from [Bos14] and non archimedeanfunctional analysis from [Sch13]. Definition 4.1.
Let U be a Hausdorff locally convex topological L -vector space. Wesay that U is a F H -space if it admits a complete metric that induces a locally convextopology on U finer than its given topology. We refer to the topological vector spacestructure on U induced by such a metric as a latent Fr´echet space structure on U . Iffurthermore, this latent Fr´echet structure is defined by a norm, then we say that U is a BH -space. Any L -Banach space is, of course, a BH -space. Let X be an affinoid rigid analyticspace defined over K . Let C an ( X , L ) denote the L -Banach algebra of L -valued rigidanalytic functions defined on X . If U is a L -Banach space, then we define the L -Banach space C an ( X , U ) of U -valued rigid analytic functions on X to be the completedtensor product C an ( X , L ) b ⊗ L U . If U is a Hausdorff locally convex topological L -vectorspace, then we define the locally convex space C an ( X , U ) of U -valued rigid analyticfunctions on X to be the locally convex inductive limit of Banach spaces C an ( X , U ) := lim −→ C an ( X , W ) , where W runs over the directed set of all BH -subspaces W of U . Furthermore, if U isFr´echet, then also C an ( X , U ) ∼ = C an ( X , L ) b ⊗ L U [Eme17, Prop. 2.1.13]. We say that X is σ -affinoid if there is an increasing sequence X ⊂ X ⊂ · · · ⊂ X n ⊂ · · · of affinoidopen subsets of X such that X = ∪ ∞ n =1 X n with { X n } n ≥ forming an admissible coverof X . Of course, an affinoid space is σ -affinoid. The basic example of a σ -affinoid rigid-analytic space, which is not affinoid, is an open ball which is a union of an increasingsequence of closed balls. (This notion of σ -affinoid space will be useful in our nextsection where we will find our rigid analytic vectors of B ( V ) under the action of a σ -affinoid rigid analytic subgroup of GL ( Z p )). If X is σ -affinoid and U is a Hausdorfflocally convex L -vector space then we define the convex L -vector space C an ( X , U ) tobe the projective limit lim ←− Y C an ( Y , U ) where Y runs over all admissible affinoid opensubsets of X . If U is a Fr´echet space and X is σ -affinoid, then Emerton also shows that C an ( X , U ) ∼ = C an ( X , L ) b ⊗ L U [Eme17, Prop. 2.1.19].Suppose G is an affinoid rigid analytic group and suppose that the group of K -rational points G := G ( K ) is Zariski dense in G . Let U be a L -Banach space with atopological action of G . We define the G -(rigid) analytic vectors U G − an of U to be thesubspace of those vectors u for which the orbit map g gu is an element of C an ( G , U ).If U is arbitrary convex L -vector space, then U G − an = lim −→ W W G − an where W runs overall G -invariant BH -subspaces of U . If U is Fr´echet, then U G − an → C an ( G , U ) is a closedembedding and, in particular, U G − an is again a Fr´echet space. As before, if G is σ -affinoid and U is Hausdorff locally convex L -vector space equipped with a topological G -action, then we define U G − an := lim ←− U H − an , where the projective limit is taken overall admissible affinoid open subgroups H of G . In particular, if H ⊂ H is an inclusionof admissible open affinoid subgroups of G , then the natural map U H − an → U H − an isan injection, because when composed with the natural injection U H − an → U , yields IGID ANALYTIC GEOMETRY AND p -ADIC LANGLANDS 11 the natural injection U H − an → U . The representation U is called a rigid analytic G ( K )-representation if the natural inclusion map U G − an → U is a bijection.4.1. Passage from rigid analytic to locally analytic vectors.
