aa r X i v : . [ m a t h . N T ] S e p MODULI OF LANGLANDS PARAMETERS
JEAN-FRANC¸ OIS DAT, DAVID HELM, ROBERT KURINCZUK, AND GILBERT MOSS
Abstract.
Let F be a non-archimedean local field of residue characteristic p ,let ˆ G be a split connected reductive group over Z [ p ] with an action of W F , andlet G L denote the semidirect product ˆ G ⋊ W F . We construct a moduli spaceof Langlands parameters W F → G L , and show that it is locally of finite typeand flat over Z [ p ], and that it is a reduced local complete intersection. We giveparameterizations of the connected components of this space over algebraicallyclosed fields of characteristic zero and characteristic ℓ = p , as well as of thecomponents of the space over Z ℓ and (conjecturally) over Z [ p ]. Finally, westudy the functions on this space that are invariant under conjugation by ˆ G (or, equivalently, the GIT quotient of this space by ˆ G ) and give a completedescription of this ring of functions after inverting an explicit finite set ofprimes depending only on G L . Contents
1. Introduction and main results 12. The space of tame parameters 93. Reduction to tame parameters 164. Moduli of Langlands parameters 255. Unobstructed points 426. The GIT quotient in the banal case 57Appendix A. Moduli of cocycles 66Appendix B. Twisted Poincar´e polynomials 75References 781.
Introduction and main results
Introduction.
Let F be a local field with residue characteristic p , and G aquasi-split connected reductive group over F . Let ℓ be a prime different from p .A Langlands parameter for G is a continuous L -homomorphism W F → G L ( Q ℓ );that is, an ℓ -adically continuous homomorphism from the Weil group W F to thegroup of Q ℓ -points of the Langlands dual group G L := ˆ G ⋊ W F of G , such that thecomposition with the natural map G L ( Q ℓ ) → W F is the identity.When G is the general linear group GL n , then G L is simply the product GL n × W F ,and a Langlands parameter for G is simply a continuous representation: W F → GL n ( Q ℓ ). Such representations vary nicely in algebraic families; in particular, givena continuous representation ϕ : W F → GL n ( F ℓ ), we can associate to it the univer-sal framed deformation ring R (cid:3) ϕ . a complete Noetherian local W ( F ℓ )-algebra that Mathematics Subject Classification. admits a continouous representation ϕ univ : W F → GL n ( R (cid:3) ϕ ) such that the pair( R (cid:3) ϕ , ϕ univ ) is universal for pairs ( R, ϕ ), where R is a complete Noetherian local W ( F ℓ )-algebra and ϕ : W F → GL n ( R ) is a lift of ϕ .Given the importance of such deformation spaces in the Langlands program, itis natural to attempt to construct corresponding “universal deformation spaces”for Langlands parameters attached to groups G other than GL n . Indeed, Bellovinand Gee [BG19] and Booher and Patrikis [BP19] independently study a closelyrelated problem. Specifically, (cf. [BP19], Section 2) define an G L -Weil-Delignerepresentation over a W ( F ℓ )-algebra A to be a triple ( D A , r, N ), where D A is an G L -bundle over Spec A , r : W F → Aut G ( D A ) is a homomorphism with open kernel ,and N is a nilpotent element of the Lie algebra of Aut G L ( D A ) such that Ad r ( w ) N = | w | N for all w ∈ W F . Both Bellovin-Gee and Booher-Patrikis construct modulispaces of such G L -Weil-Deligne representations, that are schemes locally of finitetype over W ( F ℓ ), and show that their general fibers are generically smooth andequidimensional of dimension equal to the dimension of G L .When A is complete local, and ℓ is invertible in A , Grothendieck’s monodromytheorem gives a natural bijection between G L -Weil-Deligne representations withvalues in A and Langlands parameters with values in A , so the results of Bellovin-Gee and Booher-Patrikis in some sense give a solution to the problem of findinguniversal families for Langlands parameters over G . Their method relies heavily onthe exponential and logarithm maps, which have denominators, and also involvesdivision by the order of the image of an element of inertia. There is thus reasonto question whether these constructions give the “right” objects if the prime ℓ issmall enough to divide one of these denominators. For instance, when G = GL ,and ℓ divides q − q denotes the order of the residue field of F ), the spacesconstructed by Bellovin-Gee and Booher-Patrikis fail to be flat over W ( F ℓ ). Sinceuniversal framed deformation rings are known to be flat over W ( F ℓ ), this meansthat when G = GL , the constructions of Bellovin-Gee and Booher-Patrikis failto recover the existing theory in such characteristics. It is reasonable to expectthat this failure of flatness persists for more complicated groups. Such a failuremakes the spaces they construct unsuitable for formulating analogues of Shotton’s“ ℓ = p Breuil-Mezard” results for GL n [Sho18]. We refer the reader to section 2.3for further discussion of this point.In light of these issues, it is tempting to look at alternative characterizationsof Langlands parameters over fields of characteristic zero, in the hope that theysuggest better behaved moduli problems. There are (at least) three definitions ofa “Langlands parameter over Q ℓ ” common in the literature:(1) pairs ( r, N ), where r : W F → G L ( Q ℓ ) is an L -homomorphism with openkernel and N ∈ Lie( ˆ G Q ℓ ) a nilpotent element, such that Ad r ( w ) = | w | N ,(2) maps W F × SL ( Q ℓ ) → G L ( Q ℓ ) whose restriction to the first factor is an L -homomorphism with open kernel and whose restriction to the second factoris algebraic, and(3) L -homomorphisms ϕ L : W F → G L ( Q ℓ ) that are ℓ -adically continuous.The first of these definitions generalizes in an obvious way to coefficients in anarbitrary W ( F ℓ )-algebra R , and considering the associated moduli problem leads tothe schemes considered by Bellovin-Gee and Booher-Patrikis. The second likewise ODULI OF LANGLANDS PARAMETERS 3 generalizes to such algebras R , but the associated moduli space is much less well-behaved. For instance, the moduli space of unramified pairs ( r, N ) as in (1) isconnected over Q ℓ , whereas the space of unramified maps W F × SL → G L asin (2) is, over Q ℓ , a disjoint union over the set of conjugacy classes of unipotentelements u ∈ ˆ G ( Q ℓ ), of the loci where the image of the matrix ( ) in the SL factor is conjugate to u .It is therefore tempting to try to construct a moduli space of ℓ -adically continuous L -homomorphisms from W F to G L as in (3). The notion of ℓ -adic continuity for L -homomorphisms valued in G L ( Q ℓ ) generalizes naturally to complete local ringsof residue characteristic ℓ ; this is sufficient for a well-behaved deformation theorybut is insufficient to obtain a moduli space that is locally of finite type. In orderto obtain such a space, one would need a broader notion of ℓ -adic continuity.Our approach to this question is inspired by previous work of the second au-thor in [Hel19]. That paper introduces a notion of ℓ -adic continuity for maps W F → GL n ( R ) that makes sense for arbitrary W ( F ℓ )-algebras R , and constructsuniversal families of such representations over a suitable W ( F ℓ )-scheme, which wewill denote here by X n . (This notation differs from that of [Hel19], where whatwe call the scheme X n only appears implicitly, as the disjoint union of the schemesdenoted X νq,n ). As with the constructions of Bellovin-Gee and Booher-Patrikis, thescheme X n is locally of finite type over W ( F ℓ ), but unlike their construction, thecompletion of the local ring of X n at any F ℓ -point of X n , corresponding to a map ϕ : W F → GL n ( F ℓ ), is the universal framed deformation ring R (cid:3) ϕ . In other words, X n is a locally of finite-type W ( F ℓ )-scheme that “interpolates” the universal frameddeformation rings of all n -dimensional mod ℓ representations of W F .The schemes X n constructed in [Hel19] play a central role in the formulationand proof of the “local Langlands correspondence in families” for the group GL n ,now proven by two of the authors in [HM18]. (These results, in turn, imply theexistence of the families conjectured by Emerton and the second author in [EH14].)In particular, the subring of functions on X n that are invariant under the conju-gation action on Langlands parameters is naturally isomorphic to the center of thecategory of smooth W ( F ℓ )[GL n ( F )]-modules. Morally, this means that aspects ofthe geometry of X n are reflected in the representation theory of GL n ( F ). For in-stance, the connected components of X n correspond to the “blocks” of the categoryof smooth W ( F ℓ )[GL n ( F )]-modules.In this paper our first objective is to generalize the construction of [Hel19] tothe setting of Langlands parameters for arbitrary quasi-split, connected reductivegroups, with an eye towards formulating a conjectural analogue of the local Lang-lands correspondence in families for such groups. In a departure from previouswork on the subject, we work over the base ring Z [ p ] rather than over a ring ofWitt vectors; this introduces some technical complexity but gives us the smallestpossible base ring for such a correspondence. (In particular this allows us to studychains of congruences of Langlands parameters modulo several different primes.)We refer the reader to the next subsection for precise definitions.Second, we aim to understand the geometry of these moduli spaces of Langlandsparameters. Several natural questions arise. It turns out that, as in the settingof local deformation theory of Galois representations, the spaces we obtain have aquite tractable local structure: they are reduced local complete intersections thatare flat over Spec Z [ p ], of relative dimension dim ˆ G . Moreover, we give descriptions JEAN-FRANC¸ OIS DAT, DAVID HELM, ROBERT KURINCZUK, AND GILBERT MOSS of the connected components of these moduli spaces, both over algebraically closedfields of arbitrary characteristic ℓ = p , and (conjecturally) over Z [ p ].Finally, we study the rings of functions on these moduli spaces that are invariantunder ˆ G -conjugacy (or, equivalently, the GIT quotient of the moduli space of Lang-lands parameters by the conjugation action of ˆ G .) As in the case of GL n , the ringof such functions is in general quite complicated, and does not admit an explicitdescription. (In particular, the corresponding GIT quotients are very far from beingnormal.) Nonetheless, we show that after inverting an explicit finite set of primes(depending only on G ), the GIT quotients are quite nice; indeed, they are disjointunions of quotients of tori by finite group actions. Over the complex numbers theseconnected components coincide with varieties studied by Haines [Hai14].At a late stage in the preparation of this paper, we became aware of recent re-sults of Xinwen Zhu ([Zhu20], particularly section 3.1) in which he also proposes aconstruction of moduli spaces of Langlands parameters. Zhu shows, as we do, thatthe spaces are flat, reduced local complete intersections, although he uses differenttechniques. Our global study of the connected components and our explicit descrip-tion of the GIT quotients by ˆ G do not appear in his work. Zhu is motivated by theformulation of some conjectures relating the derived category of coherent sheaveson the space of Langlands parameters for an L -group G L to categories defined viathe representation theory of G . The ℓ -adic version of these conjectures has alsobeen proposed by Laurent Fargues and Peter Scholze, and, in particular, Scholzehas also proposed another construction of the moduli spaces under consideration(over Z ℓ , for ℓ = p ), using the condensed mathematics of Clausen-Scholze.1.2. The moduli space of Langlands parameters.
We now describe in detailthe moduli problem that we study. Following [Hel19], the approach we take is to“discretize” the tame inertia group. Fix an arithmetic Frobenius element Fr in W F and a pro-generator s of the tame inertia group I F /P F . These satisfy the relationFr s Fr − = s q . We then consider the subgroup h Fr , s i = s Z [ q ] ⋊ Fr Z of W F /P F , wedenote by W F its inverse image in W F , and we endow it with the topology thatextends the profinite topology of P F and induces the discrete topology on h Fr , s i .Note that (in contrast to the subgroup W F of G F ), the subgroup W F of W F verymuch depends on the choices of Fr and s .Although the topology on W F is finer than the one induced from W F , the relationFr s Fr − = s q implies that a morphism W F −→ G L ( ¯ Q ℓ ) is continuous if and onlyif it is continuous for the topology induced from W F . It follows that restriction to W F induces a bijection between objects of type (3) and the following objects :(4) continuous morphisms ϕ L : W F −→ G L ( ¯ Q ℓ ) (with either the discrete orthe natural topology on G L ( ¯ Q ℓ )).These objects are now easy to define over any Z ℓ -algebra R since only the discretetopology of G L ( R ) is needed. Indeed, they are also defined for any Z [ p ]-algebraand their moduli space over Z [ p ] is already interesting.We therefore consider the following setting: • ˆ G is a split reductive group scheme over Z [1 /p ] endowed with a finite actionof the absolute Galois group G F (we do not assume that G F preserves apinning). ODULI OF LANGLANDS PARAMETERS 5 • W F is the inverse image in W F of the subgroup s Z [ q ] ⋊ Fr Z of W F /P F ,which depends on the choice of a generator s of the tame inertia group I F /P F and a lift of Frobenius. • ( P eF ) e ∈ N is a decreasing sequence of open subgroups of P F that are normalin W F and whose intersection is { } .Note that for any Z [ p ]-algebra R , there is a natural bijection between the contin-uous L -homomorphisms ϕ L : W F → G L ( R ) (with respect to the discrete topologyon G L ( R )) and the set of continuous 1-cocycles Z ( W F , ˆ G ( R )) on W F with valuesin ˆ G ( R ). If, given ϕ L , we denote by ϕ the corresponding cocycle, then this bijectionis characterized by the identity ϕ L ( w ) = ( ϕ ( w ) , w ) for all w ∈ W F .Since the cocycles we consider are continuous with respect to the discrete topol-ogy, we have Z ( W F , ˆ G ( R )) = S e ∈ N Z ( W F /P eF , ˆ G ( R )). It is easy to see that thefunctor R Z ( W F /P eF , ˆ G ( R )) on Z [ p ]-algebras is represented by an affine schemeof finite presentation over Z [ p ], that we denote by Z ( W F /P eF , ˆ G ). It follows thatthe functor R Z ( W F , ˆ G ( R )) is represented by a scheme Z ( W F , ˆ G ), in whicheach Z ( W F /P eF , ˆ G ) sits as a direct summand, and which is the increasing unionof all these subschemes.As a Z [ p ]-scheme, the scheme Z ( W F , ˆ G ) depends on the choices we made defin-ing W F as a subgroup of W F . Indeed, if W ′ F is the subgroup arising from a differentchoice (Fr ′ , s ′ ) then there is not typically a canonical isomorphism of Z [ p ]-schemesfrom Z ( W F , ˆ G ) to Z ( W ′ F , ˆ G ). However, there are canonical such isomorphismsover Z ℓ for each ℓ not equal to p (Corollary 4.2). Moreover, we show (Theorem 4.18)that the GIT quotient Z ( W F , ˆ G ) (cid:12) ˆ G is, up to canonical isomorphism, indepen-dent of the choices defining W F . We further suspect, but do not prove, that thecorresponding quotient stacks are also canonically isomorphic.Our study of Z ( W F , ˆ G ) relies on the restriction map Z ( W F , ˆ G ) −→ Z ( P F , ˆ G ).One crucial point is that the scheme Z ( P F , ˆ G ) is particularly well behaved, because p is invertible in our coefficient rings. Indeed, we prove the following result in theappendix. Proposition 1.1.
The scheme Z ( P F , ˆ G ) is smooth and its base change to ¯ Z [ p ] is adisjoint union of orbit schemes. More precisely, there is a set Φ ⊂ Z ( P F , ˆ G (¯ Z [ p ])) such that (1) Z ( P F , ˆ G ) ¯ Z [ p ] = ` φ ∈ Φ ˆ G · φ and ˆ G · φ represents the sheaf-theoretic (fppfor ´etale) quotient ˆ G/C ˆ G ( φ ) . (2) each centralizer C ˆ G ( φ ) is smooth over ¯ Z [ p ] with split reductive neutral com-ponent and constant π . This says in particular that any cocycle φ ′ ∈ Z ( P F , ˆ G ( R )) is, locally for the´etale topology on R , ˆ G -conjugate to a locally unique φ in Φ.Via the restriction morphism Z ( W F , ˆ G ) −→ Z ( P F , ˆ G ), the proposition inducesa decomposition Z ( W F , ˆ G ) ¯ Z [ p ] = a φ ∈ Φ ˆ G × C ˆ G ( φ ) Z ( W F , ˆ G ) φ JEAN-FRANC¸ OIS DAT, DAVID HELM, ROBERT KURINCZUK, AND GILBERT MOSS where Z ( W F , ˆ G ) φ is the closed subscheme of parameters ϕ such that ϕ | P F = φ .In Section 3 we further decompose Z ( W F , ˆ G ) φ as follows. Proposition 1.2.
For each φ ∈ Φ , there is a finite set ˜Φ φ ⊂ Z ( W F , ˆ G (¯ Z [ p ])) φ ,which is a singleton if C ˆ G ( φ ) is connected, with the following properties : (1) ∀ ˜ ϕ ∈ Φ φ , ˜ ϕ ( W F ) normalizes a Borel pair in C ˆ G ( φ ) ◦ (2) ∀ ˜ ϕ ∈ Φ φ , the map η η · ˜ ϕ defines a closed and open immersion Z ˜ ϕ ( W F /P F , C ˆ G ( φ ) ◦ ) ֒ → Z ( W F , ˆ G ) φ (3) The collection of these maps defines an isomorphism (1.1) a ( φ, ˜ ϕ ) ˆ G × C ˆ G ( φ ) ◦ Z ˜ ϕ ( W F /P F , C ˆ G ( φ ) ◦ ) ∼ −→ Z ( W F , ˆ G ) ¯ Z [ p ] These results are essentially a “geometrization” of the functoriality principleof [Dat17]. We note that if ˆ G is a classical group and p >
2, then C ˆ G ( φ ) is alwaysconnected. Moreover, if the center of ˆ G is smooth over Z [ p ], then we show that“Borel pair” can be replaced by “pinning” in (1).In general, this result shows that the crucial case to study is the space of tame parameters for a tame action of W F that preserves a Borel pair of ˆ G . This case isthoroughly studied in Section 2. Using the results of that section and the abovedecomposition we will get the following result, (Theorem 4.1) Theorem 1.3.
The scheme Z ( W F , ˆ G ) is syntomic (flat and locally a completeintersection) of pure absolute dimension dim( ˆ G ) + 1 , and generically smooth over Z ( P F , ˆ G ) . We further conjecture that the summands appearing in the decomposition of (1.1)are connected. The last proposition reduces this conjecture to proving that for anyˆ G ′ with a tame Galois action preserving a Borel pair, the summand in (1.1) corre-sponding to tame parameters is connected. In Theorem 4.29 we prove this underthe assumption that the action even preserves a pinning, i.e. when G L ′ is gen-uinely the L -group of a tamely ramified reductive group G ′ over F . This allows usto deduce our conjecture in many cases. In particular, Theorem 4.5 asserts : Theorem 1.4.
If the center of ˆ G is smooth, then all the summands in the decom-position of (1.1) are connected. For G = GL n , where all centralizers are connected, this result says that each Z ( W F , ˆ G ) φ is connected, and may be thought of as the Galois counterpart of thefact, discovered by S´echerre and Stevens [SS19], that two irreducible representationsof GL n ( F ) belong to the same endoclass if and only if they are connected by a seriesof congruences at various primes different from p .1.3. The space of parameters over Z ℓ . Let us now fix a prime number ℓ = p .For a Z ℓ -algebra R , we say that a ˆ G ( R )-valued cocycle ϕ is ℓ -adically continuous if there is some ℓ -adically separated ring R such that ϕ comes by pushforwardfrom some ˆ G ( R )-valued cocycle ϕ , all of whose pushforwards to ˆ G ( R /ℓ n ) arecontinuous for the topology inherited from W F . It is not a priori clear that thisdefinition is local for any usual topology. But the following result, extracted fromTheorem 4.1, shows it is, and may justify again our approach involving the weirdgroup W F . ODULI OF LANGLANDS PARAMETERS 7
Theorem 1.5.
The ring of functions R e G L of the affine scheme Z ( W F /P eF , ˆ G ) is ℓ -adically separated and the universal cocycle ϕ e univ extends uniquely to an ℓ -adicallycontinuous cocycle ϕ eℓ − univ : W F /P eF −→ ˆ G ( R e G L ⊗ Z ℓ ) which is universal for ℓ -adically continuous cocycles. The ℓ -adic continuity property of ϕ eℓ − univ and the ℓ -adic separateness of R e G L ⊗ Z ℓ imply that the restriction of ϕ eℓ − univ to the prime-to- ℓ inertia group I ℓF factors overa finite quotient. Since the order of this finite quotient is invertible in Z ℓ , we canuse the same strategy as before to decompose Z ( W F , ˆ G ) ¯ Z ℓ using now restrictionof parameters to I ℓF . The upshot is a decomposition similar to (1.1)(1.2) Z ( W F , ˆ G ) ¯ Z ℓ = a ( φ ℓ , ˜ ϕ ) ˆ G × C ˆ G ( φ ℓ ) ◦ Z ˜ ϕ ( W F /P F , C ˆ G ( φ ℓ ) ◦ ) IℓF
Theorem 4.8 asserts that each summand of this decomposition has a geometricallyconnected special fiber so, in particular, is connected. The collection of all theseconnectedness results for varying ℓ is used in the proof of the connectedness resultsover ¯ Z [ p ].1.4. The categorical quotient over a field.
We now fix an algebraically closedfield L of characteristic ℓ = p but we allow ℓ = 0. We consider the categoricalquotient Z ( W F , ˆ G ) L (cid:12) ˆ G L = lim −→ Spec(( R e G L ⊗ L ) ˆ G L ) . Recall that the closed points of Z ( W F , ˆ G ) L (cid:12) ˆ G L correspond to closed ˆ G ( L )-orbitsin Z ( W F , ˆ G ( L )). A theorem of Richardson tells us that a cocycle ϕ has closedorbit if and only if its image in G L = ˆ G ⋊ W F is completely reducible in the sensethat whenever it is contained in a parabolic subgroup of G L , it has to be containedin some Levi subgroup of this parabolic subgroup.When ℓ = 0, we already know from (1.2) how to parametrize its connectedcomponents, and we now wish to describe them explicitly, at least up to homeo-morphism. In order to give a unified treatment including ℓ = 0, we (re)label theconnected components of Z ( W F , ˆ G ) L (cid:12) ˆ G L by the set Ψ( L ) of ˆ G ( L )-conjugacyclasses of pairs ( φ, β ) consisting of • a completely reducible inertial cocycle φ ∈ Z ( I F , ˆ G ( L )). • an element β in { ˜ β ∈ ˆ G ( L ) ⋊ Fr , ˜ βφ ( i ) ˜ β − = φ (Fr i Fr − ) } /C ˆ G ( φ ) ◦ .For such a pair, the centralizer C ˆ G ( φ ( I F )) is a (possibly disconnected) reductivealgebraic group over L . So we fix a Borel pair ( ˆ B φ , ˆ T φ ) in C ˆ G ( φ ) ◦ and we choosea lift ˜ β of β that normalizes this Borel pair. The adjoint action of ˜ β on ˆ T φ onlydepends on β , and so does its action on the Weyl group Ω φ = Ω ◦ φ ⋊ π ( C ˆ G ( φ )).Now for all ˆ t ∈ ˆ T φ we can extend φ uniquely to a cocycle ϕ ˆ t ˜ β ∈ Z ( W F , ˆ G L ) suchthat ϕ ˆ t ˜ β (Fr) = ˆ t ˜ β . The following result is Corollary 4.22. Theorem 1.6.
The collection of maps ˆ t ϕ ˆ t ˜ β define a universal homeomorphism a ( φ,β ) ∈ Ψ( L ) ( ˆ T φ ) β (cid:12) (Ω φ ) β ≈ −→ Z ( W F , ˆ G L ) (cid:12) ˆ G L , which is an isomorphism if char( L ) = 0 . JEAN-FRANC¸ OIS DAT, DAVID HELM, ROBERT KURINCZUK, AND GILBERT MOSS
In particular, we see that each connected component of Z ( W F , ˆ G L ) (cid:12) ˆ G L isirreducible with normal reduced subscheme.When L = C , this allows us to compare in Section 6.3 our categorical quotientwith Haines’ algebraic variety constructed in [Hai14]. Corollary 1.7.
The scheme Z ( W F , ˆ G C ) (cid:12) ˆ G C is canonically isomorphic to Haines’variety. When L = ¯ F ℓ , we give in Theorem 6.8 an explicit condition on ℓ for the homeo-morphism of the above theorem to be an isomorphism. This involves the notion of G L -banal prime that we now discuss.1.5. Reducedness of fibers and G L -banal primes. The obstruction to ob-taining a description of the GIT quotients over Z [ p ] analogous to our descriptionof the GIT quotients over fields comes from non-reducedness of certain fibers of Z ( W F , ˆ G ). In Theorem 5.5 we determine an explicit finite set S of primes, de-pending only on G L , such that the fibers of Z ( W F , ˆ G ) are geometrically reducedoutside of S .The reducedness of the fibers mod ℓ , for ℓ outside S implies in particular thatgiven two distinct irreducible components of the geometric general fiber of Z ( W F , ˆ G ),their reductions mod ℓ remain distinct. Moreover, the reduction of each such com-ponent has scheme-theoretic multiplicity one.When G L is the L -group of a quasi-split connected reductive group G over F , thephilosophy underlying Shotton’s “ ℓ = p Breuil-Mezard conjecture” suggests thatthis “multiplicity-preserving” bijection between irreducible components in charac-teristic zero and characteristic ℓ should correspond, on the representation theoreticside of the local Langlands correspondence, to a lack of congruences between dis-tinct “inertial types” for G . It is well-known that such congruences do not appearwhen the prime ℓ is banal for G ; that is, when ℓ does not divide the pro-order ofany compact open subgroup of G . We therefore call the set of primes ℓ outside S “ G L -banal” primes, and we show that if G is an unramified group over F withno exceptional factors, then the G L -banal primes are precisely the primes that arebanal for G , see Corollary 5.23. On the other hand, for certain exceptional groups G there exist primes that are banal for G but not G L -banal. It would be an in-teresting question (which we do not attempt to address in this paper) to find anexplanation for this discrepancy in terms of the representation theory of G .Finally, we exploit the reducedness of fibers at primes away from S to computethe GIT quotient Z ( W F /P eF , ˆ G ) (cid:12) ˆ G over ¯ Z [ Mp ] for a suitable M , divisible byall G L -banal primes. We refer to Subsection 6.1 for more details on the followingstatement, which is essentially Theorem 6.7. Theorem 1.8.
There is a set of triples ( φ, ˜ β, T φ ) consisting of a cocycle φ ∈ Z ( I F , ˆ G (¯ Z [ pM ])) , an element ˜ β ∈ ˆ G (¯ Z [ pM ]) ⋊ Fr such that ˜ βφ ( i ) β − = φ (Fr i Fr − ) for all i ∈ I F , and an Ad ˜ β -stable maximal torus of C ˆ G ( φ ) ◦ , such that the collectionof embeddings T φ ֒ → C ˆ G ( φ ) induce an isomorphism of ¯ Z [ pM ] -schemes a ( φ,β ) ( T φ ) Ad ˜ β (cid:12) (Ω φ ) Ad ˜ β ∼ −→ ( Z ( W F /P eF , ˆ G ) (cid:12) ˆ G ) ¯ Z [ pM ] . ODULI OF LANGLANDS PARAMETERS 9
When G L is the Langlands dual group of an unramified group, M can be takenas the product of G L -banal primes. In general, a description of the integer M canbe extracted from Proposition 6.2.1.6. Acknowledgements.
The authors are grateful to the organizers of the April2018 conference on “New developments in automorphic forms” at the Instituto deMatematicas Universidad de Sevilla, where many of the ideas behind this paperwere first worked out. We are also grateful to the organizers of the October 2019Oberwolfach workshop “New developments in the representation theory of p -adicgroups” where most of the results of this paper have been announced. We thankJack Shotton, Stefan Patrikis, Sean Howe, Shaun Stevens, and Peter Scholze forhelpful conversations on the subject of the paper. The second author was partiallysupported by EPSRC grant EP/M029719/1, and the fourth author was partiallysupported by NSF grant DMS-200127.2. The space of tame parameters
We begin by considering moduli of tame Langlands parameters for tame groups.Let F be a non-archimedean local field of residue characteristic p , and let I F , P F denote the inertia group and wild inertia group of F , respectively. Let O be thering of integers in a finite extension K of Q , and ˆ G be a split connected reductivealgebraic group over O [ p ], and let ( ˆ B, ˆ T ) be a pair consisting of a Borel subgroupˆ B of ˆ G defined over O [ p ] and a split maximal torus ˆ T of ˆ G contained in ˆ B .We suppose that ˆ G is equipped with an action of W F /P F that preserves thepair ( ˆ B, ˆ T ), and factors through a finite quotient W of W F /P F . Regard W as aconstant group scheme over O [ p ], and let G L denote the semidirect product ˆ G ⋊ W ;we regard G L as a disconnected algebraic group over O [ p ]. Remark 2.1.
In most of our applications, G L will in fact be the L -group of a con-nected, quasi-split reductive F -group G that splits over a tamely ramified extensionof F . However, for a technical reason which will become clear in the next section,it is convenient (and harmless) to have the extra degree of generality allowed bythe above hypothesis.Let Fr denote a lift of arithmetic Frobenius to W F /P F , and let s be a topologicalgenerator of I F /P F . We will regard Fr and s as elements of W . We have Fr s Fr − = s q in W F /P F , where q is the order of the residue field of F .2.1. The scheme Z ( W F /P F , ˆ G ) . Recall that, in the case where G L is the L -group of a connected, quasi-split, reductive F -group G , a tame Langlands param-eter for G is a continuous homomorphism ρ : W F /P F → G L ( Q ℓ ), whose compo-sition with the projection G L ( Q ℓ ) → W is the natural quotient map W F → W .We will often refer to such a homomorphism as an L -homomorphism . Note thatif ρ is a tame Langlands parameter, there is a unique continuous cocycle ρ ◦ in Z ( W F /P F , ˆ G ( Q ℓ )) such that ρ ( w ) = ( ρ ◦ ( w ) , w ); this gives a bijection between thespace of L -homomorphisms and this space of cocycles.Let ( W F /P F ) denote the subgroup of W F /P F generated by the elements Fr and s that we fixed above, regarded as a discrete group. Let W F be the preimage of( W F /P F ) in W F . (Note that both these groups depend heavily on the choices wemade for Fr and s !) For any O -algebra R , the space of L -homomorphisms ( W F /P F ) → G L ( R ) isnaturally in bijection with the space of cocycles Z ( W F /P F , ˆ G ( R )). We will mostoften denote by ϕ a cocycle, and by L ϕ the associated L -homomorphism. The functor that sends R to Z ( W F /P F , ˆ G ( R )) is representable by an affinescheme denoted by Z ( W F /P F , ˆ G ). Concretely, a cocycle ϕ is determined by thetwo elements ϕ (Fr) and ϕ ( s ) of ˆ G ( R ). Conversely, a pair of elements F , σ arisesin this way if, and only if the following identitiy holds in G L ( R )( F , Fr)( σ , s )( F , Fr) − = ( σ , s ) q . We may thus identify Z ( W F /P F , ˆ G ) with the closed subscheme of ˆ G × ˆ G con-sisting of pairs ( F , σ ) ∈ ˆ G × ˆ G such that the above identity holds in G L . Inparticular, Z ( W F /P F , ˆ G ) is affine, with coordinate ring R G L , and we have a “uni-versal pair” ( F , σ ) of elements of ˆ G ( R G L ) satisfying the above identity. The“universal cocycle” ϕ univ on Z ( W F /P F , ˆ G ( R G L )) is then the unique cocycle suchthat ϕ univ (Fr) = F and ϕ univ ( s ) = σ . We will also let F and σ denote the uni-versal elements ( F , Fr) and ( σ , s ) of G L ( R G L ), respectively, so that the universal L -homomorphism L ϕ univ is given by L ϕ univ (Fr) = F and L ϕ univ ( s ) = σ .Given a O [ p ]-algebra R and an R -valued point x of Z ( W F /P F , ˆ G ), we will let F x , σ x , ( F ) x , ( σ ) x ϕ x denote the objects obtained by base change from F , σ , F , σ , and ϕ univ , respectively.Of course, the universal cocycle ϕ univ cannot possibly extend in any nice wayto a cocycle in Z ( W F /P F , ˆ G ( R G L )). However, we will later show that if v is anyfinite place of O of residue characteristic ℓ = p , then ϕ univ extends naturally to acocycle ϕ univ ,v in Z ( W F /P F , ˆ G ( R G L ,v )), where R G L ,v denotes the tensor product R G L ⊗ O O v . In order to prove this, we must first understand the geometry of Z ( W F /P F , ˆ G ).2.2. Geometry of Z ( W F /P F , ˆ G ) . Let L be an algebraically closed field in which p is invertible, and consider the fiber Z ( W F /P F , ˆ G ) L of Z ( W F /P F , ˆ G ) over Spec L .We have a map: π s : Z ( W F /P F , ˆ G ) L → ( G L ) L that takes a cocycle ϕ to L ϕ ( s )or, in other words, a pair ( F , σ ) to σ . Let ξ be a point of G L ( L ) in the imageof this map. We denote by X ξ the scheme-theoretic fiber of this map over ξ ; itis a closed subscheme of Z ( W F /P F , ˆ G ) L . Similarly, denote by X ( ξ ) the locallyclosed subscheme of Z ( W F /P F . ˆ G ) L that is the preimage in Z ( W F /P F , ˆ G ) L ofthe ˆ G L -conjugacy class of ξ in G L ( L ). In particular, Z ( W F /P F , ˆ G )( L ) is the(set-theoretic) union of the X ( ξ ) ( L ), as ξ runs over a set of representatives for theˆ G ( L )-conjugacy classes of G L ( L ) in the image of the map π s .Let ˆ G ξ be the ˆ G -centralizer of ξ . This is a possibly non-reduced group schemeover Spec L that acts on X ξ via g · ( F x , σ x ) = ( F x g, σ x ). Moreover, for any twopoints x = ( F x , σ x ) and y = ( F y , σ y ) of X ξ , we have σ x = σ y = ξ and F x F − y ∈ ˆ G ξ .Thus X ξ is a ˆ G ξ -torsor over Spec L .Now fix an L -point x = ( F x , ξ ) in X ξ . We then obtain a surjective morphism : π x : ˆ G L × ˆ G ξ → X ( ξ ) that sends ( g, g ′ ) to ( g F x g ′ g − , gξg − ). Moreover, we have an action of ˆ G ξ onˆ G L × ˆ G ξ given by g ′′ · ( g, g ′ ) = ( g ( g ′′ ) − , F − x g ′′ F x g ′ ( g ′′ ) − ). This action commutes ODULI OF LANGLANDS PARAMETERS 11 with π x and makes ˆ G ( L ) × ( ˆ G ξ )( L ) into a ( ˆ G ξ )( L )-torsor over X ( ξ ) ( L ). In particular,we deduce that X ( ξ ) is irreducible with smooth reduced subscheme of dimensiondim ˆ G . Lemma 2.2.
Let x be an L -point of Z ( W F /P F , ˆ G ) , and let σ x = σ ux σ ss x be the Jordan decomposition of σ x ; i.e. σ ux is a unipotent element of G L ( L ) and σ ss x is a semisimple element that commutes with σ ux . Then the order of σ ss x is primeto ℓ and divides e ( q fN − , where N is the order of the Weyl group of ˆ G , e is theorder of s in W , and f is the order of Fr in W .Proof. Let e ′ be the prime-to- ℓ part of e . The element ( σ ss x ) e ′ is then a semisimpleelement σ ′ x of ˆ G . The element F x conjugates σ ′ x to its q th power. Thus F fx is anelement of ˆ G that conjugates σ ′ x to its q f th power. Since σ ′ x is semisimple we mayassume (conjugating it and F x appropriately) that it lies in ˆ T ( L ). Thus there is anelement w of the Weyl group of ˆ G that conjugates σ ′ x to its q f th power. Since w N is the identity we have σ ′ x = ( σ ′ x ) q fN and the claim follows. (cid:3) Corollary 2.3.
The image of Z ( W F /P F , ˆ G )( L ) in G L ( L ) under the map π s is aunion of finitely many ˆ G ( L ) -conjugacy classes in G L ( L ) .Proof. Let τ be an L -point in the image of π s , and let τ = τ u τ ss be the Jordandecomposition of τ . Then τ ss is semisimple with bounded order, so lies in oneof finitely many conjugacy classes. Moreover, if we fix τ ss , then τ u lies in thecentralizer G L τ ss of τ ss in G L . By [DM94], Theorem 1.8, since s preserves theBorel pair ( ˆ B, ˆ T ), the connected component of the identity in G L τ ss is reductive.Two elements τ, τ ′ with semisimple part τ ss are ˆ G ( L )-conjugate if, and only if,their unipotent parts τ u , ( τ ′ ) u are ˆ G τ ss ( L )-conjugate. But there are only finitelymany unipotent conjugacy classes in G L τ ss ( L ) (see, for instance [FG12], Lemma2.6, for an elementary proof of this), and therefore only finitely many ˆ G τ ss ( L )-orbitsof unipotent elements of G L τ ss ( L ). The result follows. (cid:3) From this finiteness result we deduce that the scheme Z ( W F /P F , ˆ G ) L is the(set-theoretic) union of the subschemes X ( ξ ) , as ξ runs over a set of representativesfor the ˆ G ( L )-conjugacy classes of G L ( L ) in the image of the map π s . Corollary 2.4.
The scheme Z ( W F /P F , ˆ G ) is flat over O [ p ] of relative dimension dim ˆ G , and is a local complete intersection.Proof. The scheme Z ( W F /P F , ˆ G ) is isomorphic to the fiber over the identity ofthe map: ˆ G × ˆ G → ˆ G given by ( F , σ ) ( F Fr)( σ s )( F Fr) − ( σ s ) − q . In particular its irreduciblecomponents have dimension at least dim ˆ G + 1, and Z ( W F /P F , ˆ G ) is a local com-plete intersection if every irreducible component has dimension exactly dim ˆ G + 1.Suppose we have an irreducible component Y of larger dimension. Then for someprime v of O [ p ], of characteristic ℓ , the fiber of Y over v has dimension greaterthan dim ˆ G . But Z ( W F /P F , ˆ G ) F ℓ is a set-theoretic union of finitely many locallyclosed subschemes of dimension dim ˆ G , so this is impossible. Thus every irreducible component has dimension exactly dim ˆ G + 1, and in particular cannot be containedin the fiber of Z ( W F /P F , ˆ G ) over ℓ for any prime ℓ . By the unmixedness theorem,every associated prime of Z ( W F /P F , ˆ G ) has characteristic zero, so Z ( W F /P F , ˆ G )is flat over O [ p ] as claimed. (cid:3) Lemma 2.2 is a pointwise result about the order of σ ss x , but it can be turned intoa global statement. Indeed, we will say that an R -point of ˆ G is unipotent if thecorresponding map Spec R → ˆ G factors through the unipotent locus on ˆ G . If R isreduced, one can check this pointwise on Spec R . Proposition 2.5.
There exists an integer M , depending only on G L , such that σ M is a unipotent element of ˆ G . When G L = GL n , one can take M = q n ! − .Proof. We first prove this when G L = GL n . In this case Lemma 2.2 shows that ateach Q ℓ -point x of Z ( W F /P F , ˆ G ), the expression ( σ ss x ) q n ! − is equal to the identity.In particular σ q n ! − is an element of R G L [ t ] whose specialization at every point x of Z ( W F /P F , ˆ G ) is unipotent. On the other hand, by [Hel19], Proposition 6.2, when G L = GL n , Z ( W F /P F , ˆ G ) is reduced. Hence σ : Z ( W F /P F , ˆ G ) → ˆ G factorsthrough the unipotent locus of ˆ G as claimed.When G L is arbitrary, the result follows by choosing a faithful representation G L → GL n , and noting that the unipotent locus on G L is the preimage of theunipotent locus on GL n . (cid:3) We will see in the next section that in fact Z ( W F /P F , ˆ G ) is reduced for all G L ;the argument above then shows that in fact σ e ( q Nf − is unipotent.2.3. A construction of Bellovin-Gee.