Suppose G is alocally K -analytic group. Suppose that ( φ, H, H ) is a chart of G , that is H is a compactopen subgroup of G , H is an affinoid rigid analytic space over L isomorphic to a closedball and φ : H ∼ = −→ H ( K ) with the additional property that H is a subgroup of G . Werefer to such a chart an analytic open subgroup. Let U be a Hausdorff locally convextopological L -vector space equipped with a topological G -action, then for each analyticopen subgroup H of G we can form the convex space U H − an . Then the locally analyticvectors of U is the locally convex inductive limit U la = lim −→ H U H − an , where H runs over all analytic open subgroups of G . Suppose U = lim −→ n U n is a locallyanalytic representation of G , where U n are L -Banach representations of G with injectiveand compact transition maps. Then Emerton shows that U is admissible (in the sense ofSchneider and Teitelbaum) if and only if ( U H − an ) ′ b is finitely generated as a D an ( H , L )-module for any cofinal sequence of analytic open subgroups H [Eme17, Chapter 6].Here D an ( H , L ) is the rigid analytic distributions on H (the dual of C an ( H , L )).5. Wide open congruence subgroups and distribution algebra
Emerton introduced the notion of good analytic open subgroups in [Eme17, Sec.5.2]. In the case of congruence subgroups of GL ( Z p ), these are termed as wide openby Patel, Schmidt and Strauch [PSS14]. In the following we follow the presentation in[PSS14] and we recall few properties of rigid analytic distribution algebras of wide opencongruence subgroups. Let n ≥ G (0) = G = GL ( Z p ) = Spec( Z p [ a, b, c, d, / ∆]) , where ∆ = ad − bc ; comultiplication given by the usual formula. Define the affine groupscheme G ( n ) over Z p by setting O ( G ( n )) = Z p [ a n , b n , c n , d n , / ∆ n ] . Here∆ n = (1 + p n a n )(1 + p n d n ) − p n b n c n , where a n , b n , c n , d n denote indeterminants. The group scheme homomorphism G ( n ) → G ( n −
1) given on the level of algebras are a n − pa n , b n − pb n , c n − pc n , d n − pd n , if n >
1. For n = 1, we set a pa , b pb , c pc , d pd . If R is a flat Z p -algebra, the homomorphism G ( n ) → G (0) = G gives an isomorphism G ( n )( R ) with G ( R ) and G ( n )( R ) = n (cid:18) a bc d (cid:19) ∈ G ( R ) | a − , b, c, d − ∈ p n R o , where we make a formal identification by setting a = 1 + p n a n , b = p n b n , c = p n c n and d = 1 + p n d n . Let b G ( n ) be the completion of G ( n ) along its special fiber G ( n ) F p whichis a formal group scheme over Spf( Z p ). We denote its generic fiber in the sense of rigidanalytic geometry [Bos14] by G ( n ). This is an affinoid rigid analytic group. For a finiteextension K over Q p , its K -points are(5.1) G ( n )( K ) = n (cid:18) a bc d (cid:19) | a − , b, c, d − ∈ p n O K o . Let [ G ( n ) ◦ be the completion of G ( n ) in the closed point corresponding to the unitelement in G ( n ) F p . This is again a formal scheme over Spf( Z p ). Its generic fiber is theso-called ‘wide open’ rigid analytic group which we denote by G ( n ) ◦ . We have G ( n ) ◦ ( K ) = n (cid:18) a bc d (cid:19) | a − , b, c, d − ∈ p n m O K o We can identify G ( n ) ◦ with the rigid analytic four dimensional open polydisc ( B ◦ ) (which is a σ -affinoid space) via coordinates of the second kind( t , t , t , t ) exp( t p n e )exp( t p n h )exp( t p n h )exp( t p n f ) , where e = (cid:18) (cid:19) , h = (cid:18) (cid:19) , h = (cid:18) (cid:19) , f = (cid:18) (cid:19) are elements of the enveloping algebra U ( g ) with g = Lie( GL ( Q p )). Then, we canidentify functions on G ( n ) ◦ as functions on ( B ◦ ) via pull-back. This justifies that G ( n ) ◦ is ‘good analytic’ in the sense of Emerton [Eme17, Sec. 5.2]. By Proposition5 . . D an ( G ( n ) ◦ , L ) is of compact type and can be written as an inductive limit D an ( G ( n ) ◦ , L ) = lim −→ m D an ( G ( n ) ◦ , L ) ( m ) where each D an ( G ( n ) ◦ , L ) ( m ) is a noetherian L -Banach algebra; hence D an ( G ( n ) ◦ , L ) is acoherent ring. This coherent property of D an ( G ( n ) ◦ , L ) makes it interesting to study thecategory of finitely presented modules (which now becomes an abelian category) over D an ( G ( n ) ◦ , L ). Following ideas of Schneider and Teitelbaum of studying coadmissiblemodules over the locally analytic distribution algebra of compact p -adic Lie groups,Emerton, Patel, Schmidt and Strauch have also studied the category of modules M IGID ANALYTIC GEOMETRY AND p -ADIC LANGLANDS 13 over D an ( G ( n ) ◦ , L ) which are of the form M ∼ = lim −→ m M m where each M m is finitelygenerated over the noetherian ring D an ( G ( n ) ◦ , L ) ( m ) with compatibility condition M m ′ ∼ = D an ( G ( n ) ◦ , L ) ( m ′ ) ⊗ D an ( G ( n ) ◦ ,L ) ( m ) M m for m ′ > m, (see Lemma A. D an ( G ( n ) ◦ , L ) which will be useful in the next section when we find G ( n ) ◦ -analytic vectors of B ( V ) when V is irreducible, crystalline, Frobenius semisimplewith distinct Hodge-Tate weights. The following facts can be found in the Appendix ofEmerton’s paper [Eme07]. Proposition 5.1. If U is an admissible locally analytic GL ( Z p ) -representation over L . Then(1) ( U G ( n ) ◦ − an ) ′ is finitely presented as a D an ( G ( n ) ◦ , L ) -module.(2) There is a natural isomorphism ( U G ( n ) ◦ − an ) ′ ∼ = D an ( G ( n ) ◦ , L ) b ⊗ D la ( G ( n ) ◦ ( Q p ) ,L ) U ′ . Proposition 5.2.
The functor U U G ( n ) ◦ − an on the category of admissible locallyanalytic GL ( Z p ) -representations is exact in the strong sense (that is, it takes an ex-act sequence of admissible locally analytic representations to a strict exact sequence ofnuclear Fr´echet spaces). Rigid analytic vectors of crystalline representations
Suppose that V is irreducible, crystalline, Frobenius semisimple with distinct Hodge-Tate weights 0 and k −
1. With notations as in Section 3.1, B ( V ) la ∼ = LA ( α ) ⊕ π ( β ) LA ( β ) . Our goal in this section is to find out (and provide an explicit description of) the G ( n ) ◦ -analytic vectors of B ( V ) la , denoted by (cid:0) B ( V ) la (cid:1) G ( n ) ◦ − an . Let us first search for G ( n ) ◦ -analytic vectors of LA ( α ). Recall that the locally analytic induced principalseries LA ( α ) is (cid:0) Ind GL ( Q p ) B ( Q p ) χ α (cid:1) la where χ α = unr( α − ) ⊗ x k − unr( pβ − ). The Iwasawadecomposition [OS10, Sec. 3.2.2] gives(6.1) (cid:0) Ind GL ( Q p ) B ( Q p ) χ α (cid:1) la ∼ = (cid:0) Ind GL ( Z p ) B ( Z p ) χ α (cid:1) la as GL ( Z p )-equivariant topological isomorphism. Let I ( Z p ) be the Iwahori sub-group (cid:18) Z × p p Z p Z p Z × p (cid:19) , W be the ordinary Weyl group of GL ( Q p ) with respect to B ( Z p ), P w ( Z p ) = I ( Z p ) ∩ wB ( Z p ) w − for w ∈ W . By the Bruhat-Tits decomposition ( loc.cit and [Car79, Sec. 3.5]), GL ( Z p ) = ⊔ w ∈ W I ( Z p ) wB ( Z p ) , we obtain the decomposition(6.2) (cid:0) Ind GL ( Z p ) B ( Z p ) χ α (cid:1) la ∼ = ⊕ w ∈ W (cid:0) Ind I ( Z p ) P w ( Z p ) ( χ wα ) (cid:1) la , an I -equivariant decomposition of topological vector spaces, where the action of χ wα isgiven by χ wα ( h ) = χ α ( w − hw ) with h ∈ P w ( Z p ).We first consider w = Id and find out G ( n ) ◦ -analytic vectors in (cid:0) Ind I ( Z p ) P ( Z p ) ( χ α ) (cid:1) la ; here P ( Z p ) is P Id ( Z p ). In the following, since we are always over Z p , by abuse of notation,we will write I for I ( Z p ) and P for P ( Z p ). The space (cid:0) Ind IP ( χ α ) (cid:1) la := n f ∈ C la ( I, L ) : f ( gb ) = χ − α ( b ) f ( g ) o , where g ∈ I, b ∈ P . The action of I on (cid:0) Ind IP ( χ α ) (cid:1) la is given by left translation. Let I (1) be the pro- p Iwahori subgroup of I . Note that, since χ α is fixed, the restriction ofthe functions of (cid:0) Ind IP ( χ α ) (cid:1) la to I (1) ⊂ I is injective. Therefore, we see that the spaceof (cid:0) Ind IP ( χ α ) (cid:1) la is isomorphic to (cid:0) Ind I (1) Q ( χ α ) (cid:1) la := n f ∈ C la ( I (1) , L ) : f ( gb ) = χ − α ( b ) f ( g ) o , where Q = P ∩ I (1). The pro- p Iwahori I (1) has a natural triangular decomposition I (1) = U Q where U = n (cid:18) z (cid:19) , z ∈ Z p o . As a vector space, (cid:0)