The scheme Z ( W F /P F , ˆ G ) is very closelyrelated to certain affine schemes studied by Bellovin-Gee in section 2 of [BG19].More precisely, for any finite Galois extension L/F they define a scheme Y L/F,φ, N ([BG19], Definition 2.1.2) parameterizing tuples (Φ , N , τ ) where Φ is an element of G L , N is a nilpotent element of Lie( ˆ G ), and τ : I L/F → G L is a homomorphism,that satisfy:(1) Ad(Φ) N = q N ,(2) For all w ∈ I L/F , Φ τ ( w )Φ − = τ ( w q ), and(3) For all w ∈ I L/F , Ad( τ ( w )) N = N .Let Y ◦ L/F,φ, N denote the closed subscheme of Y L/F,φ, N for which the images ofΦ and τ ( s ) under the map G L → W are Fr and s , respectively. Then Y ◦ L/F,φ, N isa union of connected components of Y L/F,φ, N .We then have: Proposition 2.6.
Fix M such that σ M is unipotent, and let L/F be a finite, tamelyramified Galois extension whose ramification index is divisible by M . Then there isa natural isomorphism Z ( W F /P F , ˆ G ) Q ℓ → ( Y ◦ L/F,φ, N ) Q ℓ .Proof. We give maps in both directions that are inverse to each other. On theone hand, without any hypotheses on
L/F , there is always a map Y ◦ L/F,φ, N → Z ( W F /P F , ˆ G ) over Q ℓ that takes a triple (Φ , N , τ ) to the L -homomorphism L ϕ defined by L ϕ (Fr) = Φ and L ϕ ( s ) = τ ( s ) exp( N ). In the other direction, given acocycle ϕ we can set Φ = L ϕ (Fr), N = M log( L ϕ ( s ) M ), and let τ : I F → G L ( R G L ) ODULI OF LANGLANDS PARAMETERS 13 be the map taking s a to L ϕ ( s ) a exp( − a N ); the latter factors through I L/F under ourramification condition on L . These two maps are clearly inverse to each other. (cid:3) As this isomorphism involves exponentiation, and division by M , it does notextend to the special fiber modulo small primes. In fact the space Y ◦ L/F,φ, N can bequite badly behaved at small primes: for instance, if ˆ G = GL , and we take L/F to be a finite, tamely ramified Galois extension of ramification index M divisibleby q − σ M is unipotent), then at any prime ℓ dividing q + 1 the fiberof Y ◦ L/F,φ, N has dimension five, whereas the generic fiber has dimension four. Thatis, Y ◦ L/F,φ, N fails to be flat in this setting. One could attempt to remedy this byreplacing Y ◦ L/F,φ, N by the closure of its generic fiber, but even then, at primes ℓ asabove, there is not a bijection between the irreducible components of ( Y ◦ L/F,φ, N ) F ℓ and those of Z ( W F /P F , ˆ G ) F ℓ . Indeed, one can verify that the irreducible compo-nents of the latter behave in a manner consistent with the ℓ = p Breuil-Mezardconjecture of Shotton [Sho18], whereas those of the former do not.Bellovin-Gee show ([BG19], Theorem 2.3.6) that Y L/F,φ, N (and hence Z ( W F /P F , ˆ G ))is generically smooth, by constructing a smooth point on each irreducible compo-nent of Y L/F,φ, N in characteristic zero. We sketch their construction here (or rather,its adaptation to Z ( W F /P F , ˆ G )), both in the interests of being self-contained andbecause we will need it for other purposes.Fix a prime ℓ and a Q ℓ point ξ of G L in the image of the map Z ( W F /P F , ˆ G ) → G L taking ϕ to ϕ ( s ). As Z ( W F /P F , ˆ G ) Q ℓ is (set-theoretically) the union ofthe smooth schemes X ( ξ ) for such ξ , it suffices to construct a smooth point of Z ( W F /P F , ˆ G ) on each connected component of X ( ξ ) .Let ξ = ξ ss ξ u be the Jordan decomposition of ξ . Since we are in characteristic,zero ξ u is a unipotent element of ˆ G , and we may consider its logarithm N , whichis a nilpotent element of Lie algebra of the centralizer ˆ G ξ ss of ξ ss .Let λ be a cocharacter of ˆ G ξ ss that is an associated cocharacter of N , in thesense of [BG19], section 2.3. In particular, for all t we have Ad( λ ( t )) N = t N . SetΛ = λ ( q ) for some square root q of q , so that Ad(Λ) N = q N . Then Λ ξ u Λ − = ξ qu .Note that as q is an ℓ -unit for all ℓ = p , we may regard Λ as a Z ℓ -point of ˆ T .Further let H denote the normalizer, in G L , of the subgroup of G L generatedby ξ ss . Let Y be the set of g ∈ H such that gξ u g − = ξ qu . Note that in particularthe map Z ( W F /P F , ˆ G ) → G L that takes ϕ to L ϕ (Fr) identifies X ξ with a unionof connected components of Y .On the other hand Y = Λ · ( H ∩ G L N ). By [Bel16], Proposition 4.9, the inclusionof H ∩ G L N ∩ G L λ into H ∩ G L N is a bijection on connected components, andby [Bel16], Lemma 5.3 there is a point of finite order on each connected componentof H ∩ G L N ∩ G L λ . Thus on each connected component of Y there is a point ofthe form Λ c , where c has finite order and commutes with Λ. Then Bellovin andGee show, via a cohomology calculation, that when (Λ c, ξ ) lies in Z ( W F /P F , ˆ G )it is a smooth point of Z ( W F /P F , ˆ G ) Q ℓ . We immediately deduce: Proposition 2.7.
The scheme Z ( W F /P F , ˆ G ) is generically smooth (and thereforereduced.) Proof.
Generic smoothness is immediate since there is a point of the form (Λ c, ξ )on every connected component of X ( ξ ) for all ξ . Since Z ( W F /P F , ˆ G ) is a localcomplete intersection there is no embedded locus; that is, Z ( W F /P F , ˆ G ) is re-duced. (cid:3) Remark 2.8.
We will later give an argument that in fact the fibers Z ( W F /P F , ˆ G ) F ℓ are generically smooth outside of an explicit finite set. This argument is indepen-dent of (though partially inspired by) the above argument of Bellovin-Gee, andcertainly implies the above proposition, as well as the separatedness results be-low. We include the Bellovin-Gee argument here for convenience of exposition, andbecause the comparison with their construction is interesting in its own right. Proposition 2.9.
For any prime ℓ = p , and any irreducible component Y of Z ( W F /P F , ˆ G ) Q ℓ , there exists a Z ℓ -point of Z ( W F /P F , ˆ G ) on Y .Proof. We have shown that Y contains a point of the form (Λ c, ξ ) constructedabove. We must show that this point is conjugate to a Z ℓ -point. Note that Λ, c and ξ are contained in G L ( L ) for some L ⊂ Q ℓ finite over Q ℓ . Since c has finiteorder and Λ is a Z ℓ -point of ˆ T , the subgroup of G L ( L ) generated by Λ c has compactclosure. Moreover, since Λ c normalizes the subgroup of G L ( L ) generated by ξ , andsome power of ξ is unipotent, the subgroup of G L ( L ) generated by Λ c and ξ hascompact closure. Thus it normalizes a facet of the semisimple building B ( ˆ G, L )and fixes its barycenter x . There is a finite extension L ′ of L such that x becomesan hyperspecial point in B ( ˆ G, L ′ ) and is conjugate to the “canonical” hyperspe-cial point o fixed by ˆ G ( O L ) under some element g ∈ ˆ G ( L ). The fixator of o in G L ( L ) is Z ˆ G ( L ) . G L ( O L ), hence the L -homomorphism L ϕ : W F /P F −→ G L ( Q ℓ )associated to the pair ( g (Λ c ) , g ξ ) takes values in Z ˆ G ( Q ℓ ) . G L ( Z ℓ ). Consider itscomposition with the quotient map to ( Z ˆ G ( Q ℓ ) . G L ( Z ℓ )) / ˆ G ( Z ℓ ) = Q ⋊ W with Q := ( Z ˆ G ( Q ℓ ) . ˆ G ( Z ℓ )) / ˆ G ( Z ℓ ) = Z ˆ G ( Q ℓ ) /Z ˆ G ( Z ℓ ). Since it has relatively compactimage and Q is discrete, it factors over a finite quotient W ′ of W F /P F . But since Q is a Q -vector space of finite dimension, we have H ( W ′ , Q ) = { } , so the above com-position is conjugate, under some element of q ∈ Q , to the trivial L -homomorphism W F −→ Q ⋊ W . So if ˜ q is any lift of q in Z ˆ G ( Q ℓ ), the conjugate ˜ q ( L ϕ ) associatedto the pair ( ˜ qg (Λ c ) , ˜ qg ξ ) is G L ( Z ℓ )-valued, as desired. (cid:3) Corollary 2.10.
For any prime ℓ = p , the ring R G L is ℓ -adically separated.Proof. Since R G L is reduced and flat over O , we have an embedding: R G L → Y Y O Y , where Y runs over the irreducible components of R G L . Each O Y is affine, integral,and flat over O , and by Proposition 2.9 contains an integral point. In particular ℓ is not invertible on O Y . Thus it suffices to show that any integral flat Z ℓ -algebra A in which ℓ is not invertible is ℓ -adically separated. Indeed, suppose a is a nonzeroelement of A in the intersection of the ideals generated by ℓ i . Then for each i ,there is an a i ∈ A such that ℓ i a i = a . Each a i is unique since A is integral, so a i − = ℓa i . Since the ascending chain of ideals generated by the a i stabilizes, wehave a i = ua i − for some unit u and integer i . Then, as A is integral, we have uℓ = 1, contradicting the fact that ℓ is not invertible in A . (cid:3) ODULI OF LANGLANDS PARAMETERS 15
The universal family.
Now that we have shown that R G L is ℓ -adicallyseparated, we return to the question of extending the parameter ϕ univ to an L -homomorphism defined on all of W F . As we have already remarked, this is onlypossible after tensoring with O v for some finite place v of O of residue characteristic ℓ = p . The key point is the following notion of continuity, first introduced in [Hel19]in the case G L = GL n : Definition 2.11.
Let R be a Noetherian O [ p ]-algebra, and let ρ : W F → G L ( R )be a group homomorphism. We say that ρ is ℓ -adically continuous if one of thefollowing two conditions hold:(1) The ring R is ℓ -adically separated, and for each n >
0, the preimage of U n under ρ is open in W F , where U n is the kernel of the map G L ( R ) → G L ( R/ℓ n R ).(2) There exists a Noetherian, ℓ -adically separated O [ p ]-algebra R ′ , a map f : R ′ → R , and an ℓ -adically continuous map ρ ′ : W F → G L ( R ′ ) such that ρ = f ◦ ρ ′ .If R is ℓ -adically separated and condition (2) in the above definition holds, it iseasy to check that condition (1) holds as well, so the two conditions are consistentwith each other. We will say that a cocycle ϕ ∈ Z ( W F , ˆ G ( R )) is ℓ -adically contin-uous if its associated L -homomorphism L ϕ is ℓ -adically continuous as in the abovedefinition. Theorem 2.12.
For each finite place v of O of residue characteristic ℓ = p , thereexists a unique ℓ -adically continuous cocycle ϕ univ ,v : W F /P F → ˆ G ( R G L ⊗ O O v ) whose restriction to ( W F /P F ) is equal to ϕ univ . Moreover, if R is any Noetherian O v -algebra, and ϕ : W F /P F → ˆ G ( R ) is an ℓ -adically continuous cocycle, then thereis a unique map: f : R G L ⊗ O O v → R such that ϕ = ϕ univ ,v ◦ f .Proof. When G L = GL n , this is proved in [Hel19], Proposition 8.2; we reduce tothis case. Choose a faithful representation τ : G L → GL n defined over O [ p ]. Then τ ◦ L ϕ univ ∈ Hom( W F /P F , GL n ( R G L )) = Z ( W F /P F , GL n ( R G L ))where GL n is equipped with the trivial action of W F . There is thus a unique map f : R GL n → R G L that takes the universal cocycle on Z ( W F /P F , GL n ) (actually ahomomorphism) to τ ◦ L ϕ univ . Since this universal cocycle extends to an ℓ -adicallycontinuous cocycle on W F /P F , with values in R GL n ⊗ O O v , composing this extensionwith f gives an extension of τ ◦ L ϕ univ to an ℓ -adically continuous homomorphism W F /P F −→ GL n ( R G L ⊗ O O v ). Denote this homomorphism by L ϕ univ ,v . Its re-striction to W F /P F factors through G L ( R G L ⊗ O O v ) and is equal to L ϕ univ , soit only remains to prove that L ϕ univ ,v factors through G L ( R G L ⊗ O O v ) too. Butthis follows from the ℓ -adic separateness of R G L ⊗ O O v and the fact that for each n ∈ N , we know that the image of L ϕ univ ,v ( W F ) in GL n ( R G L ⊗ O O v / ( ℓ n )) coin-cides with the image of L ϕ univ ,v ( W F ), which is contained in G L ( R G L ⊗ O O v / ( ℓ n )).Uniqueness and the universal property are now straightforward. (cid:3) In light of this, we define a “good coefficient ring” to be a ring R that is an O⊗ Z ℓ -algebra for some ℓ = p , and a “good coefficient field” to be a good coefficient ring that is also a field. Theorem 2.12 then implies that for any good coefficient ring R ,and any cocycle ϕ : ( W F /P F ) → G L ( R ), there is a unique ℓ -adically continuouscocycle ϕ : W F /P F → G L ( R ) extending ϕ .In particular, if R is a complete local O -algebra with maximal ideal m , of residuecharacteristic ℓ , then any ℓ -adically continuous cocycle ϕ : W F /P F → G L ( R ) isclearly m -adically continuous. Conversely, given an m -adically continous cocycle ϕ : W F /P F → G L ( R ), Theorem 2.12 shows that there is a unique ℓ -adically continuouscocycle ϕ ′ extending the restriction of ϕ to ( W F /P F ) . Then ϕ ′ and ϕ are both m -adically continuous and agree on ( W F /P F ) , so ϕ is also ℓ -adically continuous.Thus the notions of ℓ -adic and m -adic continuity coincide for cocycles valued in R .3. Reduction to tame parameters
In this section, we broaden the setting as follows. We still consider a splitreductive group scheme ˆ G over Z [ p ] endowed with a finite action of W F , but we nolonger assume that this action is tame, nor that it stabilizes a Borel pair. For any Z [ p ]-algebra R , we denote by Z ( W F , ˆ G ( R )) the set of 1-cocycles whichare continuous for the natural topology of the source and the discrete topology onthe target. We use similar notation for W F and any closed subgroup thereof.It will be handy to switch between 1-cocycles and their associated L -morphisms.In this regard, we usually denote by G L a group scheme of the form ˆ G ⋊ W with W any finite quotient of W F through which the given action on ˆ G factors. Notethat W may be allowed to change according to our needs, but we prefer to keepit finite in order to work with algebraic group schemes. For the sake of clarity,we will most often distinguish a 1-cocycle ϕ from its associated L -homomorphism ϕ L := ϕ ⋊ id : W F −→ G L ( R ), although occasionally it will be more handy towrite ϕ for the L -homomorphism.3.1. Overview.
Our aim is to show how the study of moduli of 1-cocycles W F −→ ˆ G (and subsequently, moduli of ℓ -adically continuous 1-cocycles W F −→ ˆ G ) can bereduced to the particular case considered in the previous section, namely the case oftame 1-cocycles valued in a reductive group scheme with a tame Galois action thatstabilizes a Borel pair. The principle is very simple ; suppose R is a Z [ p ]-algebraand ϕ : W F −→ ˆ G ( R ) is a 1-cocycle, and denote by φ : P F −→ ˆ G ( R ) its restrictionto P F . Then the conjugation action of W F on ˆ G ( R ) through L ϕ stabilizes thecentralizer C ˆ G ( R ) ( L φ ( P F )) and the restricted action on this subgroup factors over W F /P F . Denoting by Ad ϕ this action, an elementary computation shows that themap η η · ϕ sets up a bijection Z ϕ ( W F /P F , C ˆ G ( R ) ( L φ ( P F ))) ∼ −→ { ϕ ′ ∈ Z ( W F , ˆ G ( R )) , ϕ ′| P F = φ } . By Lemma A.1 in the appendix, the functor on R -algebras R ′ C ˆ G ( R ′ ) ( L φ ( P F ))is representable by a smooth group scheme that we denote by C ˆ G ( φ ). Moreover,by [PY02, Thm 2.1], its connected geometric fibers are reductive. Therefore, one istempted to see the set Z ϕ ( W F /P F , C ˆ G ( R ) ( L φ ( P F ))) as an instance of the type oftame parameters that were studied in the previous section. However, making thisidea work requires addressing the following issues : • The group scheme C ˆ G ( φ ) may have non-connected fibers. • Its neutral component C ˆ G ( φ ) ◦ may not be split. ODULI OF LANGLANDS PARAMETERS 17 • The action Ad ϕ may neither be finite nor preserve a Borel pair of C ˆ G ( φ ) ◦ .In order to address these issues, the first step is to find a nice set of representativesof conjugacy classes of continuous cocycles with source P F . Since we prefer to workwith finitely presented objects, we choose a decreasing sequence ( P eF ) e ∈ N of opennormal subgroups of P F whose intersection is { } . Then we fix e ∈ N such that P eF acts trivially on ˆ G , and we restrict attention to cocycles that are trivial on P eF .The following theorem follows from Theorems A.9, A.12 and Proposition A.13 inthe appendix. Theorem 3.1.
There is a number field K e and a finite set Φ e ⊂ Z (cid:16) P F /P eF , ˆ G ( O K e [1 /p ]) (cid:17) , such that (1) For any O K e [ p ] -algebra R , any cocycle φ : P F /P eF −→ ˆ G ( R ) is ´etale-locally ˆ G -conjugate to a locally unique φ ∈ Φ e . (2) For any φ ∈ Φ e , the reductive group scheme C ˆ G ( φ ) ◦ is split over O K e [ p ] and the component group π ( φ ) := π ( C ˆ G ( φ )) is constant. Some definitions and constructions.
Let φ ∈ Φ e . For any O K e [ p ]-algebra R we denote by Z ( W F , ˆ G ( R )) φ the set of 1-cocycles W F −→ ˆ G ( R ) that extend φ . The functor R Z ( W F , ˆ G ( R )) φ is visibly representable by an affine schemeof finite type over O K e [ p ], namely a closed subscheme of ˆ G × ˆ G . We denote thisscheme by Z ( W F , ˆ G ) φ . Definition 3.2.