Ind I (1) Q ( χ α ) (cid:1) la is isomor-phic to C la ( U, L ) ∼ = C la ( Z p , L ). This later identification is given by f (cid:18) z (cid:19) f ( z )where z ∈ Z p , identified as points of the rigid analytic closed affinoid unit ball B . Theaction of I (1) on C la ( Z p , L ) is given by the following lemma. Lemma 6.1. [Clo18, Lemma 3.3]
For y ∈ Z p , x ∈ p Z p , s, t ∈ p Z p , we have(i) (cid:18) y (cid:19) f ( z ) = f ( z − y ) ,(ii) (cid:18) s t (cid:19) f ( z ) = f ( st − ) χ α ( s, t ) , (iii) (cid:18) x (cid:19) f ( z ) = f ( z − xz ) χ α ((1 − xz ) − , − xz ) . We want to find the subspace (cid:0)
Ind I (1) Q ( χ α ) la (cid:1) G ( n ) ◦ − an where G ( n ) ◦ is a σ -affinoid rigidanalytic space defined in Section 5. Therefore, by Section 4, we have (cid:0) Ind I (1) Q ( χ α ) la (cid:1) G ( n ) ◦ − an ∼ = lim ←− m>n (cid:0) Ind I (1) Q ( χ α ) la (cid:1) G ( m ) − an Let U ( m ) and U be the rigid analytic affinoids such that U ( m ) := U ( m )( Z p ) = G ( m )( Z p ) ∩ U ( Z p ) where U ( Z p ) = U . (Here G ( m ) is as in equation (5.1)). Notethat, we have realized (cid:0) Ind I (1) Q ( χ α ) (cid:1) la as a space of locally analytic functions on U . Let f ∈ C la ( U, L ). Suppose f ∈ (cid:0) C la ( U, L ) (cid:1) U ( m ) − an , then u uf ∈ C an ( U ( m ) , C la ( U, L ))where u ∈ U ( m ). So u uf (1) is in C an ( U ( m ) , L ), because evaluation at 1 ∈ U ( m )is a continuous map. Now, by part (i) of Lemma 6.1, this gives that u f ( u ) is IGID ANALYTIC GEOMETRY AND p -ADIC LANGLANDS 15 in C an ( U ( m ) , L ). Hence (cid:0) Ind I (1) Q ( χ α ) la (cid:1) G ( m ) − an can be identified with a subspace of C la ( U, L ) ∩ C an ( U ( m ) , L ). In the following we will show that any member of C la ( U, L ) ∩C an ( U ( m ) , L ) is actually G ( m )-analytic by using Lemma 6.1. Proposition 6.2.
We have (cid:0)
Ind I (1) Q ( χ α ) la (cid:1) G ( m ) − an ∼ = C la ( U, L ) ∩ C an ( U ( m ) , L )Note that the space C an ( U ( m ) , L ) is an L -Banach space with the following norm.Suppose f ∈ C an ( U ( m ) , L ), then(6.3) f ( z ) = ∞ X l =0 a l z l , a l ∈ L, is convergent in the ball p m Z p . That is,(6.4) val p ( a l ) + ml → ∞ as l → ∞ . Then, we define the C an ( U ( m ) , L )-valuation of f by val C ( f ) = inf l { val p ( a l ) + ml } . Proof of Proposition 6.2.
Consider the relation (cid:18) y (cid:19) f ( z ) = f ( z − y ) where y ∈ p m Z p and f ∈ C an ( U ( m ) , L ). Then, f ( z − y ) = ∞ X l =0 a l ( z − y ) l = ∞ X l =0 a l l X v =0 (cid:18) lv (cid:19) z l − v ( − v y v = ∞ X v =0 y v (cid:16) X l ≥ v a l (cid:18) lv (cid:19) ( − v z l − v (cid:17) = ∞ X v =0 y v f v . For each fixed v , val C ( f v ) ≥ inf l ≥ v { val p ( a l ) + m ( l − v ) } = inf l ≥ v { val p ( a l ) + ml } − mv .Therefore, by equation 6.4,val C ( f v ) + mv ≥ inf l ≥ v { val p ( a l ) + ml } → ∞ as v → ∞ . This implies that the action (cid:18) y (cid:19) (cid:18) y (cid:19) f is analytic on U ( m ), i.e. it belongsto C an ( U ( m ) , C an ( U ( m ) , L )).Next consider the action in part (iii) of Lemma 6.1. This is given by (cid:18) x (cid:19) f ( z ) = f ( z − xz ) χ α ((1 − xz ) − , − xz ) , where now x, z ∈ p m Z p . Note that the character χ α = unr( α − ) ⊗ x k − unr( pβ − ), α, β ∈ O L ; the unramified character unr( − ) : Q × p → L × is given by unr( λ ) : y λ val p ( y ) .Therefore, for s, t ∈ p Z p , χ α (cid:18) s x t (cid:19) = t k − . Hence χ α ((1 − xz ) − , − xz ) = (1 − xz ) k − . This being a polynomial in x , is alwaysrigid analytic in x when x varies over B ( p m Z p ). So we can forget about the factor χ α ((1 − xz ) − , − xz ) and it will be enough to show that the action (cid:18) x (cid:19) f ( z − xz )is rigid analytic for the action of the upper unipotent rigid analytic subgroup of G ( m ).Now, f ( z − xz ) = ∞ X l =0 a l z l (1 − xz ) l = ∞ X l =0 a l z l ∞ X q =0 (cid:18) l + q − q (cid:19) x q z q = ∞ X q =0 x q (cid:16) ∞ X l =0 a l (cid:18) l + q − q (cid:19) z l + q (cid:17) = ∞ X q =0 x q f q , where we have used the fact that (1 − v ) − l = P ∞ q =0 (cid:0) l + q − q (cid:1) v q for | v | p <
1. For each q ,val C ( f q ) ≥ inf l { val p ( a l ) + m ( l + q ) } = inf l { val p ( a l ) + ml } + mq = val C ( f ) + mq Therefore, of course, val C ( f q ) + mq ≥ val C ( f ) + 2 mq → ∞ as q → ∞ .Next, for s ∈ p m Z p and z ∈ p m Z p , consider (cid:18) s
00 1 (cid:19) f ( z ) = f ( sz ) χ α ( s,
1) = f ( sz ) . Write s = 1 + s ′ , where s ′ ∈ p m Z p , and the analyticity has to be checked in the variable s ′ . f ( sz ) = ∞ X l =0 a l ( z + s ′ z ) l = ∞ X l =0 a l l X q =0 (cid:18) lq (cid:19) z l − q ( s ′ z ) q = ∞ X q =0 ( s ′ ) q (cid:16) X l ≥ q a l (cid:18) lq (cid:19) z l (cid:17) = ∞ X q =0 ( s ′ ) q f q . Therefore, val C ( f q ) ≥ inf l ≥ q { val p ( a l ) + ml } → ∞ as q → ∞ by equation (6.4). Hence,obviously, val C ( f q ) + mq → ∞ as q → ∞ . Finally, consider the action (cid:18) t (cid:19) f ( z ) = f (cid:0) zt (cid:1) χ α (1 , t ) , IGID ANALYTIC GEOMETRY AND p -ADIC LANGLANDS 17 for t = 1 + t ′ , t ′ ∈ p m Z p . By the same argument as before, χ α (1 , t ′ ) is a polynomialin t ′ and so it is, of course, analytic when t ′ ∈ B ( p m Z p ) .f (cid:0) z t ′ (cid:1) = ∞ X l =0 a l z l (1 + t ′ ) l = ∞ X l =0 a l z l ∞ X q =0 (cid:18) l + q − q (cid:19) ( − q ( t ′ ) q = ∞ X q =0 ( t ′ ) q (cid:16) ∞ X l =0 a l (cid:18) l + q − q (cid:19) ( − q z l (cid:17) = ∞ X q =0 ( t ′ ) q f q . Therefore, val C ( f q ) ≥ inf l { val p ( a l ) + ml } = val C ( f ). This implies that val C ( f q ) + mq →∞ as q → ∞ . This completes the proof of Proposition 6.2. (cid:3) This gives the following corollary.