An element φ ∈ Φ e is called admissible if the scheme Z ( W F , ˆ G ) φ is not empty.In the sequel, it will be convenient to choose our “ L -group” G L in the form G L =ˆ G ⋊ W e where W e is a finite quotient of W F into which P F /P eF maps injectively .For example, we may choose our sequence ( P eF ) e such that P eF = P F e for someGalois extension F e of F and put W e = Gal( F e /F ). Then the L -homomorphism ϕ L associated to ϕ ∈ Z ( W F , ˆ G ( R )) φ factors through the subgroup C G L ( R ) ( φ ) := (cid:8) ( g, w ) ∈ G L ( R ) , ( g, w ) L φ ( w − pw )( g, w ) − = L φ ( p ) , ∀ p ∈ P F (cid:9) . Writing the functor C G L ( φ ) : R C G L ( R ) ( φ ) on O K e [ p ]-algebras as a disjointunion F w ∈ W e T ˆ G ( φ, w φ ) of transporters in ˆ G , we see from Lemma A.1 that thisfunctor is represented by a smooth group scheme that sits in an exact sequence1 → C ˆ G ( φ ) → C G L ( φ ) → W e . Actually, it follows from the uniqueness of φ in i) of Theorem 3.1 that T ˆ G ( φ, w φ )is either empty or is a C ˆ G ( φ )-torsor for the ´etale topology. Therefore, C G L ( φ ) isan extension of the constant subgroup W e,φ := { w ∈ W e , T ˆ G ( φ, w φ ) = ∅} of W e by C ˆ G ( φ ). Since C G L ( φ ) ◦ = C ˆ G ( φ ) ◦ is a split reductive group scheme over O K e [ p ], weknow by general results [Con14, Prop. 3.1.3] that˜ π ( φ ) := π ( C G L ( φ ))is a separated ´etale group scheme over O K e [ p ]. Since it is an extension of W e,φ by π ( φ ), we see that ˜ π ( φ ) is actually finite ´etale. Therefore, after maybe enlarging K e , we may assume that ˜ π ( φ ) is constant over O K e [ p ] . Now, let us assume that φ is admissible. Then we have W e,φ = W e and an exact sequence of abstract groups1 → π ( φ ) → ˜ π ( φ ) → W e → . Therefore, the affine scheme Z ( W F , ˆ G ) φ decomposes as a disjoint union Z ( W F , ˆ G ) φ = G α ∈ Σ( φ ) Z ( W F , ˆ G ) φ,α , where • Σ( φ ) denotes the set of cross-sections W F −→ ˜ π ( φ ) that extend the map P F −→ ˜ π ( φ ) given by the composition of L φ with the projection to ˜ π ( φ ). • Z ( W F , ˆ G ) φ,α ( R ) = Z ( W F , ˆ G ( R )) φ,α is the subset of extensions ϕ of φ such that the composition of ϕ L with the projection to ˜ π ( φ ) is α . Definition 3.3.
We will say that α ∈ Σ( φ ) is admissible if the scheme Z ( W F , ˆ G ) φ,α is not empty.We now note that two elements ϕ, ϕ ′ ∈ Z ( W F , ˆ G ( R )) φ,α differ by a tame cocyclevalued in C ˆ G ( φ ) ◦ (beware the ◦ ). More precisely, if we write ϕ ′ ( w ) = η ( w ) ϕ ( w ),then w η ( w ) belongs to Z ϕ ( W F /P F , C ˆ G ( φ ) ◦ ( R )). In other words, the map η η · ϕ sets up an isomorphism of R -schemes Z ϕ ( W F /P F , C ˆ G ( φ ) ◦ ) R ∼ −→ Z ( W F , ˆ G ) φ,α,R . At this point we have dealt with the first two issues mentioned in the beginning ofthis section. The next result deals with the third issue and will allow us to reduceto the tame parameters that were studied in the previous section.
Theorem 3.4.
There is a finite extension K ′ e of K e such that for any admissible φ ∈ Φ e and any admissible α ∈ Σ( φ ) , there is some ϕ α ∈ Z ( W F , ˆ G ( O K ′ e [ p ])) φ,α such that ϕ L α ( W F ) is finite and Ad ϕ α preserves a Borel pair of the split reductivegroup scheme C ˆ G ( φ ) ◦ . Fix φ, α and ϕ α as in the theorem. Since ϕ L α ( W F ) is finite, ϕ α extends canon-ically to W F with ϕ L α ( W F ) = ϕ L α ( W F ). So the conjugation action Ad ϕ α of W F on the reductive group C ˆ G ( φ ) ◦ extends to a finite action of W F , and it has to betrivial on P F . Since this action stabilizes a Borel pair, we see that the O K ′ e [ p ]-scheme Z ϕ (( W F /P F ) , C ˆ G ( φ ) ◦ ) is (a base change of) an instance of those tamemoduli schemes studied in Section 2. Remark 3.5.
It is natural to ask whether we can find ϕ α so that Ad ϕ α preservesa pinning of C ˆ G ( φ ) ◦ . Our techniques can achieve this when the center of C ˆ G ( φ ) ◦ issmooth, see Remark 3.9. In Theorem 3.12, we give a sufficient condition on ˆ G foreach C ˆ G ( φ ) ◦ to have smooth center.Before we can prove the theorem, we need some preparation. Let us fix a Borelpair B φ = ( B φ , T φ ) in C ˆ G ( φ ) ◦ and let us denote by T φ the normalizer in C G L ( φ )of this Borel pair. By [Con14, Prop. 2.1.2], this is again a smooth group schemeover O K e [ p ]. Since the normalizer of a Borel pair in a connected reductive groupover an algebraically closed field is the torus of the Borel pair, we have ( T φ ) ◦ = C ˆ G ( φ ) ◦ ∩T φ = T φ . Since any two Borel pairs in a connected reductive group over analgebraically closed field are conjugate, we also have π ( T φ ) = π ( C G L ( φ )) = ˜ π ( φ ). ODULI OF LANGLANDS PARAMETERS 19
Moreover, since T φ is abelian, the conjugation action of T φ on T φ factors throughan action ˜ π ( φ ) −→ Aut O Ke [ p ] − gp.sch. ( T φ ) . In particular, any section α ∈ Σ( φ ) provides us with an action of W F /P F on thetorus T φ . Therefore, the subsetΣ( W F , T φ ( R )) φ := { ϕ ∈ Z ( W F , ˆ G ( R )) φ , ϕ L ( W F ) ⊂ T φ ( R )) } = { ϕ ∈ Z ( W F , ˆ G ( R )) φ , Ad ϕ preserves B φ } decomposes as a disjoint unionΣ( W F , T φ ( R )) φ = G α ∈ Σ( φ ) Σ( W F , T φ ( R )) φ,α where each Σ( W F , T φ ( R )) φ,α is either empty or is a principal homogeneous setunder the abelian group Z α (( W F /P F ) , T φ ( R )). Varying R , we get a closed affinesubscheme Σ( W F , T φ ) φ of Z ( W F , ˆ G ) φ which decomposes as a coproduct of affine O K e [ p ]-schemes Σ( W F , T φ ) φ = G α ∈ Σ( φ ) Σ( W F , T φ ) φ,α where each Σ( W F , T φ ) φ,α carries an action of the abelian group O K e [ p ]-scheme Z α (( W F /P F ) , T φ ), and is a pseudo-torsor for this action, in the sense of [TheStacks Project, Tag 0497]. Theorem 3.6.
Suppose φ and α are admissible. (1) Z α (( W F /P F ) , T φ ) is a diagonalisable group scheme over O K e [ p ] . (2) Σ( W F , T φ ) φ,α is a fppf torsor under Z α (( W F /P F ) , T φ ) .Moreover, these two statements still hold with W F replaced by a suitable finitequotient. Before we prove this result, let us see how it implies Theorem 3.4. To this aim,it suffices to show that any fppf torsor under a diagonalisable group over O K e [ p ]becomes trivial over O K ′ e [ p ] for some finite extension K ′ e . Since a diagonalisablegroup is a product of copies of G m and µ m ’s, we may treat each of these groupsseparately. As long as G m is concerned, since any fppf G m -torsor is also an ´etale G m -torsor, it suffices to take K ′ e equal to the Hilbert class field K he of K e . On theother hand, when base changed to O K he [ p ], a µ m -torsor is given as the torsor of m -th roots of some element f ∈ O K he [ p ] × , because of the exact sequence O K e [1 /p ] × ( . ) m −→ O K e [1 /p ] × −→ H fppf ( S, µ m ) −→ H fppf ( S, G m ) ( . ) m −→ H fppf ( S, G m )where S denotes Spec( O K e [ p ]). Thus we can take for K ′ e a splitting field of X m − f over K he in this case. Proof. (1) Consider the map Z α (( W F /P F ) , T φ ( R )) −→ T φ ( R ) × T φ ( R ) that sends a1-cocycle η to the pair of elements ( η (Fr) , η ( σ )). It identifies Z α (( W F /P F ) , T φ ( R ))with the subset of elements ( F, s ) in T φ ( R ) × T φ ( R ) defined by the equation F · α (Fr)( s ) · α ( σ ) q ( F ) − = s · α ( σ )( s ) · · · α ( σ q − )( s ) . This identifies in turn Z α (( W F /P F ) , T φ ) with the kernel of the morphism of groupschemes T φ × T φ −→ T φ defined by the ratio of both sides of the equation. But akernel of a morphism of diagonalisable groups is diagonalisable.(2) We already know that Σ( W F , T φ ) φ,α is finitely presented over O K e [ p ], so itremains to find a faithfully flat O K e [ p ]-algebra R such that Σ( W F , T φ ( R )) φ,α is notempty. Existence of a point over a closed geometric point.
By the admissibility assump-tion, the scheme Z ( W F , ˆ G ) φ,α is not empty. Since it has finite presentation over O K e [ p ], Chevalley’s constructibility theorem ensures that it has a non-empty closedfiber, which in turn ensures that it has a point with finite residue field k of char-acteristic = p . Note that the associated L -morphism W F −→ G L ( k ) has to factorover a finite quotient of W F , hence it is continuous for the topology of W F inducedby the usual topology on W F , and the discrete topology on G L ( k ). Therefore,Proposition 3.7 below ensures that Σ( W F , T φ (¯ k )) φ,α is not empty. Pick a point inthis set and let ¯ ϕ : W F −→ T φ (¯ k ) be the L -morphism corresponding to this point(here and in the remaining of this proof, we denote 1-cocycles and their associated L -morphisms by the same symbol in order to lighten notation a bit). Note that¯ ϕ also has to factor through a finite quotient of W F , so it extends uniquely to acontinuous morphism from W F . Lifting this point to characteristic . Let us try to lift ¯ ϕ to a Witt-vectors valuedpoint ϕ : W F −→ T φ ( W (¯ k )). By smoothness of T φ , the map T φ ( W (¯ k )) −→ T φ (¯ k )is surjective, so we may choose lifts ˜ ϕ ( w ) ∈ T φ ( W (¯ k )) of ¯ ϕ ( w ) and we may do it insuch a way that • ˜ ϕ ( w ) only depends on ¯ ϕ ( w ) and ˜ ϕ ( w ) = 1 if ¯ ϕ ( w ) = 1. • ˜ ϕ ( pw ) = φ ( p ) ˜ ϕ ( w ) for all w ∈ W F and p ∈ P F .Note that ˜ ϕ ( w ) belongs to the transporter T ˆ G ( φ, w φ ), so that we also have ˜ ϕ ( wp ) = φ ( wpw − ) ˜ ϕ ( w ) = ˜ ϕ ( w ) φ ( p ) for all w ∈ W F and p ∈ P F . Moreover, the automor-phism (Ad ˜ ϕ ( w ) ) | T φ only depends on the image of ˜ ϕ ( w ) in π ( T φ ), which is the sameas that of ¯ ϕ ( w ). Hence this automorphism is the one given by the action of α ( w ).It follows that the map c : ( w, w ′ ) ∈ W F × W F ˜ ϕ ( w ) ˜ ϕ ( w ′ ) ˜ ϕ ( ww ′ ) − ∈ ker (cid:0) T φ ( W (¯ k )) −→ T φ (¯ k ) (cid:1) has finite image, factors over W F /P F × W F /P F , and is a 2-cocycle from W F /P F into A := ker( T φ ( W (¯ k )) −→ T φ (¯ k )) endowed with the action α . If this cocycleis cohomologically trivial, that is, if there is some continuous map t : W F /P F → A such that c ( w, w ′ ) = t ( w )( ˜ ϕ ( w ) t ( w ′ )) t ( ww ′ ) − , then the map w ϕ ( w ) := t ( w ) − ˜ ϕ ( w ) is a continuous lift of ¯ ϕ . Now, if ℓ denotes the characteristic of ¯ k , thegroup A is certainly ℓ ′ -divisible, but not ℓ -divisible, so that H ( W F /I F , A ) is apriori not trivial. However, if ¯ O denotes the ring of integers of an algebraic closureof W (¯ k ), then the group A ′ = ker( T φ ( ¯ O ) −→ T φ (¯ k )) is divisible hence, by Lemma3.8, c is cohomologically trivial there, and we get a lift ϕ of ¯ ϕ valued in G L ( ¯ O ).We now modify this lift ϕ so that it has finite image. To do so we introduce themaximal subtorus C φ of T φ on which W F /P F acts trivially. This is the split torusover W (¯ k ) whose group of characters is the torsion-free quotient of the W F /P F -coinvariants of the group of characters of T φ . Now, pick an integer m such that¯ ϕ (Fr m ) = 1 and ϕ (Fr m ) is central in ϕ ( W F ) (this is possible since ϕ ( I F ) is finite).The element ϕ (Fr m ) ∈ A ′ then belongs to T φ ( ¯ O ) W F /P F . Since the group scheme ODULI OF LANGLANDS PARAMETERS 21 T W F /P F φ has neutral component C φ , a power of ϕ (Fr m ), say ϕ (Fr m ′ ), belongs to C φ ( ¯ O ) ∩ ker( T φ ( ¯ O ) → T φ (¯ k )) = ker( C φ ( ¯ O ) → C φ (¯ k )). But the latter is a divisiblegroup so we may pick there an element c such that c m ′ = ϕ (Fr m ′ ). Consider then ϕ ′ : w c − ν ( w ) ϕ ( w ). This is still a G L ( ¯ O )-valued lift of ¯ ϕ and it has finite image. A section over a quasi-finite flat extension.
Now, the existence of such a liftshows that the morphism of finite presentation Σ( W F , T φ ) φ,α −→ Spec( O K e [ p ])is dominant and, even better, that there is a finite quotient W of W F such thatΣ( W, T φ ) φ,α −→ Spec( O K e [ p ]) is dominant (with obvious notation). Therefore, wecan find a finite extension K of K e and an integer N such that Σ( W F , T φ ) φ,α has asection over O K [ N ] that corresponds to a morphism ϕ : W F −→ T φ ( O K [ N ]) whichfactors over a finite quotient of W F . Sections over the missing points.
Let us fix a prime λ of K that divides N butnot p , and denote by K λ the completion of K at λ and by O λ its ring of integers.Using the inclusion O K [ N ] ֒ → K λ we get a morphism ϕ : W F −→ T φ ( K λ ). Wewould like to conjugate it, so that it factors though T φ ( O λ ). We will show that thisis possible after maybe passing to a ramified extension of K λ . Indeed, the problemis to find some t ∈ T φ ( K λ ) such that tϕ ( w ) t − ∈ T φ ( O λ ) for all w ∈ W F . Observethat T φ ( K λ ) = T φ ( K λ ) T φ ( O λ ), so that T φ ( O λ ) is a normal subgroup of T φ ( K λ )with quotient of the form T φ ( K λ ) /T φ ( O λ ) = ( T φ ( K λ ) /T φ ( O λ )) ⋊ ˜ π ( φ ) . So we see that the existence of t as above is equivalent to the existence of ¯ t ∈ T φ ( K λ ) /T φ ( O λ ) such that ¯ tϕ ¯ t − coincides with the trivial section W F α −→ ˜ π ( φ ) −→T φ ( K λ ) /T φ ( O λ ) (we have denoted by ϕ the composition of ϕ with the projectionto the above quotient). Therefore, the existence of t as above is equivalent to thevanishing of ϕ in H ( W ′ , T φ ( K λ ) /T φ ( O λ )), where W ′ is any finite quotient of W F through which ϕ (hence also α ) factors.Now, let v λ be the normalized valuation on K λ and let X ∗ ( T φ ) be the group ofcocharacters of T φ . The pairing ( t, µ ) v λ ( µ ( t )) for t ∈ T φ ( K λ ) and µ ∈ X ∗ ( T φ )induces an isomorphism of abelian groups T φ ( K λ ) /T φ ( O λ ) ∼ −→ Hom( X ∗ ( T φ ) , Z )which shows that T φ ( K λ ) /T φ ( O λ ) is a free abelian group of rank dim( T φ ) and that H ( W ′ , T φ ( K λ ) /T φ ( O λ )) has no reason to vanish. However, let ¯ K λ be an algebraicclosure of K λ with ring of integers ¯ O λ and denote by v λ the unique extension of v λ to ¯ K λ . Then the same pairing as above induces an isomorphism T φ ( ¯ K λ ) /T φ ( ¯ O λ ) ∼ −→ Hom( X ∗ ( T φ ) , Q )which shows that T φ ( ¯ K λ ) /T φ ( ¯ O λ ) is a Q -vector space, and therefore that the group H ( W ′ , T φ ( ¯ K λ ) /T φ ( ¯ O λ )) vanishes. It follows that there is some finite extension K ′ λ of K λ with ring of integers O ′ λ , and some element t ′ ∈ T φ ( K ′ λ ) such that ϕ λ := t · ϕ ( w ) · t − defines a section of Σ( W ◦ F , T φ ) φ,α over O ′ λ . Conclusion.
With ϕ and the ϕ λ , we have found a section over the finite fpqccovering S λ | N Spec( O ′ λ ) ∪ Spec( O K [ N ]) of Spec( O K e [ p ]). Since Σ( W F , T φ ) φ,α isfinitely presented, there also exists a section over a fppf covering. (cid:3) In the above proof, we have used the following result in order to pass from thenon-emptyness of Z ( W F , ˆ G ) φ,α to that of Σ( W F , T φ ) φ,α . Proposition 3.7.
Let K be an algebraically closed field of characteristic differentfrom p , let ϕ : W F −→ L G ( K ) be a continuous L -morphism, and let φ := ϕ | P F .Then there is another extension ϕ ′ = η · ϕ of φ , with η ∈ Z ϕ ( W F /P F , C ˆ G ( φ ) ◦ ) ,and whose conjugation action Ad ϕ ′ on C ˆ G ( φ ) preserves a Borel pair of C ˆ G ( φ ) ◦ .Proof. Fix a Borel pair B φ of C ˆ G ( φ ) ◦ . Since C ˆ G ( φ ) ◦ acts transitively on its Borelpairs, we may choose for all ¯ w ∈ W F /P F an element α ( ¯ w ) ∈ C ˆ G ( φ ) ◦ ( K ) suchthat Ad α ( ¯ w ) ◦ Ad ϕ ( w ) stabilizes B φ . Moreover, we may and will choose α ( ¯ w ) sothat it only depends on Ad ϕ ( w ) , ensuring in turn that the map ¯ w α ( ¯ w ) iscontinuous. Since the stabilizer of B φ in C ˆ G ( φ ) ◦ is T φ , we see that the automorphismAd α ( ¯ w ) ◦ Ad ϕ ( w ) of T φ does not depend on the choice of α ( ¯ w ), and this definesan action of W F /P F on T φ by algebraic automorphisms. Note that this actionis the same as the one given by the image of Ad ϕ ( w ) in Out( C ˆ G ( φ ) ◦ ) throughthe canonical identification of T φ with the “abstract” torus of the root datum of C ˆ G ( φ ) ◦ . In particular, this action is finite since it factors through the quotient ofthe normalizer of φ ( P F ) in L G by C ˆ G ( φ ) ◦ , which is a finite group. Now we remarkthat the map( ¯ w, ¯ w ′ ) α ( ¯ w ) ϕ ( w ) α ( ¯ w ′ ) ϕ ( w ′ )( α ( ¯ w ¯ w ′ ) ϕ ( ww ′ )) − = α ( ¯ w ) Ad ϕ ( w ) ( α ( w ′ )) α ( ww ′ ) − defines a continuous 2-cocycle from W F /P F to T φ ( K ) with respect to the actiondescribed above. If this cocycle is a coboundary, that is, if there is a continuousmap β : W F /P F −→ T φ ( K ) such that α ( ¯ w ) Ad ϕ ( w ) ( α ( w ′ )) α ( ww ′ ) − = β ( ¯ w )(Ad α ( ¯ w ) ◦ Ad ϕ ( w ) ( β ( ¯ w ′ ))) β ( ¯ w ¯ w ′ ) − , then the map η : ¯ w β − ( ¯ w ) α ( ¯ w ) is in Z ϕ ( W F /P F , C ˆ G ( φ ) ◦ ( K )) and the pa-rameter ϕ ′ = η · ϕ normalizes the Borel pair B φ as desired.Hence the obstruction to finding η as desired lies in H ( W F /P F , T φ ( K )). How-ever, since T φ ( K ) is a divisible group, the following lemma shows that this coho-mology group vanishes. (cid:3) Lemma 3.8.
Denote W = W F /P F and I = I F /P F , and let A be an abelian groupwith a finite action of W . We consider only continuous cohomology of W withrespect to the discrete topology on the coefficients. (1) There is a short exact sequence → H ( W/I, H ( I, A )) → H ( W, A ) → H ( I, A ) W/I → . (2) We have H ( I, A ) = colim ( n,p )=1 ( A I /N M ( A ) n ) where • M is the order of the action of a pro-generator σ of I and N M ( a ) = aσ ( a ) · · · σ M − ( a ) • { n ∈ N , ( n, p ) = 1 } is ordered by divisibility and the transition map A I /N M ( A ) n → A I /N M ( A ) n ′ for n | n ′ is induced by the map a a n ′ /n .In particular, H ( I, A ) = { } whenever A contains a p ′ -divisible group offinite index. (3) We have H ( W/I, H ( I, A )) = H ( I, A ) Fr = [ N − M ( A [ p ′ ]) /A ( σ )] Fr where • A [ p ′ ] is the prime-to- p torsion of A and A ( σ ) = { aσ ( a ) − , a ∈ A } . • Fr is a Frobenius lift in W . Moreover, if m is the order of the actionof Fr on A , then Fr m acts on H ( I, A ) by raising to the power q m .In particular, H ( W/I, H ( I, A )) = { } whenever A is a p ′ -divisible group. ODULI OF LANGLANDS PARAMETERS 23
Proof. (1) follows from the Hochschild-Serre spectral sequence with the facts that
W/I = Z and H n ( Z , M ) = 1 for any n ≥ Z [ Z ]-module M .(2) By identifying I with the inverse limit of Z /nM Z for ( n, p ) = 1, we canwrite H ( I, A ) as the direct limit of H ( Z /nM Z , A ), indexed by integers n coprimeto p and ordered by divisibility. The standard formula for the H of a cyclicgroup tells us that H ( Z /nM Z , A ) = A σ /N M ( A ) n and that the transition map A σ /N M ( A ) n → A σ /N M ( A ) n ′ for n | n ′ is induced by the map a a n ′ /n . Now,suppose B is a p ′ -divisible subgroup of A such that ( A/B ) N = 1 for some integer N >
1. Then N M ( B ) is a p ′ -divisible subgroup of A σ hence it is contained in N M ( A ) n for all n > n, p ) = 1. Since N M ( A ) contains ( A σ ) M , we seethat each A σ /N M ( A ) n has exponent dividing N M and prime to p . It follows that,denoting by N ′ the prime-to- p part of N , the transition maps vanish whenever nN ′ M | n ′ , showing that the colimit vanishes, whence H ( I, A ) = 1.(3) By the continuity constraint on cocycles, the map Z ( I, A ) → A , η η ( σ )identifies Z ( I, A ) with the subgroup { a ∈ A, ∃ n ∈ N , ( n, p ) = 1 , N n ( a ) = 1 } .Since N nM ( a ) = N n ( N M ( a )) = N M ( a ) n , this is also the subgroup { a ∈ A, ∃ n ∈ N , ( n, p ) = 1 , N M ( a ) n = 1 } . In other words, with the notation of the lemma wehave Z ( I, A ) = N − M ( A [ p ′ ]) . As a consequence H ( I, A ), being by definition thequotient of Z ( I, A ) by σ -conjugacy under A , is also the quotient by the subgroup A ( σ ), and the formula of (3) follows.Let us make the action of Fr on H ( I, A ) more explicit. Note first that theaction of Fr on Z ( I, A ) is given by Fr( η )( σ ) = Fr( η (Fr − σ Fr)) = Fr( η ( σ q )) =Fr( N q ( η ( σ ))). If m is such that Fr m acts trivially on A , then Fr m ( η )( σ ) = N q m ( η ( σ )).But since the image N q m ( η ( σ )) of N q m ( η ( σ )) in Z ( I, A ) /A ( σ ) is η ( σ ) q m , we seethat the action of Fr m on H ( I, A ) is simply given by the q m -th power map. There-fore, the space of Fr m co-invariants is the quotient H ( I, A ) Fr m = H ( I, A ) / (cid:0) H ( I, A ) (cid:1) q m − , which is trivial whenever H ( I, A ) is p ′ -divisible. The latter holds if Z ( I, A ) is p ′ -divisible, and this holds in turn if A is p ′ -divisible. (cid:3) Remark 3.9.
This lemma is the main point in proving the existence of L -morphismsthat preserve a Borel pair. When the center Z φ of C ˆ G ( φ ) ◦ is a torus, and more gen-erally when it is smooth over O K e [ p ], then Z φ ( K ) is a p ′ -divisible group for any alge-braically closed field K of characteristic not p , so that the same lemma implies that H ( W F /P F , Z φ ( K )) vanishes. In this case, fix a pinning ε φ of C ˆ G ( φ ) ◦ and considerits normalizer Z φ in C G L ( φ ), which is an extension of π ( G L ) by Z φ . Thanks to thisvanishing result, the same argument as in Theorem 3.6 shows that Σ( W ◦ F , Z φ ) φ,α is a fppf torsor under the diagonalisable group scheme Z α (( W F /P F ) , Z φ ), andtherefore that we can find ϕ α as in Theorem 3.4 with the additional property that Ad ϕ α preserves the pinning ε φ . In this case, the group scheme C ˆ G ( φ ) ◦ · ϕ ( W F ) isisomorphic to a suitable quotient of the Langlands dual group scheme over O K ′ e [ p ]of some tamely ramified reductive group over F , namely “the” quasi-split reductivegroup G φ,α dual to C ˆ G ( φ ) ◦ over ¯ F and whose F -structure is induced by the outeraction W F α −→ ˜ π ( φ ) −→ Out( C ˆ G ( φ ) ◦ ) . In particular, when C ˆ G ( φ ) is connected, Σ( φ ) is trivial so we get a single associatedquasi-split reductive group G φ over F and, under the hypothesis of this Remark, we have an isomorphism over O K ′ e [ p ] G L φ = C ˆ G ( φ ) ⋊ Ad ϕ W e ∼ −→ C G L ( φ ) . Example 3.10 (Classical groups) . Let us assume that p > G L is a Langlands dual group of a quasi-split classical group G over F , sothat ˆ G is one of Sp n , SO n +1 or SO n . Then the following holds :(1) C ˆ G ( φ ) is connected for all φ ∈ Φ e . More precisely, it is isomorphic to aproduct ˆ G ′ × GL n × · · · × GL n r with ˆ G ′ of the same type as ˆ G . This follows from the fact that the only self-dual irreducible represen-tation of a p -group is the trivial representation. Indeed, decomposing theunderlying symplectic or orthogonal space as a sum of φ ( P F )-isotypic com-ponents, this fact shows that each pair of dual non-trivial irreducible rep-resentations contributes a factor GL to the centralizer, while the trivialrepresentation contributes a classical group of the same sign. We thendeduce the following :(2) G φ is a (possibly non split) classical group times a product of restrictionsof scalars of general linear groups and unitary groups. (3) If G is symplectic, then we can find an extension ϕ of φ such that Ad ϕ preserves a pinning of C ˆ G ( φ ) . In particular we get an isomophism G L φ ∼ −→ C G L ( φ ) as above. Recall that even though G is split here, we take G L = G × W e where P F /P eF injects into W e .The next lemma provides many examples to which Remark 3.9 applies. Lemma 3.11.
Let H be a reductive group scheme over ¯ Z [ p ] and let P be a finite p -group of automorphisms of H . If the center Z ( H ) of H is smooth over ¯ Z [ p ] , thenso is the center Z ( H P, ◦ ) of the connected centralizer H P, ◦ of P Proof.
Recall that the center Z of a reductive group scheme is a group of multiplica-tive type, associated to an ´etale sheaf X ( Z ) of finitely generated abelian groups.In particular, Z is flat over the base, and it is smooth if and only if the order ofthe torsion subgroups of all stalks of X ( Z ) are invertible on the base. In our case,since Spec(¯ Z [ p ]) is connected, it suffices to check the ¯ Q -stalk. Hence we see that Z is smooth if and only if the torsion subgroup of X ( Z ¯ Q ) has p -power order, if andonly if π ( Z ¯ Q ) has p -power order.As a consequence, we are reduced to prove a statement for reductive groups over¯ Q : if H is a reductive algebraic over ¯ Q with an action of a p -group P and such that π ( Z ( H ◦ )) has p -power order, then π ( Z ( H P, ◦ )) has also p -power order .By using a central series of P , we may argue by induction and we see that itsuffices to treat the case where P is cyclic of order p . Moreover, the surjectiveadjoint quotient morphism H −→ H ad induces a surjective morphism H P, ◦ ։ ( H ad ) P, ◦ whose kernel is K := Z ( H ) P ∩ H P, ◦ . So π ( K ) is dual to the torsionsubgroup of X ( K ), which is a quotient of the torsion subgroup of the coinvariants X ( Z ( M )) P , which has p -power order. Since Z ( H P, ◦ ) is an extension of Z (( H ad ) P, ◦ )by K , we see that it suffices to prove that π ( Z (( H ad ) P, ◦ )) has p -power order. ButTheorem 3.11 of [DM18] asserts that this order even divides the order p of the cyclicgroup P . (cid:3) Using Remark 3.9, we can now strengthen Theorem 3.4 for a certain class ofgroups, by replacing “Borel pair” by “pinning”.
ODULI OF LANGLANDS PARAMETERS 25
Theorem 3.12.
Suppose that the center of ˆ G is smooth over Z [ p ] . Then there is afinite extension K ′ e of K e such that for any admissible φ ∈ Φ e and any admissible α ∈ Σ( φ ) , there is some ϕ α ∈ Z ( W F , ˆ G ( O K ′ e [ p ])) φ,α such that ϕ L α ( W F ) is finiteand Ad ϕ α preserves a pinning of the split reductive group scheme C ˆ G ( φ ) ◦ . Moduli of Langlands parameters
We maintain the setup and notation of the previous section. In particular, ˆ G is a reductive group scheme over Z [ p ] endowed with a finite action of W F , and G L = ˆ G ⋊ W is an adjustable associated “ L -group” of finite type.4.1. The moduli space of cocycles.
Let us fix a “depth” e ∈ N such that theaction of P eF on ˆ G is trivial. The functor R Z ( W F /P eF , ˆ G ( R )) is representable byan affine scheme of finite presentation over Z [ p ] that we denote by Z ( W F /P eF , ˆ G ),and whose affine ring we denote by R e G L . By construction, it comes with a universal1-cocycle ϕ e univ : W F /P eF −→ ˆ G ( R e G L ) . Restriction to P F provides us with a morphism of Z [ p ]-schemes(4.1) Z ( W F /P eF , ˆ G ) −→ Z ( P F /P eF , ˆ G )with the notation of appendix A. Using the notation of Theorem 3.1, we have adecomposition of the right hand side over O K e [ p ] as follows Z ( P F /P eF , ˆ G ) O Ke [ p ] = a φ ∈ Φ e ˆ G · φ, where ˆ G · φ denotes the orbit of φ , which in this context is a smooth affine schemethat represents the quotient sheaf ˆ G/C ˆ G ( φ ) on the big ´etale site of Z [ p ] (see RemarkA.10). This induces in turn a decomposition(4.2) Z ( W F /P eF , ˆ G ) O Ke [ p ] = a φ ∈ Φ adm e ˆ G × C ˆ G ( φ ) Z ( W F , ˆ G ) φ with the notation of the last section. Here the summand ˆ G × C ˆ G ( φ ) Z ( W F , ˆ G ) φ isan affine scheme that represents the quotient sheaf of ˆ G × Z ( W F , ˆ G ) φ by the actionof C ˆ G ( φ ) by right translations on ˆ G and by (twisted) conjugation on Z ( W F , ˆ G ) φ .Recall that φ is called “admissible” if this summand is non-empty and we havedenoted by Φ adm e the subset of admissible elements. In terms of rings, we have thedecomposition(4.3) R e G L ⊗ Z [ p ] O K e [1 /p ] = Y φ ∈ Φ adm e R G L , [ φ ] = Y φ ∈ Φ adm e (cid:16) O ˆ G ⊗ Z [ p ] R G L ,φ (cid:17) C ˆ G ( φ ) . The [ φ ]-part of the universal 1-cocycle ϕ [ φ ]univ : W F /P eF −→ ˆ G ( R G L , [ φ ] )is universal for 1-cocycles ϕ : W F −→ ˆ G ( R ) such that ϕ | P F is ´etale-locally (over R )ˆ G -conjugate to φ . Over R G L ,φ we have an extension of φϕ φ univ : W F /P eF −→ ˆ G ( R G L ,φ ) which is universal for 1-cocycles ϕ : W F −→ ˆ G ( R ) such that ϕ | P F = φ . The1-cocycles ϕ [ φ ]univ and ϕ φ univ determine each other in the following ways. • ϕ φ univ is deduced from ϕ [ φ ]univ by pushing out along the morphism (cid:16) O ˆ G ⊗ Z [ p ] R G L ,φ (cid:17) C ˆ G ( φ ) −→ (cid:16) O ˆ G ⊗ Z [ p ] R G L ,φ (cid:17) ε ˆ G ⊗ id −→ R G L ,φ • ϕ [ φ ]univ is deduced from ϕ φ univ by the formula ϕ [ φ ]univ ( w ) : O ˆ G Ad ∗ w −→ O ˆ G ⊗ Z [ p ] O ˆ G id ⊗ ϕ φ univ ( w ) −→ O ˆ G ⊗ Z [ p ] R G L ,φ where Ad ∗ w is induced by the w -twisted conjugation action of ˆ G on itself,and the composition lands into (cid:16) O ˆ G ⊗ Z [ p ] R G L ,φ (cid:17) C ˆ G ( φ ) .We now recall the decomposition of the previous section(4.4) Z ( W F , ˆ G ) φ = a α ∈ Σ( φ ) adm Z ( W F , ˆ G ) φ,α and we fix, for each α ∈ Σ( φ ) adm , a 1-cocycle ϕ α : W F −→ ˆ G ( O K ′ e [ p ]) as inTheorem 3.4. Then we have an isomorphism ρ ρ · ϕ α (4.5) Z ϕα (( W F /P F ) , C ˆ G ( φ ) ◦ ) ∼ −→ Z ( W F , ˆ G ) φ,α × O Ke [ p ] O K ′ e [1 /p ]where the LHS is a space of tame parameters as studied in Section 2. Accordingly,we have a decomposition of O K e [ p ]-algebras R G L ,φ = Q α R G L ,φ,α and, for each α ,the α -component of ϕ φ univ is given, over R G L ,φ,α ⊗ O Ke [ p ] O K ′ e [ p ] by(4.6) ϕ φ,α univ = ρ G L ϕα · ϕ α : W F −→ ˆ G (cid:16) R G L ,φ,α ⊗ O Ke [ p ] O K ′ e [1 /p ] (cid:17) . We are now in position to prove :
Theorem 4.1. i) The scheme Z ( W F /P eF , ˆ G ) is syntomic (flat and local completeintersection) of relative dimension dim( ˆ G ) over Z [ p ] , and generically smooth.ii) For any prime ℓ = p , the ring R e G L is ℓ -adically separated and the pushforwardof ϕ L e univ to R e G L ⊗ Z ℓ extends uniquely to a ℓ -adically continuous L -morphism ϕ L eℓ − univ : W F /P eF −→ G L ( R e G L ⊗ Z ℓ ) which is universal for ℓ -adically continuous L -morphisms as in Definition 2.11.Proof. i) Since O K ′ e [ p ] is a syntomic cover of Z [ p ], it suffices to prove i) after basechange to this ring. In what follows, we implicitly base-change ˆ G and all schemesintroduced above to this ring, but we omit it in the notation to keep it readable. So,it suffices to prove i) for each summand ˆ G × C ˆ G ( φ ) Z ( W F , ˆ G ) φ of the decomposition(4.2). Consider the morphism(4.7) ˆ G × C ˆ G ( φ ) Z ( W F , ˆ G ) φ −→ ˆ G · φ obtained by restriction of (4.1). Its base change along the orbit morphism ˆ G −→ ˆ G · φ is the first projection(4.8) ˆ G × Z ( W F , ˆ G ) φ −→ ˆ G ODULI OF LANGLANDS PARAMETERS 27
Thanks to the decomposition (4.4) and the isomorphisms (4.5), we may applyCorollary 2.4 and Proposition 2.7 to deduce that Z ( W F , ˆ G ) φ is syntomic of relativedimension dim( C ˆ G ( φ )) over O K ′ e [ p ], and generically smooth. It follows that themorphism (4.8) is syntomic of pure relative dimension dim( C ˆ G ( φ )) and that thesource space is generically smooth since the target is smooth. Since the orbitmorphism is surjective and smooth (because C ˆ G ( φ ) is smooth), the same propertyholds for the morphism (4.7) by descent. But the orbit ˆ G.φ itself is smooth over O K ′ e [ p ] (since it is a summand of Hom( P F /P eF , ˆ G )) and has relative dimensiondim( ˆ G ) − dim( C ˆ G ( φ )). So i) follows.ii) The ℓ -adic separatedness of R e ˆ G follows from Corollary 2.10 and (4.3). More-over, (4.6) together with Theorem 2.12 show that for each φ ∈ Φ e , the univer-sal L -morphism ϕ L φ univ extends uniquely and ℓ -adically continuously to an L -morphism ϕ L φℓ − univ : W F −→ G L ( R ′ G L ,φ ⊗ Z ℓ ). Here we have written R ′ G L ,φ := R G L ,φ ⊗ O Ke [ p ] O K ′ e [1 /p ], and we have used the fact that the ϕ α occuring in (4.6) hasfinite image, hence extends uniquely to W F by continuity. Using the relation be-tween ϕ [ φ ]univ and ϕ φ univ , we see ultimately that ϕ L e univ extends to an ℓ -adically contin-uous L -morphism ϕ L eℓ − univ : W F −→ G L ( R ′ eG L ⊗ Z ℓ ) where R ′ eG L = R e G L ⊗ O K ′ e [ p ].We now claim that ϕ L eℓ − univ factors through G L ( R e G L ⊗ Z ℓ ). Indeed, its pushfor-ward to G L ( R ′ eG L ⊗ Z /ℓ n Z ) has the same image as the pushforward of ϕ L e univ bycontinuity, hence it factors through G L ( R e G L ⊗ Z /ℓ n ) for all n . But since R ′ eG L islocally free of finite rank over R e G L , the claim follows. The universal property isstraightforward. (cid:3) Statement ii) clarifies a bit the dependence of our moduli space Z ( W F /P eF , ˆ G )on our initial choices of a topological generator s of I F /P F and of a lift of FrobeniusFr in W F /P F when defining the subgroup W F of W F . Corollary 4.2.
For any prime ℓ = p , the base change Z ( W F /P eF , ˆ G ) Z ℓ is canoni-cally independent of the choices made to define the subgroup W F . Namely, let W ′ F be another choice of subgroup, then there is a unique isomorphism of Z ℓ -schemes Z ( W F /P eF , ˆ G ) Z ℓ ∼ −→ Z ( W ′ F /P eF , ˆ G ) Z ℓ compatible with the universal ℓ -adicallycontinuous -cocycles on each side. Besides the above result, our main conjecture over ¯ Z [ p ] states that the decom-position (4.4) is the decomposition into connected components. Conjecture 4.3.
For any pair ( φ, α ) , the O K e [ p ] -scheme Z ( W F , ˆ G ) φ,α is con-nected and remains connected after any finite flat integral base change. The isomorphisms in (4.5) reduce this conjecture to the following one :
Conjecture 4.4.
For any split ˆ G over Z [ p ] with a tamely ramified Galois actionthat preserves a Borel pair, the tame summand Z ( W F /P F , ˆ G ) ¯ Z [ p ] is connected. In Subsection 4.6, we prove the last statement under the additional assumptionthat ˆ G has smooth center, or the Galois action preserves a pinning. Thanks toTheorem 3.12, this is enough to get the following result towards Conjecture 4.3 : Theorem 4.5.