Corollary 6.3. (cid:0)
Ind I (1) Q ( χ α ) la (cid:1) G ( n ) ◦ − an ∼ = lim ←− m>n C la ( U, L ) ∩ C an ( U ( m ) , L ) ∼ = C an ( U ( n ) ◦ , L ) ∩ C la ( U, L ) . Recall that the above Corollary gives us the analytic vectors when the Weyl element w is identity. Having found out the analytic vectors for the Weyl orbit w = Id , ageneral argument in appendix A of [Ray18] gives us the analytic vectors for all othernon-trivial Weyl orbits coming from equation (6.2). Hence, it follows that (cid:16) Ind I ( Z p ) P w ( Z p ) ( χ wα ) la (cid:17) G ( n ) ◦ − an ∼ = lim ←− m>n C an ( w U ( m ) w − , L ) ∩ C la ( wU w − , L ) ∼ = C an ( w U ( n ) ◦ w − , L ) ∩ C la ( wU w − , L ) , where w U ( m ) w − and w U ( n ) ◦ w − have usual meaning. (They are rigid analyticspaces such that w U ( m ) w − ( Z p ) = w U ( m )( Z p ) w − ). Therefore, we have shown thatthe following Theorem holds. Theorem 6.4.
Let GA ( α ) be the space of functions ⊕ w ∈ W C an ( w U ( n ) ◦ w − , L ) ∩ C la ( wU w − , L ) where the action of the torus of U ( n ) ◦ ( Z p ) on each of the direct sum is given by χ wα .Then GA ( α ) is stable under the action of G ( n ) ◦ ( Z p ) and is a rigid analytic G ( n ) ◦ ( Z p ) -representation in the sense of Emerton. Furthermore, we have (cid:0) LA ( α ) (cid:1) G ( n ) ◦ − an ∼ = GA ( α ) as rigid analytic G ( n ) ◦ ( Z p ) -representations. Exactly, by a similar argument, replacing α by β we can find the space GA ( β ) of G ( n ) ◦ -analytic vectors of LA ( β ).Recall the locally algebraic representation π ( β ) with a closed GL ( Q p )-equivariantembeddding into LA ( β ) (cf. equation (3.2)), where π ( β ) = Sym k − L ⊗ L (cid:0) Ind GL ( Q p ) B ( Q p ) unr( β − ) ⊗ unr( pα − ) (cid:1) sm (cf. equation (2.3)). As before, by Iwasawa decomposition and Bruhat-Tits decompo-sition (cf. (6.1) and (6.2)), we have (cid:0) Ind GL ( Q p ) B ( Q p ) unr( β − ) ⊗ unr( pα − ) (cid:1) sm ∼ = ⊕ w ∈ W (cid:0) Ind I ( Z p ) P w ( Z p ) (unr( β − ) ⊗ unr( pα − )) w (cid:1) sm , Similar to the case of locally analytic vectors considered before, the smooth vectorsof the space Ind I ( Z p ) P w ( Z p ) (unr( β − ) ⊗ unr( pα − )) w can be identified with locally constantfunctions on wU w − . Therefore, the U ( m )-analytic subspace of (cid:0) Ind I ( Z p ) P w ( Z p ) (unr( β − ) ⊗ unr( pα − )) w (cid:1) sm is the set of functions which are constant on w U ( m )( Z p ) w − and locallyconstant on its complement space, i.e. wU w − \ w U ( m )( Z p ) w − ; we denote this spaceby C ( β ) w U ( m ) w − . (The ‘ C ’ stands for constant on w U ( m ) w − and ‘ β ’ means we aredealing with the character corresponding to β ). Then the following corollary is obvious,as the polynomials Sym k − L are always, of course, rigid analytic. Corollary 6.5.
Let C ( β ) m denote ⊕ w ∈ W C ( β ) w U ( m ) w − . Then G ( n ) ◦ -analytic vectorsof π ( β ) are given by (cid:0) π ( β ) (cid:1) G ( n ) ◦ − an ∼ = Sym k − L ⊗ lim ←− m>n C ( β ) m . Replacing α by β we obtain (cid:0) π ( α ) (cid:1) G ( n ) ◦ − an ∼ = Sym k − L ⊗ lim ←− m>n C ( α ) m . From thediscussion after equation 4 . (cid:0) LA ( α ) ⊕ π ( β ) LA ( β ) (cid:1) ∗ ∼ = ker (cid:16) LA ( α ) ∗ ⊕ LA ( β ) ∗ → π ( β ) ∗ (cid:17) . But the locally analytic principal series LA ( α ) or LA ( β ) are spaces of compact type,hence reflexive [Sch13, Prop. 16.10]. Also closed subspace (and quotient by a closedsubspace) of a reflexive space is reflexive. Therefore, π ( β ) is reflexive and note that themap LA ( α ) ∗ ⊕ LA ( β ) ∗ → π ( β ) ∗ in (6.5) is a surjection, since it comes from dualizingthe natural injection π ( β ) ֒ → LA ( β ) ֒ → LA ( α ) ⊕ LA ( β ). Therefore, dualizing equation(6.5) and using Proposition 5.2, we get the following Theorem. Theorem 6.6.