Conjecture 4.3 holds if the center of ˆ G is smooth over Z [ p ] . Decomposition after localization at a prime ℓ = p . For each choice ofa prime ℓ = p , statement ii) of Theorem 4.1 allows us to refine the decomposition(4.3) after tensoring by Z ℓ . Indeed, denote by I ℓF the maximal closed subgroup of I F with prime-to- ℓ pro-order. Then, since ϕ L eℓ − univ is ℓ -adically continuous, thekernel I ℓ,eF of ( ϕ L eℓ − univ ) | I ℓF is open in I ℓF . It follows that restriction to I ℓF providesa morphism of Z ℓ -schemes Z ( W F /P eF , ˆ G ) Z ℓ −→ Z ( I ℓF /I ℓ,eF , ˆ G ) Z ℓ . But since the finite group I ℓF /I ℓ,eF has order invertible in Z ℓ , we can apply the resultsof appendix A. In particular, there is a finite ´etale extension Λ e of Z ℓ and a finiteset Φ ℓe ⊂ Z ( I ℓF /I ℓ,eF , ˆ G (Λ e )) such that Z ( I ℓF /I ℓ,eF , ˆ G ) Λ e = a φ ℓ ∈ Φ ℓe ˆ G · φ ℓ , from which we deduce a decomposition similar to (4.2) Z ( W F /P eF , ˆ G ) Λ e = a φ ℓ ∈ Φ ℓe ˆ G × C ˆ G ( φ ℓ ) Z ( W F /P eF , ˆ G ) Λ e ,φ ℓ , where Z ( W F /P eF , ˆ G ) Λ e ,φ ℓ denotes the closed subscheme of Z ( W F /P eF , ˆ G ) Λ e de-fined by ( ϕ eℓ − univ ) | I ℓF = φ ℓ . Then we can play the same game as in Subsection3.2. Namely, taking an L -group ˆ G ⋊ W e such that I ℓF /I ℓ,eF injects into W e , wedefine C G L ( φ ℓ ), ˜ π ( φ ℓ ) and Σ( φ ℓ ) exactly as in that subsection. This allows us todecompose further Z ( W F /P eF , ˆ G ) Λ e ,φ ℓ = a α ℓ ∈ Σ( φ ℓ ) Z ( W F /P eF , ˆ G ) Λ e ,φ ℓ ,α ℓ . We will say again that φ ℓ and α ℓ are admissible if the corresponding summand isnon empty. Moreover, we have an analogue of Theorem 3.4 with the same proof(actually, the proof simplifies a bit since we work here over a DVR). Theorem 4.6.
There is an integral finite flat extension Λ ′ e of Λ e such that, foreach admissible φ ℓ , α ℓ , we can find a cocycle ϕ α ℓ ∈ Z ( W F /P eF , ˆ G ) Λ ′ e ,φ ℓ ,α ℓ withfinite image and such that Ad ϕ αℓ normalizes a Borel pair of C ˆ G ( φ ℓ ) ◦ . As in Remark 3.9, this can be improved in certain circumstances. Namely, if thecenter Z ( C ˆ G ( φ ℓ ) ◦ ) is smooth over Λ e – equivalently, if ℓ does not divide the orderof X ∗ ( Z ( C ˆ G ( φ ℓ ) ◦ )) tors – then one can find ϕ α ℓ such that Ad ϕ αℓ stabilizes a pinningof C ˆ G ( φ ℓ ) ◦ . Using a version of Lemma 3.11 where ¯ Z [ p ] is replaced by ¯ Z ℓ and P is replaced by any solvable group of order prime to ℓ , one obtains the followinganalogue of Theorem 3.12. Theorem 4.7.
Assume that the center of ˆ G is smooth over Z ( ℓ ) . Then there is anintegral finite flat extension Λ ′ e of Λ e such that, for each admissible φ ℓ , α ℓ , we canfind a cocycle ϕ α ℓ ∈ Z ( W F /P eF , ˆ G ) Λ ′ e ,φ ℓ ,α ℓ with finite image and such that Ad ϕ αℓ fixes a pinning of C ˆ G ( φ ℓ ) ◦ . In particular, this result applies to classical groups whenever ℓ = 2. ODULI OF LANGLANDS PARAMETERS 29
Fix ϕ α ℓ as in one of the above theorems. Having finite image, it extends to W F and the conjugation action Ad ϕ αℓ factors over W F /I ℓF . Then the usual map ρ ρ · ϕ α ℓ provides an isomorphism of Λ ′ e -schemes Z ϕαℓ (cid:0) ( W F /P F ) , C ˆ G ( φ ℓ ) ◦ (cid:1) Λ ′ e , ℓ ∼ −→ Z ( W F /P eF , ˆ G ) Λ ′ e ,φ ℓ ,α ℓ where the subscript 1 ℓ on the left hand side denotes the closed and open subschemeof Z ϕαℓ (cid:0) ( W F /P F ) , C ˆ G ( φ ℓ ) ◦ (cid:1) Λ ′ e where the universal ℓ -adically continuous tameparameter restricts trivially to I ℓF . Theorem 4.8.
For each pair ( φ ℓ , α ℓ ) , the Λ e -scheme Z ( W F /P eF , ˆ G ) Λ e ,φ ℓ ,α ℓ hasa geometrically connected special fiber. In particular, it is connected and its basechange to any integral finite flat extension of Λ e remains connected We will prove this result after some preparation on categorical quotients. Mean-while, we note that the second part of the statement follows from the first one since Z ( W F /P eF , ˆ G ) Λ e ,φ ℓ ,α ℓ is the spectrum of an ℓ -adically separated Z ℓ -algebra. Thecollection of these results for all ℓ = p will be the main ingredient in the proof ofTheorem 4.5.4.3. Quotients, moduli spaces of parameters.
The group scheme ˆ G acts byconjugation on Z ( W F /P eF , ˆ G ). There are several type of quotients which can beconsidered here : the stacky quotient, the quotient as fppf sheaves, or the quotientin the category of affine schemes, which is simply Spec(( R e G L ) ˆ G ). Whatever typeof quotient is considered, let us denote it by H ( W F /P eF , ˆ G ). Then, (4.2) inducesa decomposition H ( W F /P eF , ˆ G ) O Ke [ p ] = a φ ∈ Φ adm e Z ( W F , ˆ G ) φ /C ˆ G ( φ ) , where the quotients on the right hand side are of the same type. Next, (4.4) givesfor each φ a decomposition Z ( W F , ˆ G ) φ /C ˆ G ( φ ) ◦ = a α ∈ Σ( φ ) adm Z ( W F , ˆ G ) φ,α /C ˆ G ( φ ) ◦ (beware the ◦ ) while (4.5) provides for each α an isomorphism H ϕα ( W F /P F , C ˆ G ( φ ) ◦ ) ∼ −→ (cid:16) Z ( W F , ˆ G ) φ,α /C ˆ G ( φ ) ◦ (cid:17) O K ′ e [ p ] . Now, let us denote by Σ( φ ) a set of representatives of π ( φ )-orbits in Σ( φ ) andby π ( φ ) α the stabilizer of α in π ( φ ). Let C ˆ G ( φ ) α be the closed subgroup schemeof C ˆ G ( φ ) inverse image of π ( φ ) α . It stabilizes the summand Z ( W F , ˆ G ) φ,α of Z ( W F , ˆ G ) φ , whence an action of π ( φ ) α on H ϕα ( W F /P F , C ˆ G ( φ ) ◦ ) through thelast isomorphism. We thus have obtained an isomorphism H ( W F /P eF , ˆ G ) O K ′ e [ p ] = a φ ∈ Φ adm e a α ∈ Σ( φ ) adm0 H ϕα ( W F /P F , C ˆ G ( φ ) ◦ ) /π ( φ ) α . In the case of the affine categorical quotient, we will use the familiar notation Z ( W F /P eF , ˆ G ) (cid:12) ˆ G := Spec(( R e G L ) ˆ G ) . From the above discussion we deduce :
Proposition 4.9.
The affine categorical quotient Z ( W F /P eF , ˆ G ) (cid:12) ˆ G is a flat,reduced, ℓ -adically separated affine scheme of finite presentation over Z [ p ] and itsring of functions decomposes as ( R e G L ) ˆ G ⊗ O K ′ e [1 /p ] = Y φ ∈ Φ adm e Y α ∈ Σ( φ ) adm0 (cid:18)(cid:16) R G L ϕα (cid:17) C ˆ G ( φ ) ◦ (cid:19) π ( φ ) α . With similar notation, we also have local decompositions for each prime ℓ = p ( R e G L ) ˆ G ⊗ Λ ′ e = Y φ ℓ ∈ Φ ℓ, adm e Y α ℓ ∈ Σ( φ ℓ ) adm0 (cid:18) R G L ϕαℓ , ℓ (cid:19) C ˆ G ( φ ℓ ) ◦ ! π ( φ ℓ ) αℓ . Proof.
The first decomposition has been explained above and the second one is sim-ilar, based on section 4.2. The claimed properties of ( R e G L ) ˆ G follow from Theorem4.1 except for its finite generation as a Z [ p ]-algebra, which is a difficult result ofThomason [Tho87, Thm 3.8]. (cid:3) Closed orbits over an algebraically closed field.
Let L be an alge-braically closed field of characteristic ℓ different from p . Let us consider the affinecategorical quotient Z ( W F /P eF , ˆ G ) L (cid:12) ˆ G L = Spec(( R e G L ⊗ L ) ˆ G L ) . Its relation with the affine quotient over Z [ p ] can be extracted from Alper’s paper[Alp14], which builds on the work of Seshadri [Ses77] and Thomason [Tho87] onGeometric Invariant Theory over arbitrary bases. Proposition 4.10.
The canonical map ( R e G L ) ˆ G ⊗ L −→ ( R e G L ⊗ L ) ˆ G L is injective.It is surjective if ℓ = 0 and, when ℓ > , there is an integer r such that its imagecontains { f ℓ r , f ∈ ( R e G L ⊗ L ) ˆ G L } . In particular the canonical morphism of L -schemes Z ( W F /P eF , ˆ G ) L (cid:12) ˆ G L −→ (cid:16) Z ( W F /P eF , ˆ G ) (cid:12) ˆ G (cid:17) L is a universal homeomorphism, and even an isomorphism when ℓ = 0 .Proof. The case ℓ = 0 is easy, so we assume that ℓ is prime. Consider first the map( R e G L ) ˆ G /ℓ ( R e G L ) ˆ G −→ ( R e G L /ℓR e G L ) ˆ G . It is injective because R e G L is ℓ -torsion free.Moreover, since ˆ G is geometrically reductive over Z [ p ] in the sense of [Alp14, Def.9.1.1] (by Theorem 9.7.5 of [Alp14]), it follows from [Alp14, Rk 5.2.2] that this mapis an “adequate” homeomorphism, in the sense of [Alp14, Def 3.3.1]. In particularit is “universally adequate”, hence the map of the proposition is adequate too, and[Alp14, Lemma 3.2.3] insures the existence of r as claimed in the proposition. (cid:3) Remark 4.11.
In Theorem 6.8, we will get an explicit bound on the set of primes ℓ for which the canonical morphism of this proposition is an isomorphism.By classical Geometric Invariant Theory, we know that the L -points of the affinequotient Z ( W F /P eF , ˆ G ) L (cid:12) ˆ G L correspond bijectively to Zariski closed ˆ G ( L )-orbitsin Z ( W F /P eF , ˆ G ( L )). On the other hand, a theorem of Richardson provides acriterion to decide when the ˆ G ( L )-orbit of ϕ ∈ Z ( W F /P eF , ˆ G ( L )) is closed. ODULI OF LANGLANDS PARAMETERS 31
Definition 4.12.
We say that ϕ ∈ Z ( W F /P eF , ˆ G ( L )) is G L -semisimple if theZariski closure ϕ L ( W F ) of the image of ϕ L in G L ( L ) is a completely reducible subgroup of G L ( L ) in the sense of [BMR05].Let us recall the definition from [BMR05] : a closed subgroup Γ of G L ( L ) is called completely reducible if for all R -parabolic subgroups P ( L ) of G L ( L ) containing Γ,there exists a R -Levi subgroup of P ( L ) containing Γ. Here, and as in [BMR05],we use Richardson’s definition of parabolic and Levi subgroups via cocharacters,which makes perfect sense for non-connected reductive groups.It wouldn’t be difficult to check directly that for a continuous 1-cocycle ϕ : W F → ˆ G ( L ), the property of being G L -semisimple neither depends on the choiceof an integer e such that ϕ L factors through W F /P eF , nor on the particular choice of L -group we make. Anyway, this fact is also a consequence of Richardson’s theoremthat we now state. Theorem 4.13 (Richardson) . The ˆ G ( L ) -orbit of ϕ ∈ Z ( W F /P eF , ˆ G ( L )) is closedif and only if ϕ is G L -semisimple.Proof. Recall that the map ϕ ϕ L identifies the set Z := Z ( W F /P eF , ˆ G ( L ))with the set of L -homomorphisms W F /P eF −→ G L ( L ), which is contained in theset H of all homomorphisms W F /P eF −→ G L ( L ). Both Z and H have a naturalreduced L -scheme structure, and Z is open and closed in H . In particular, theˆ G ( L )-orbit of ϕ ∈ Z is closed in Z if and only the ˆ G ( L )-orbit of ϕ L is closed in H . Now, on H the action of ˆ G ( L ) extends to G L ( L ) and, since ˆ G has finite indexin G L , we see that the ˆ G ( L )-orbit of ϕ L is closed if and only if its G L ( L )-orbit isclosed.Now, let w , · · · , w n be a finite set of generators of the group W F /P eF . Thenthe map ψ ( ψ ( w ) , · · · , ψ ( w n )) is an G L L -equivariant closed embedding of H into G L nL . So we see that the G L ( L )-orbit of ϕ L in H is closed if and only ifthe G L ( L )-orbit of ( ϕ L ( w ) , · · · , ϕ L ( w n )) ∈ G L ( L ) n is closed in G L ( L ) n . Now,Richardson’s theorem (see Cor 3.7 and § G L ( L ) generated by( ϕ L ( w ) , · · · , ϕ L ( w n )) is completely reducible in the sense recalled above. (cid:3) In view of this result, we may drop the G L and simply say that “ ϕ is semisimple”.The following result will be crucial in our study of the affine quotient. Proposition 4.14.
Any semisimple -cocycle ϕ : W F → ˆ G ( L ) extends continu-ously and uniquely to W F . Moreover, the prime-to- p part | L ϕ ( I F ) | p ′ of the cardi-nality of ϕ L ( I F ) is bounded independently of ϕ and of the field L . More precisely, | ϕ L ( I F ) | p ′ divides e.χ ˆ G, Fr ( q ) where • e is the tame ramification of the finite extension F ′ of F given by the kernelof the map W F −→ π ( G L ) , • Fr is any lift of Frobenius in W F . • χ ˆ G, Fr ∈ Z [ T ] is introduced in the appendix B.2. Remark 4.15.
If we restrict attention to fields of characteristic ℓ >
0, then thestatement that ϕ extends continuously to W F is true for all 1-cocycles, by (ii) ofTheorem 4.1. However, there is obviously no uniform bound on | ϕ L ( I F ) | p ′ withoutthe semisimplicity hypothesis, when we vary the field L . Proof of the proposition.
Recall from [Iwa55, Thm. 2 (iii)] that there exist lifts˜ s and ˜Fr of s and Fr in W F such that ˜Fr . ˜ s. ˜Fr − = ˜ s q and that W F decomposesas a semi-direct product W F = P F ⋊ h ˜ s, ˜Fr i . Accordingly, W F decomposes as W F = P F ⋊ h ˜ s, ˜Fr i . Then we see that a continuous 1-cocycle ϕ from W F extendscontinuously to W F if and only if ϕ L (˜ s ) has finite order, in which case this orderis prime-to- p , the extension is unique, and it satisfies ϕ ( W F ) = ϕ ( W F ).Let us now assume that ϕ is G L -semisimple and show that ϕ L (˜ s ) then has finiteorder. Let F ′ and e be as in the statement of the proposition. Note that ˜ s e ∈ W F ′ and thus ϕ L (˜ s ) e ∈ ˆ G ( L ). Since ϕ L ( P F ′ ) is finite, there certainly is an integer m such that ϕ L (˜ s ) em commutes with ϕ L ( P F ′ ). This means that h ϕ L (˜ s ) em i is a normalsubgroup of ϕ L ( I F ′ ), which is a normal subgroup of ϕ L ( W F ). Now recall from[BMR05, Thm 3.10] that any normal subgroup of a completely reducible subgroupof G L ( L ) is completely reducible. So we infer that h ϕ L (˜ s ) em i is a completelyreducible subgroup of G L ( L ), hence also a completely reducible subgroup of ˆ G ( L ).This means that ϕ L (˜ s ) em is a semi-simple element of ˆ G ( L ). Since it is conjugateto its q -power under ϕ L ( ˜Fr) ∈ ˆ G ( L ), Proposition B.3 (2) shows that ϕ L (˜ s ) em hasfinite order, and this order divides χ ˆ G, Ad ϕ ( ˜Fr) ( q ) = χ ˆ G, ˜Fr ( q ).It now remains to estimate m and prove that m = χ ˆ G, ˜Fr ( q ) works. For this, wemay assume that ϕ belongs to some Z ( W F , ˆ G ) φ,α and write ϕ = ρ · ϕ α as in (4.5).By construction ϕ L α has finite image in G L (¯ Z [1 /p ]). Let m be the order of theelement ϕ L α (˜ s ) e , which lies in ˆ G (¯ Z [1 /p ]). Then we have ϕ L (˜ s ) em = ρ (˜ s ) . Ad ϕ α (˜ s ) ( ρ (˜ s )) · · · Ad ϕ α (˜ s ) em ( ρ (˜ s )) ∈ C ˆ G ( φ ) ◦ ( L ) , hence ϕ L (˜ s ) em commutes with ϕ ( P F ′ ) as desired. But observe now that ϕ L α (˜ s ) e isalso a semisimple element of ˆ G ( ¯ Q ) that is conjugate to its q th -power under ϕ L α ( ˜Fr).Hence, as above, its order m divides χ ˆ G, ˜Fr ( q ). (cid:3) Let us denote by N ˆ G the l.c.m of all | ϕ L ( I F ) | p ′ for ϕ and L varying as in theproposition, and where G L is the minimal L -group. Corollary 4.16.
For each “depth” e , there is an open normal subgroup I eF of I F with index dividing N ˆ G such that any semisimple cocycle ϕ : W F /P eF −→ ˆ G ( L ) istrivial on I eF ∩ W F , and therefore extends canonically to W F /I eF . With I eF as in this corollary, we may consider the ˆ G -stable closed subscheme Z ( W F /I eF , ˆ G ) of Z ( W F /P eF , ˆ G ) consisting of 1-cocycles that are trivial on I eF .We will denote by S eG L its affine ring, which is thus a quotient of R e G L . Proposition 4.17. i) The homomorphism of rings (4.9) ( R e G L ) ˆ G −→ ( S eG L ) ˆ G is injective and its image contains { f N , f ∈ ( S eG L ) ˆ G } for some integer N > .ii) The corresponding morphism of schemes (4.10) Z ( W F /I eF , ˆ G ) (cid:12) ˆ G −→ Z ( W F /P eF , ˆ G ) (cid:12) ˆ G is a finite universal homeomorphism and becomes an isomorphism after tensoringby Q . ODULI OF LANGLANDS PARAMETERS 33
Proof. i) Injectivity of (4.9) . For any algebraically closed field L in which p isinvertible, the last corollary tells us that all closed orbits of Z ( W F /P eF , ˆ G )( L ) arecontained in Z ( W F /I eF , ˆ G )( L ), hence the morphism (4.10) is bijective on L -points.It follows that the kernel of (4.9) is contained in the nilradical of ( R e G L ) ˆ G , which istrivial since R e G L is reduced, being syntomic over Z [ p ] and generically smooth byTheorem 4.1. Image of (4.9) . By [Alp14, Thm 9.7.5], ˆ G is geometrically reductive in the senseof [Alp14, Def. 9.1.1]. By the characterization of this property given in [Alp14,Lem 9.2.5 (2)’], it follows that the map (4.9) is “universally adequate”. Then,Proposition 3.3.4 of [Alp14] provides the desired N .ii) now follows from Lemmas 3.1.4 and 3.1.5 of [Alp14]. (cid:3) Note that the ring S eG L may not share the nice properties of R e G L ; it may notbe reduced nor be flat over Z [1 /p ]. However, the last proposition implies that thenilradical of ( S eG L ) ˆ G coincides with its Z [1 /p ]-torsion ideal. Moreover, the fact that(4.10) is an isomorphism after tensoring by Q shows that ( Z ( W F /P eF , ˆ G ) (cid:12) ˆ G ) Q iscanonically independent of our initial choice of subgroup W F in W F . Actually, wecan do better with a little more work : Theorem 4.18.
The affine quotient Z ( W F /P eF , ˆ G ) (cid:12) ˆ G is canonically independentof the choice of a topological generator s of I F /P F and a lift of Frobenius Fr in W F /P F in the definition of the subgroup W F .Proof. Let (Fr ′ , s ′ ) be another choice, leading to a subgroup W ′ F of W F . Denoteby R e ′ G L the affine ring of Z ( W ′ F /P eF , ˆ G ). Denote by ι the embedding (4.9) andby ι ′ the analogous embedding ( R e ′ G L ) ˆ G ֒ → ( S eG L ) ˆ G . For each prime ℓ = p , wehave a canonical isomorphism ( R e ′ G L ) ˆ G ⊗ Z ℓ ≃ ( R e G L ) ˆ G ⊗ Z ℓ from Corollary 4.2. Byconstruction it commutes with the base changes of ι and ι ′ to Z ℓ , which means that ι (( R e G L ) ˆ G ) ⊗ Z ℓ = ι ′ (( R e ′ G L ) ˆ G ) ⊗ Z ℓ inside ( S eG L ) ˆ G ⊗ Z ℓ . This implies that ℓ is notin the support of the quotient Z [ p ]-module ( ι (( R e G L ) ˆ G ) + ι ′ (( R e ′ G L ) ˆ G )) /ι (( R e G L ) ˆ G ).Since this is true for all ℓ = p , it follows that this quotient is 0, hence ι (( R e G L ) ˆ G ) = ι ′ (( R e ′ G L ) ˆ G ) inside ( S eG L ) ˆ G . (cid:3) We may wonder whether such an independence result still holds for the stackyquotient. We believe that, at least, the categories of quasi-coherent sheaves on suchstacks might be equivalent.4.5.
Geometric connected components in positive characteristic.
We main-tain our setup of an algebraically closed field L of characteristic ℓ = p , andwe assume that ℓ >
0. In order to parametrize the connected components of Z ( W F /P eF , ˆ G ) L , we first observe that, since ˆ G is connected, the canonical mor-phism Z ( W F /P eF , ˆ G ) L −→ Z ( W F /P eF , ˆ G ) L (cid:12) ˆ G L induces a bijection on the respective sets of connected components. Hence, Propo-sition 4.17 invites us to study the connected components of Z ( W F /I eF , ˆ G ) L (cid:12) ˆ G L . Note : in order to lighten the notation a bit we will sometimes denote by H thecategorical quotient of cocycles modulo the relevant group action. Using restriction to I ℓF , we have already obtained a decomposition H ( W F /I eF , ˆ G L ) = a φ ℓ ∈ Φ ℓ, adm e a α ℓ ∈ Σ( φ ℓ ) (cid:16) H ϕαℓ (cid:0) W F /I eF I ℓF , C ˆ G ( φ ℓ ) ◦ L (cid:1)(cid:17) (cid:12) π ( φ ℓ ) αℓ . The following result shows that each summand is connected and will provide atopological description of these summands.
Proposition 4.19.
Assume that the action of W F on ˆ G is trivial on I ℓF and sta-bilizes a Borel pair ( ˆ B, ˆ T ) . Then the following holds. (1) The reduced fixed-points subgroup ( ˆ G L ) I F is a connected reductive subgroupof ˆ G L and the reduced fixed-points subgroup ( ˆ T L ) I F is a maximal torus of ( ˆ G L ) I F whose Weyl group is the I F -fixed subgroup Ω I F of the Weyl group Ω of ˆ T in ˆ G . (2) The closed immersion Z ( W F /I F , ˆ G I F L ) ֒ → Z ( W F /I eF I ℓF , ˆ G L ) induces anhomeomorphism Z ( W F /I F , ˆ G I F L ) (cid:12) ˆ G I F L −→ Z ( W F /I eF I ℓF , ˆ G L ) (cid:12) ˆ G L . (3) The map t ( ϕ : Fr t ⋊ Fr) induces an isomorphism ( ˆ T I F L ) Fr (cid:12) Ω W F ∼ −→ Z ( W F /I F , ˆ G I F L ) (cid:12) ˆ G I F L . Proof. (1) The group I F acts on ˆ G through a cyclic ℓ -power quotient, any generatorof which is a quasi-semisimple automorphism of ˆ G (in the sense of Steinberg).Therefore, the first assertion follows from Cor. 1.33 of [DM94] and its proof.(2) We first make the following observation. If φ : I F −→ ˆ G ( L ) is a semisimplecocycle trivial on I ℓF , then it is ˆ G ( L )-conjugate to the trivial cocycle (note thatthe latter is indeed semisimple by the characterization given in [BMR05, Prop 3.5(v)]). To prove this, let us use the “minimal” L -group ˆ G ⋊ Γ, where Γ is theimage of W F in Aut( ˆ G ). Then the image C of I F in Γ is a cyclic ℓ -group. Let ¯ s be the image of the pro-generator s of I F /P F in C . By [Ste68, 7.2], the element L φ ( s ) := ( σ, ¯ s ) normalizes a Borel subroup of ˆ G . After conjugating by some elementof ˆ G ( L ) we may assume that it normalizes ˆ B , thus L φ factors trough the minimal R -parabolic subgroup ˆ B ⋊ C of ˆ G ⋊ C . Since φ is assumed to be semisimple, L φ should factor through some R -Levi subgroup of ˆ B ⋊ C . But these Levi subgroupsare ˆ B -conjugated to ˆ T ⋊ C . Therefore we may conjugate again L φ so that it factorsthrough ˆ T ⋊ C , which means that φ ∈ Z ( C, ˆ T ( L )). But since ˆ T ( L ) is a ℓ -torsionfree divisible group, we have H ( C, ˆ T ( L )) = { } , which means that φ is conjugateto the trivial cocycle.We deduce that the subset Z ( I F /I eF I ℓF , ˆ G ( L )) ss of Z ( I F /I eF I ℓF , ˆ G ( L )) thatconsists of semisimple cocycles is closed, since it is a single orbit and this orbitis closed by definition of semisimple. Moreover this closed subset identifies withˆ G ( L ) / ˆ G ( L ) I F . By pull-back, we deduce that the subset Z ( W F /I eF I ℓF , ˆ G ( L )) I F − ss of Z ( W F /I eF I ℓF , ˆ G ( L )) that consists of all cocycles ϕ : W F −→ ˆ G ( L ) such that ϕ | I F is semisimple, is closed and identifies with ˆ G ( L ) × ˆ G ( L ) IF Z ( W F /I F , ˆ G ( L ) I F ).Now by [BMR05, Thm 3.10] we know that any semisimple ϕ : W F −→ ˆ G ( L )has semisimple restriction to I F , so that the above closed subset contains all closedorbits of Z ( W F /I eF I ℓF , ˆ G ( L )). So denote by Z ( W F /I eF I ℓF , ˆ G L ) I F − ss the (reduced) ODULI OF LANGLANDS PARAMETERS 35 closed subscheme of Z ( W F /I eF I ℓF , ˆ G L ) associated to this closed subset. Then thesame argument as in Proposition 4.17 shows that the canonical morphism Z ( W F /I eF I ℓF , ˆ G L ) I F − ss (cid:12) ˆ G L −→ Z ( W F /I eF I ℓF , ˆ G L ) (cid:12) ˆ G L is a finite universal homeomorphism. Using that Z ( W F /I eF I ℓF , ˆ G ) I F − ss L = ˆ G L × ˆ G IFL Z ( W F /I F , ˆ G I F L )we infer statement (2).iii) This is [DM15, Prop. 7.1] (see also [Bor79, Prop. 6.7]). (cid:3) We now use the results and notation of Subsection 4.2 to spread out this result.
Corollary 4.20.
Let φ ℓ ∈ Φ ℓe and α ℓ ∈ Σ( φ ℓ ) be admissible, and fix ϕ := ϕ α ℓ ∈ Z ( W F /P eF , ˆ G (Λ ′ e )) φ ℓ ,α ℓ with finite image and such that Ad ϕ αℓ normalizes a Borelpair ( ˆ B φ ℓ , ˆ T φ ℓ ) of C ˆ G ( φ ℓ ) ◦ . We denote by Ω ◦ φ ℓ the Weyl group of ˆ T φ ℓ in C ˆ G ( φ ℓ ) ◦ and by Ω φ ℓ = Ω ◦ φ ℓ ⋊ π ( φ ℓ ) its “Weyl group” in C ˆ G ( φ ℓ ) . (1) Let C ˆ G L ( ϕ | I F ) = ( ˆ G L ) ϕ ( I F ) be the reduced centralizer of ϕ ( I F ) in ˆ G L . (a) C ˆ G L ( ϕ | I F ) ◦ is reductive with maximal torus ( ˆ T φ ℓ ,L ) ϕ ( I F ) and Weylgroup (Ω ◦ φ ℓ ) ϕ ( I F ) (b) π ( C ˆ G L ( ϕ | I F )) = π ( φ ℓ ) ϕ ( I F ) and the “Weyl group” of ( ˆ T φ ℓ ,L ) ϕ ( I F ) in C ˆ G L ( ϕ | I F ) is (Ω φ ℓ ) ϕ ( I F ) ≃ (Ω ◦ φ ℓ ) ϕ ( I F ) ⋊ π ( φ ℓ ) ϕ ( I F ) . (2) The natural closed immersion induces an homeomorphism H ϕ ( W F /I F , C ˆ G L ( ϕ | I F ) ◦ ) −→ H ϕ ( W F /I eF I ℓF , C ˆ G L ( φ ℓ ) ◦ ) which is equivariant for the natural actions of π ( φ ℓ ) α ℓ = π ( φ ℓ ) ϕ ( W F ) . (3) The map t ( ϕ t : Fr t ⋊ Fr) induces an isomorphism ( ˆ T ϕ ( I F ) φ ℓ ,L ) ϕ (Fr) (cid:12) (Ω ◦ φ ℓ ) W F ∼ −→ H ϕ ( W F /I F , C ˆ G L ( ϕ | I F ) ◦ ) and subsequently an isomorphism ( ˆ T ϕ ( I F ) φ ℓ ,L ) ϕ (Fr) (cid:12) (Ω φ ℓ ) W F ∼ −→ (cid:16) H ϕ ( W F /I F , C ˆ G L ( ϕ | I F ) ◦ ) (cid:17) (cid:12) π ( φ ℓ ) αℓ . Proof. (1)(a) follows directly from (1) of the previous proposition. For (1)(b),observe first that the fact that Ω φ ℓ is a split extension Ω ◦ φ ℓ ⋊ π ( φ ℓ ) of π ( φ ℓ ) byΩ ◦ φ ℓ comes from the fact it contains the subgroup N C ˆ G ( φ ℓ ) ( ˆ T φ ℓ , ˆ B φ ℓ ) / ˆ T φ ℓ ≃ π ( φ ℓ ).Since the action of W F through Ad ϕ on C ˆ G ( φ ℓ ) stabilizes ˆ T φ ℓ and ˆ B φ ℓ , the inducedaction on Ω φ ℓ preserves the semi-direct product decomposition, hence in particularthe ϕ ( I F )-invariants are given by (Ω φ ) ϕ ( I F ) = (Ω ◦ φ ) ϕ ( I F ) ⋊ π ( φ ℓ ) ϕ ( I F ) . Moreoverwe have H ( I F , ˆ T φ ℓ ( L )) = 1 since I F acts on the uniquely ℓ -divisible abelian groupˆ T φ ℓ ( L ) through a cyclic ℓ -group, therefore the map N C ˆ G ( φ ℓ ) ( ˆ T φ ℓ , ˆ B φ ℓ ) ϕ ( I F ) −→ π ( φ ℓ ) ϕ ( I F ) is surjective, which shows that π ( C ˆ G L ( ϕ ( I F ))) = π ( φ ℓ ) ϕ ( I F ) .(2) follows from (2) of the last proposition except for the equivariance under thegroup π ( φ ℓ ) α ℓ = π ( φ ℓ ) ϕ ( I F ) which is straightforward.The first statement of (3) follows directly from (3) of the last proposition, and weinfer the second statement from the equality (Ω φ ) ϕ ( W F ) = (Ω ◦ φ ) ϕ ( W F ) ⋊ π ( φ ℓ ) ϕ ( W F ) ,which we already explained above. (cid:3) Applying this corollary to L = ¯ F ℓ we see that the special fiber of the Λ e -scheme Z ( W F /P eF , ˆ G ) Λ e ,φ ℓ ,α ℓ of subsection 4.2 is geometrically connected. This finishesthe proof of Theorem 4.8. Corollary 4.21.
There are natural bijections between the following sets : (1)
The set of connected components of Z ( W F /P eF , ˆ G ) Λ e (2) The set of connected components of Z ( W F /P eF , ˆ G ) ¯ F ℓ = Z ( W F /P eF , ˆ G ) ¯ F ℓ (3) The set of pairs ( φ ℓ , [ α ℓ ]) with φ ℓ ∈ Φ ℓ, adm e and [ α ℓ ] a π ( φ ℓ ) -conjugacyclass in Σ( φ ℓ ) adm . (4) The set of ˆ G (¯ F ℓ ) -conjugacy classes of admissible pairs ( φ ℓ , α ℓ ) where φ ℓ ∈ Z ( I ℓF /I ℓ,eF , ˆ G (¯ F ℓ )) and α ℓ ∈ Σ( φ ℓ ) . (5) The set of ˆ G (¯ F ℓ ) -conjugacy classes of pairs ( φ, α ) where φ ∈ Z ( I F /I eF , ˆ G (¯ F ℓ )) ss is G L -semisimple and α ∈ Σ( φ ) . (6) The set of ˆ G (¯ F ℓ ) -conjugacy classes of pairs ( φ, β ) where φ ∈ Z ( I F /I eF , ˆ G (¯ F ℓ )) ss is G L -semisimple and β ∈ ˜ π ( φ ) is the image of some element in C G L ( φ ) ∩ ( ˆ G ( L ) ⋊ Fr) = { ˜ β ∈ ˆ G ( L ) ⋊ Fr , ˜ βφ ( i ) ˜ β − = φ (Fr .i. Fr − ) } . (7) The set of equivalence classes in Z ( W F /P eF , ˆ G (¯ F ℓ )) for the relation definedby ϕ ∼ ϕ ′ if and only if there is ˆ g ∈ ˆ G (¯ F ℓ ) such that ϕ | I ℓF = ˆ g ϕ ′| I ℓF and π ◦ ϕ = π ◦ ˆ g ϕ ′ with π the map C G L ( ϕ | I ℓF ) ։ π ( C G L ( ϕ | I ℓF )) . Moreover, one can replace ¯ F ℓ by any algebraically closed field L of characteristic ℓ .Proof. The bijections between (1), (2) and (3) follow from Theorem 4.8 which wehave just proved. The bijection between (3) and (4) follows from the definitions,and so does the bijection between (4) and (7). We now describe bijections between(4), (5) and (6) in a circular way.(4) → (5). Start with an admissible pair, ( φ ℓ , α ℓ ). Choose an extension ϕ of φ ℓ that preserves some chosen Borel pair of C ˆ G ( φ ℓ ) ◦ . Then φ := ϕ | I F is certainly G L -semisimple and α := π ◦ ϕ is an element of Σ( φ ) (here π is the projection C G L ( φ ) −→ ˜ π ( φ ) as usual). We need to check that any other choice ϕ ′ leads to aconjugate of ( φ, α ). Since all Borel pairs are conjugate, we may assume that ϕ ′ and ϕ fix the same Borel pair, and denote it by ( ˆ B φ ℓ , ˆ T φ ℓ ) . Then ϕ ′ = η · ϕ for some η ∈ Z ϕ ( W F /I ℓF , ˆ T φ ℓ ). Since H ϕ ( I F /I ℓF , ˆ T φ ℓ ) = 0 (because ˆ T φ ℓ is uniquely ℓ -divisible), we have H ϕ ( W F /I ℓF , ˆ T φ ℓ ) = H ϕ ( W F /I F , ( ˆ T φ ℓ ) I F ), which meansthat we can “Ad ϕ -conjugate” η by an element t ∈ ˆ T φ ℓ so that it factors through acocycle in Z ϕ ( W F /I F , ˆ T I F φ ℓ ). So, after conjugating ϕ ′ by t , it has the form η · ϕ with η ∈ Z ϕ ( W F /I F , ˆ T I F φ ℓ ). We now certainly have ( η · ϕ ) | I F = ϕ | I F and, sinceˆ T I F φ ℓ is connected, we also have π ◦ ( η · ϕ ) = π ◦ ϕ .(5) → (6). To a pair ( φ, α ) we associate ( φ, β ) with β := α (Fr).(6) → (4). Start with a pair ( φ, β ) and put φ ℓ := φ | I ℓF . Choose a lift ˜ β of β in C G L ( φ ) ∩ ( ˆ G ⋊ Fr). Then there is a unique extension ϕ of φ such that ϕ (Fr) = ˜ β .This extension certainly factors through C G L ( φ ℓ ) and we put α ℓ := π ℓ ◦ ϕ with π ℓ : C G L ( φ ℓ ) −→ ˜ π ( φ ℓ ). Note that any other choice of lift of β is of the form c ˜ β with c ∈ C ˆ G ( φ ) ◦ . Since C ˆ G ( φ ) ◦ is contained in C ˆ G ( φ ℓ ) ◦ , such a choice defines thesame α ℓ . ODULI OF LANGLANDS PARAMETERS 37
The composition of these three applications, starting from any set (4), (5) and(6) is easily seen to be the identity. (cid:3)
We finish this paragraph with another view on the topological description ofthe affine categorical quotient Z ( W F /P eF , ˆ G L ) (cid:12) ˆ G L that we have obtained above,which makes it strikingly similar to what we will get over fields of characteristic 0.For a pair ( φ, β ) as in (6) of the last corollary and a Borel pair ( ˆ B φ , ˆ T φ ) of thereductive algebraic group C ˆ G L ( φ ) ◦ we have an action of β on the torus ˆ T φ and onits Weyl group Ω φ = Ω ◦ φ ⋊ π ( φ ) in C ˆ G ( φ ) (namely, the conjugation action of anylift ˜ β of β in C G L ( φ ) that preserves ( ˆ B φ , ˆ T φ )). Putting together the last corollaryand the previous proposition, we get : Corollary 4.22.
Let Ψ e ( L ) be a set of representatives of ˆ G L -conjugacy classesof pairs ( φ, β ) as in (6) of the last corollary. For each such pair, choose a lift ˜ β of β in C G L ( φ ) that normalizes a Borel pair ( ˆ B φ , ˆ T φ ) of C ˆ G ( φ ) ◦ , and denote by ϕ L ˜ β : W F −→ G L ( L ) the corresponding extension of L φ . Then the collection ofmorphisms Z β ( W F /I F , ˆ T φ ) −→ Z ( W F /P eF , ˆ G L ) , η η · ϕ ˜ β induce an homeo-morphism a ( φ,β ) ∈ Ψ e ( L ) ( ˆ T φ ) β (cid:12) (Ω φ ) β ≈ −→ Z ( W F /P eF , ˆ G L ) (cid:12) ˆ G L . In Theorem 6.8 we will see that these homeomorphisms are actually isomor-phisms when ℓ is “ G L -banal”.4.6. Connected components over Z [ p ] . In this subsection, we assume that theaction of W F on ˆ G is trivial on P F and stabilizes a Borel pair ( ˆ B, ˆ T ), and we studythe connectedness of the depth 0 scheme Z ( W F /P F , ˆ G ) considered in Section 2,and of all its base changes to finite flat integral extensions of Z [ p ]. Our generalstrategy relies on what we already know about the connected components of thebase change to ¯ Z ℓ for all ℓ = p . The following result implements this strategy undersome additional hypothesis, that are fulfilled for example if the action of W F onˆ G is unramified. After proving it, we will show that this additional hypothesis ismore generally satisfied when the action of I F stabilizes a pinning. Proposition 4.23.