Suppose V = V ( α, β ) is crystalline, irreducible, Frobenius semisim-ple Galois representation with distinct Hodge-Tate weights. Let B ( V ) la be the locallyanalytic vectors of the GL ( Q p ) -representation B ( V ) associated to V via p -adic localLanglands constructed by Berger and Breuil. Then, (cid:0) B ( V ) la (cid:1) G ( n ) ◦ − an ∼ = Coker (cid:16) π ( β ) G ( n ) ◦ − an ֒ → LA ( α ) G ( n ) ◦ − an ⊕ LA ( β ) G ( n ) ◦ − an (cid:17) is a rigid-analytic G ( n ) ◦ ( Z p ) -representation. Furthermore,(i) π ( β ) G ( n ) ◦ − an ∼ = Sym k − L ⊗ lim ←− m>n C ( β ) m ,(ii) LA ( α ) G ( n ) ◦ − an ∼ = GA ( α ) and LA ( β ) G ( n ) ◦ − an ∼ = GA ( β ) .In particular, it follows that (cid:0) B ( V ) la (cid:1) G ( n ) ◦ − an = 0 . IGID ANALYTIC GEOMETRY AND p -ADIC LANGLANDS 19 The fact that (cid:0) B ( V ) la (cid:1) G ( n ) ◦ − an = 0 is obvious because we note that π ( β ) and π ( α )are isomorphic via the intertwining operator I sm (cf. Lemma 2.1) and we note that π ( α ) (resp. π ( β )) are embedded into LA ( α ) (resp. LA ( β )) (see (3.2)). But in Theorem6.6, we have only cut out the direct sum of G ( n ) ◦ -analytic vectors of LA ( α ) ⊕ LA ( β ) bythe G ( n ) ◦ -analytic vectors of π ( β ). In particular, an isomorphic copy of π ( α ) G ( n ) ◦ − an certainly lives inside (cid:0) B ( V ) la (cid:1) G ( n ) ◦ − an .7. Langlands base change of rigid analytic vectors and futurequestions
Let K be a finite extension of Q p and K be an extension of K of degree r . Givena formal O K -scheme X O K , Bertapelle constructs a Weil restriction functor which as-sociates to X O K another formal scheme over O K [Ber00]. Suppose X K is the rigidanalytic space associated to the formal scheme X O K . This gives a Weil restrictionfunctor which associates to X K another rigid analytic space R K/K X K over K [Ber00,Prop. 1.8]. We fix a choice of a basis of K over K . In the following Lemma, we recallhow Weil restriction behaves with affinoid closed unit balls. Lemma 7.1. [Clo18, Lemma 1.1]
Suppose K is unramified over K . Let B K be theaffinoid closed unit ball over K . Then, R K/K B K is isomorphic to the r -th power of B K . Therefore, R K/K ( B K ) d ∼ = ( B K ) dr as rigid analytic spaces. Consider a rigid analytic group G K over K isomorphic as a rigid analytic space to( B K ) d . Assume that G K is actually defined over K , i.e., is obtained by extension ofscalars from K . Then C an ( G K , K ) ∼ = C an ( G K , K ) ⊗ K . There is a natural comultipli-cation map m : C an ( G K , K ) → C an ( G K , K ) b ⊗ C an ( G K , K )which is defined by group multiplication in G K . This comultiplication map extends fromthe natural comultiplication map defined for G K and defines a natural comultiplicationmap R ( m ) : C an ( R K/K G K , K ) → C an ( R K/K G K , K ) b ⊗ C an ( R K/K G K , K ) . Here C an ( R K/K G K , K ) = C an ( R K/K G K , K ) ⊗ K . Suppose K is Galois over K . There-fore, the Galois group Gal( K/K ) acts naturally on G K by g -linear automorphisms ofthe Tate algebra and acts on R K/K G K by K -automorphisms. Then, Clozel showedthe following Lemma relating rigid analytic functions on G K with those of R K/K G K [Clo18, Sec. 1]. Lemma 7.2. (i) There exists a natural map b : C an ( G K , K ) → C an ( R K/K G K , K ) which commutes with natural comultiplication maps.(ii) There is a natural map b = Y g ∈ Gal(
K/K ) b g : C an ( G K , K ) → C an ( R K/K G K , K ) which commutes with comultiplication.(iii) Part (ii) induces an isomorphism C an ( R K/K G K , K ) ∼ = b ⊗ g ∈ Gal(
K/K ) C an ( G K , K ) . The map b is a tensor product b = b ⊗ g ∈ Gal(
K/K ) b g . Its g -component sends a powerseries P a l x m , x = ( x , ..., x d ) to the series P a l g ( x ) m . (iv) The maps b and b are continuous in canonical topologies (it respects sup norm onthe Tate algebra of power series). Now suppose that U is a rigid analytic representation of G K ( K ) over L where ι : K ֒ → L is the natural inclusion. Then, by [Clo18, Prop. 3.1], U extends naturallyto a rigid analytic representation of G K ( K ). If g ∈ Gal(
K/K ), we have the injection ι ◦ g : K → L . We write U g for the representation of G K ( K ) associated to ι ◦ g . Itis K -analytic for ι ◦ g . Then, Clozel defines the base change of U compatible withLanglands functoriality. Definition 7.3. [Clo18, Defn. 3.2]
The Langlands base change of U is the globallyanalytic representation of R K/K G K ( K ) on W = b ⊗ g ∈ Gal(
K/K ) U g . There are several future project that branches out from Theorem 6.6, the main resultof this article. We conclude this section by mentioning some of them. The first step isto generalize Weil restriction functor for the case of rigid analytic σ -affinoid represen-tations. Let us denote this functor by R K/K G ◦ K . This functor has to be constructedin such a way that it satisfies the natural analogue of Clozel’s result (cf. Lemma 7.2).In particular, we must show that C an ( R K/K G ◦ K , K ) ∼ = b ⊗ g ∈ Gal(
K/K ) C an ( G ◦ K , K ) . Then we can construct the Langlands base change of (cid:0) B ( V ) la (cid:1) G ( n ) ◦ − an analogous to thatof Clozel’s, by showing that the corresponding tensor product in Definition 7.3 will be aglobally analytic representation over the Weil restriction of sigma affinoid rigid analyticspaces. This is a future project and we will solve this in a separate paper.7.1. Why σ -affinoids and not affinoids? In this article, the reader will notice thatwe compute rigid analytic vectors for σ -affinoid rigid analytic groups (open balls) in-stead of affinoids (closed balls). It is a natural question to ask, why such a choice? Thisis mainly due to Proposition 5.2, that is, the functor U U G ( n ) ◦ − an on the category ofadmissible locally analytic GL ( Z p )-representations is exact. It is still an open questionto study if U U G ( n ) − an is exact. This seems to require some new idea. Further,the distribution algebra D an ( G ( n ) ◦ , L ) is a coherent ring (Section 5) which makes iteasy to study modules over this algebra. However, the distribution algebra of the affi-noid group D an ( G ( n ) , L ) is not so well behaved. More precisely, Clozel shows that theanalytic distribution algebra of affinoids does not respect cartesian product, D an ( X × Y, L ) ≇ D an ( X, L ) b ⊗D an ( Y, L ) , IGID ANALYTIC GEOMETRY AND p -ADIC LANGLANDS 21 where X and Y are rigid analytic spaces (cf. [Clo18, Appendix A.1]). Furthermore, D an ( G ( n ) , L ) is not noetherian [Clo18, Appendix A.2]. Therefore, it remains open todetermine the algebra structure of D an ( G ( n ) , L ).7.2. Representations non-crystalline.
In this article, we show the existence of rigidanalytic vectors in B ( V ) la where V is crystalline. A natural question is to ask if thereexists rigid analytic vectors for non-crystalline representations, that is, V = V ( s ) when s ∈ S ng ∗ ⊔ S st ∗ ⊔ S ord ∗ ⊔ S ncl ∗ . This is too an open question which needs further research.
Achnowledgements
The author heartily thanks Laurent Clozel for introducing him to the theory of rigidanalytic p -adic representations and for suggesting to look at the problem discussed inthis article. The author also thanks him for suggesting the open question presented inthis article. The author also benefited from numerous conversations with Filippo Nuccioon his visit to the University of British Columbia during September 2019 - August 2020.The author is also grateful to Matthew Emerton for bringing his attention to Emerton’spaper [Eme07]. References [BB04] Laurent Berger and Christophe Breuil. Toward a p -adic Lang-lands programme. Cours au C.M.S. de Hangzhou , 2004. Weblink: [BB10] Laurent Berger and Christophe Breuil. Sur quelques repr´esentations potentiellementcristallines de GL ( Q p ). Ast´erisque , (330):155–211, 2010.[Ber00] Alessandra Bertapelle. Formal N´eron models and Weil restriction.
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