Assume that there is a prime ℓ = p such that, for each sub-group I of finite index of I F , the Z [ p ] -group scheme ˆ G I has connected geometricfibers, and is smooth over Z [ ℓ p ] . Then the Z [ p ] -scheme Z ( W F /P F , ˆ G ) is con-nected, and so are all its base changes to finite flat integral extensions of Z [ p ] .Proof. Let C be a connected component of Z ( W F /P F , ˆ G ). Since C is flat and offinite type over Z [ p ], we certainly have C ( ¯ Q ) = ∅ . Let us consider the set I of opensubgroups I of I F such that C ( ¯ Q ) ∩ Z ( W F /I, ˆ G I )( ¯ Q ) = ∅ . By Proposition 4.17 (ii), the set I is not empty, so we may pick a maximal I ∈ I .We claim that I has ℓ -power index in I F . Indeed, suppose the contrary, let ℓ = ℓ be a prime that divides [ I F : I ] and let I ′ ⊃ I be the unique subgroup of I F such that I ′ /I is the ℓ -primary part of I F /I . Since I ∈ I , there is a connectedcomponent C I of Z ( W F /I, ˆ G I ) contained in C and such that C I ( ¯ Q ) = ∅ . Our hypothesis on ˆ G implies that ˆ G I is reductive over Z [ ℓ p ], hence Lemma 4.24 (2)applies and ensures that C I (¯ F ℓ ) is not empty. Moreover, by looking fibrewise, wesee that the pair ( ˆ B I , ˆ T I ) is a Borel pair of ˆ G I over Z [ ℓ p ]. Now, since I ′ /I has ℓ -power order, we can repeat the argument of the proof of (2) of Proposition 4.19 anddeduce that the injective map Z ( W F /I ′ , ˆ G I ′ (¯ F ℓ )) I F − ss ֒ → Z ( W F /I, ˆ G I (¯ F ℓ )) I F − ss induces a bijection between the respective sets of conjugacy classes, which in turnimplies that the morphism Z ( W F /I ′ , ˆ G I ′ ) ¯ F ℓ (cid:12) ˆ G I ′ ¯ F ℓ −→ Z ( W F /I, ˆ G I ) ¯ F ℓ (cid:12) ˆ G I ¯ F ℓ is a homeomorphism, and consequently that the morphism Z ( W F /I ′ , ˆ G I ′ ) ¯ F ℓ ֒ → Z ( W F /I, ˆ G I ) ¯ F ℓ induces a bijection on π . Therefore, there is a component C I ′ of Z ( W F /I ′ , ˆ G I ′ ) that maps into C I , hence also into C , and such that C I ′ (¯ F ℓ ) = ∅ .But since the index of I ′ in I F is prime to ℓ , Lemma 4.24 (3) ensures that C I ′ ( ¯ Q ) = ∅ , which contradicts the maximality of I unless I ′ = I .Now that we know that I has ℓ -power index in I F , we shrink it so that it still has ℓ -power index in I F and its image in Aut( ˆ G ) has prime-to- ℓ order. For this new I , the group scheme ˆ G I is also smooth at ℓ , hence, by Lemma 4.24 (2) again, wehave C I (¯ F ℓ ) = ∅ . But the map Z ( W F /I F , ˆ G I F ) ¯ F ℓ ֒ → Z ( W F /I, ˆ G I ) ¯ F ℓ induces abijection on π , by the same argument as above, hence C I contains a component C I F of Z ( W F /I F , ˆ G I F ), and so does C . So we have shown that the closed immersion Z ( W F /I F , ˆ G I F ) ֒ → Z ( W F /P F , ˆ G ) is surjective on π , and our statement followsfrom the fact that Z ( W F /I F , ˆ G I F ) ≃ ˆ G I F is connected, under our assumption.Moreover, the same argument works similarly after base change to any integralfinite flat extension of R of Z [ p ] by reducing modulo prime ideals of R rather thanprime numbers. (cid:3) Lemma 4.24.
Let ℓ = p be a prime, and let I ⊂ I F be a subgroup of finite indexsuch that the group scheme ˆ G I is reductive over Z ( ℓ ) . Then, for any connectedcomponent C of Z ( W F /I, ˆ G I ) , we have : (1) If L is an algebraically closed field over Z ( ℓ ) with C ( L ) = ∅ , then C ( L ) contains a semisimple cocycle valued in N ˆ G I ( ˆ T I )( L ) . (2) C ( ¯ Q ) = ∅ ⇒ C (¯ F ℓ ) = ∅ . (3) If ℓ does not divide the index [ I F : I ] , then C (¯ F ℓ ) = ∅ ⇒ C ( ¯ Q ) = ∅ .Proof. We lighten the notation a bit by putting ˆ H := ˆ G I , B ˆ H := ˆ B I and T ˆ H := ˆ T I .The action of W F /I on ˆ H stabilizes the Borel pair ( B ˆ H , T ˆ H ) and factors over somefinite quotient W . We also put H L := ˆ H ⋊ W and we still denote by s the imageof s in W .(1) If C ( L ) is not empty, then C ( L ) certainly contains a semisimple 1-cocycle ϕ ∈ Z ( W F /I, ˆ H ( L )). As usual, we denote by ϕ L the associated L -homomorphism W F /I −→ H L ( L ). By [Ste68, 7.2] the element ϕ L ( s ) of H L ( L ) normalizes a Borelsubgroup of ˆ H . Since C is stable under conjugation by ˆ H , we may conjugate ϕ sothat ϕ L ( s ) normalizes B ˆ H . Then ϕ L ( s ) belongs to the Borel subgroup B ˆ H ( L ) ⋊ W of H L ( L ). Since ϕ is semisimple, ϕ L ( s ) generates a completely reducible subgroupof H L ( L ), hence it belongs to a Levi subgroup of B ˆ H ⋊ W . Since all these Levi sub-groups are B ˆ H -conjugate to T ˆ H ⋊ W , we may conjugate further ϕ so that ϕ L ( s ) ∈ T ˆ H ( L ) ⋊ s . In this situation, ( T ˆ H ( L ) ϕ L ( s ) ) ◦ is a maximal torus of ( ˆ H ( L ) ϕ L ( s ) ) ◦ ODULI OF LANGLANDS PARAMETERS 39 whose centralizer in ˆ H ( L ) is T ˆ H ( L ), [DM94, Thm 1.8 iv)]. Now, ϕ L (Fr) normal-izes ( ˆ H ( L ) ϕ L ( s ) ) ◦ = ( ˆ H ( L ) ϕ L ( s ) q ) ◦ , hence it conjugates ( T ˆ H ( L ) ϕ L ( s ) ) ◦ to anothermaximal torus therein. Pick c ∈ ( ˆ H ( L ) ϕ L ( s ) ) ◦ that conjugates back this torus to( T ˆ H ( L ) ϕ L ( s ) ) ◦ . So c. ϕ L (Fr) normalizes ( T ˆ H ( L ) ϕ L ( s ) ) ◦ , hence also its centralizer T ˆ H ( L ) in ˆ H ( L ). Hence the unique 1-cocycle ϕ c : W F /I −→ ˆ H ( L ) such that (cid:26) L ( ϕ c )( s ) := L ϕ ( s ) ∈ T ˆ H ( L ) × s L ( ϕ c )(Fr) := c. ϕ L (Fr) ∈ N ˆ H ( T ˆ H )( L ) ⋊ Fr . is valued in N ˆ H ( T ˆ H )( L ) as desired, and it remains to prove that ϕ c belongs to C ( L ). But the cocycle ϕ c makes sense for any c ∈ ( ˆ H ( L ) ϕ L ( s ) ) ◦ , so it is an elementin the image of an algebraic morphism ( ˆ H ( L ) ϕ L ( s ) ) ◦ −→ Z ( W F /I, H L ( L )). Sincethe source of this morphism is connected, its image is contained in C ( L ).(2) Let us first prove that C ( ¯ Q ) contains a 1-cocycle ϕ such that ϕ L has finiteimage, which is here equivalent to ϕ L (Fr) having finite order. By (1), we may startwith ϕ such that ϕ L is N ˆ H ( T ˆ H )( L )-valued. Then, a convenient power ϕ L (Fr) r of ϕ L (Fr) belongs to ( T ˆ H ( L ) ϕ L ( s ) ) ◦ . But the latter is a divisible group, so it containsan element t such that t − r = ϕ L (Fr) r . Then, the cocycle ϕ t defined as above hasfinite image. Now, we argue as in Proposition 2.9 with the building of ˆ H ( ¯ Q ℓ ) to seethat ϕ t can be ˆ H ( ¯ Q ℓ ) conjugate so that it becomes ˆ H (¯ Z ℓ )-valued. Then its imagein Z ( W F /I, ˆ H (¯ F ℓ )) belongs to C (¯ F ℓ ).(3) By (1) we may start with ϕ ∈ C (¯ F ℓ ) taking values in N ˆ H ( T ˆ H )(¯ F ℓ ), andwe will show that it can be lifted to a 1-cocycle W F /I −→ N ˆ H ( T ˆ H )(¯ Z ℓ ). As inthe proof of Theorem 3.6, the obstruction to lifting ϕ belongs to H ( W F /I, K )where K is the kernel of the reduction map N ˆ H ( T ˆ H )(¯ Z ℓ ) −→ N ˆ H ( T ˆ H )(¯ F ℓ ), whichis also the kernel of the reduction map T ˆ H (¯ Z ℓ ) −→ T ˆ H (¯ F ℓ ). In particular, K is auniquely ℓ ′ -divisible abelian group. Since I F /I has prime to ℓ order, it follows that H ( I F /I, K ) = H ( I F /I, K ) = { } . Since we also have H ( W F /I F , K I F ) = { } ,we see that H ( W F /I, K ) = 0 and there is no obstruction to lift ϕ . (cid:3) Proposition 4.23 shows in particular that Z ( W F /P F , ˆ G ) is connected in the casewhere I F acts trivially on ˆ G . We will now show this connectedness property in themore general case where I F preserves a pinning of ˆ G . The next lemma starts witha particular subcase. Lemma 4.25.
Assume that ˆ G is semi-simple and simply connected, and that I F stabilizes a pinning of ˆ G . Then, for any subgroup I of finite index of I F , the closedsubgroup scheme ˆ G I has connected geometric fibers and is smooth over Spec( Z [ p ]) .Proof. The connectedness of geometric fibers is Steinberg’s theorem in [Ste68, Thm8.2], so we focus on the smoothness of ˆ G I . Consider the action of I on the set ofsimple factors of ˆ G . By treating distinct I -orbits seperately, we may assume that I has a single orbit, so that ˆ G = ind II ′ ˆ G ′ where ˆ G ′ is simple and I ′ has finiteindex in I . Then ˆ G I = ( ˆ G ′ ) I ′ , so we are reduced to the case where ˆ G is simple.In this case, the image of I in Aut( ˆ G ) has order either 2 or 3. If it has order 2,the smoothness of ˆ G I over Z [ p ] follows from Lemma A.1. If it has order 3 thenˆ G = Sp . One might probably compute explicitly that ˆ G I = G . Alternatively,the big cell C = ˆ U − ˆ T ˆ U associated to the Borel pair ( ˆ T , ˆ B ) is stable under I , and it suffices to prove smoothness of C I = ( ˆ U − ) I ˆ T I ˆ U I . Since ˆ G is simply connected,ˆ T I is a torus, hence is smooth. By symmetry, it remains to prove smoothness ofˆ U I . Choose an ordering of the set Φ + /I of I -orbits of positive roots and, for eachorbit ¯ α ∈ Φ + /I , choose an ordering of this orbit. To these choices is associateda decomposition ˆ U = Q ¯ α ∈ Φ + /I ˆ U ¯ α with ˆ U ¯ α = Q α ∈ ¯ α ˆ U α . Now, the point here isthat there is no pair of I -conjugate positive roots whose sum is again a root. Thisimplies that ˆ U α and ˆ U α ′ commute with each other if α, α ′ ∈ ¯ α , and it follows thatˆ U I = Q ¯ α ˆ U I ¯ α . This also implies that the I -invariant pinning ( X α ) α ∈ ∆ (with X α a basis of Lie( ˆ U α )) can be extended to an I -invariant pinning ( X α ) α ∈ Φ + for allpositive roots. Then, to each X α corresponds an isomorphism G a ∼ −→ ˆ U α and theproduct of these isomorphisms induces ( G a ) diag ∼ −→ ˆ U I ¯ α . Whence the smoothnessof ˆ U I ¯ α . (cid:3) Remark 4.26. (1) The same lemma holds with “adjoint” instead of “simply con-nected”. The reference to Steinberg’s result has to be replaced by a reference to[DM94, Remarque 1.30] for example.(2) If ˆ G = SL n +1 with a topological generator of I F acting by the non-trivialautomorphism that preserves the standard pinning, then (SL n +1 ) I F is not smoothover Z (2) . For example, with standard coordinates x = x , y = x , z = x forthe upper unipotent subgroup ˆ U of SL , the invariants ˆ U I are given by equations x = y and xy = 2 z . However, it is likely that in any simple simply connected casenot of type A n , the I -invariants are smooth over Z .This lemma, together with Proposition 4.23, shows that Z ( W F /P F , ˆ G ) ¯ Z [ p ] isconnected for ˆ G as in the lemma. In order to spread a bit this result, we will usethe next two lemmas. Lemma 4.27.
Assume given another split reductive group ˆ G ′ over Z [ p ] equippedwith an action of W F /P F and with an equivariant surjective morphism ˆ G ′ f −→ ˆ G whose kernel is a torus. Then the morphism Z ( W F /P F , ˆ G ′ ) f ∗ −→ Z ( W F /P F , ˆ G ) induces a surjection on π .Proof. Put ˆ S := ker f . For any prime ℓ = p and ϕ ∈ Z ( W F /P F , ˆ G (¯ F ℓ )), theobstruction to lifting ϕ to an element of Z ( W F /P F , ˆ G ′ (¯ F ℓ )) lies in the group H ( W F /P F , ˆ S (¯ F ℓ )), which vanishes by Lemma 3.8 since ˆ S (¯ F ℓ ) is a divisible group.Therefore the map f ∗ : Z ( W F /P F , ˆ G ′ (¯ F ℓ )) −→ Z ( W F /P F , ˆ G (¯ F ℓ ))is surjective. Since any connected component of Z ( W F /P F , ˆ G ) has ¯ F ℓ -points forsome ℓ (and even for all ℓ ), this implies the lemma. (cid:3) Lemma 4.28.
There exists a split reductive group ˆ G ′ over Z [ p ] equipped with anaction of W F /P F and an equivariant surjective morphism ˆ G ′ −→ ˆ G whose kernelis a torus, and such that, for all open subgroups I of I F , we have : (1) ( ˆ G ′ ) I has geometrically connected fibers, and (2) ( ˆ G ′ ) I is smooth over Z [ p ] if I F preserves a pinning of ˆ G .Proof. By Theorem 5.3.1 of [Con14] (or [SGA3-III, Exp XXII, § G der of ˆ G over Z [ p ] that represents ODULI OF LANGLANDS PARAMETERS 41 the fppf sheafification of the set-theoretical derived subgroup presheaf. Further, byExercise 6.5.2 of [Con14], there is a canonical central isogeny ˆ G sc −→ ˆ G der over R ,such that all the geometric fibers of ˆ G sc are simply connected semi-simple groups.Being canonical, the action of W F /P F on ˆ G der lifts uniquely to ˆ G sc and still pre-serves a Borel pair or a pinning, depending on the case. Now denote by R ( ˆ G ) theradical of ˆ G , which is a split torus. Then the natural morphism R ( ˆ G ) × ˆ G sc −→ ˆ G is a W F -equivariant central isogeny. We already know that ( ˆ G sc ) I satisfies prop-erties (1) and (2) for any subgroup I ⊂ I F , by Lemma 4.25. On the other hand, R ( ˆ G ) I is the diagonalisable group associated to the abelian group X ∗ ( R ( ˆ G )) I of I -coinvariants in X ∗ ( R ( ˆ G )), which may have torsion. Let W be a finite quotient of W F /P F through which the action of W F on ˆ G factors. Choosing a dual basis of thelattice X ∗ ( R ( ˆ G )), we get a W F -equivariant embedding X ∗ ( R ( ˆ G )) ֒ → Z [ W ] dim R ( ˆ G ) where the target has torsion-free I -coinvariants for all I . Dually we get a surjec-tive W F -equivariant morphism of tori ˆ S ։ R ( G ) such that ( ˆ S ) I is a torus, hence issmooth with connected geometric fibers. Thus we have a W F -equivariant surjectivemorphism ˆ G ′′ := ˆ S × ˆ G sc ։ ˆ G whose source satisfies both properties (1) and (2),but whose kernel D , a diagonisable subgroup, is not necessarily a torus.Now let us choose a surjective W F -equivariant morphism X −→ X ∗ ( D ) suchthat X is a permutation module (i.e. W F permutes a Z -basis of X ) and such thatfor every I F ⊂ I ⊂ I F , the map on I -invariants X I −→ X ∗ ( D ) I is surjective. Forexample, one can take X = L I Z [ W/I ] ⊗ Y I where Y I is any free abelian groupmapping surjectively to X ∗ ( D ) I . Dually, we have a W F -equivariant embedding D ֒ → ˆ S ′ of D into the split torus ˆ S ′ over Z [ p ] with character group X ∗ ( ˆ S ′ ) = X . Since its character group is a permutation module, the torus ˆ S ′ satisfies bothproperties (1) and (2). Namely, ( ˆ S ′ ) I is a torus for all I ⊂ I F . Now, by [SGA3-III,Exp. XXII § G ′ := ˆ S ′ × D ˆ G ′′ is representable by a split reductivegroup scheme, which by construction is a W F -equivariant extension of ˆ G by thetorus ˆ S ′ . Let us prove that ˆ G ′ has property (1). Fix a prime ℓ = p and look at theexact sequence( ˆ S ′ (¯ F ℓ ) × ˆ G ′′ (¯ F ℓ )) I −→ ˆ G ′ (¯ F ℓ ) I −→ H ( I, D (¯ F ℓ )) −→ H ( I, ˆ S ′ (¯ F ℓ ) × ˆ G ′′ (¯ F ℓ )) . The map H ( I, D (¯ F ℓ )) −→ H ( I, ˆ S ′ (¯ F ℓ )) identifies with the mapHom( X ∗ ( D ) I , ¯ F × ℓ ) −→ Hom( X ∗ ( ˆ S ′ ) I , ¯ F × ℓ ) , hence it is injective since the map X ∗ ( S ′ ) I −→ X ∗ ( D ) I is surjective. It followsthat the map ( ˆ S ′ (¯ F ℓ ) × ˆ G ′′ (¯ F ℓ )) I −→ ˆ G ′ (¯ F ℓ ) I is surjective. But the source of this map is a connected variety since X ∗ ( S ′ ) is apermutation module, so the target is also connected as desired. Let us now provethat ˆ G ′ has property (2). So we assume that I F preserves a pinning of ˆ G , whichimplies that it also preserves a pinning of ˆ G ′′ and ˆ G ′ , and we shall prove smoothnessof ( ˆ G ′ ) I over Z [ p ]. Since the big cell ( ˆ U ′− ˆ T ′ ˆ U ′ ) I = ( ˆ U ′− ) I ( ˆ T ′ ) I ( ˆ U ′ ) I is non-emptyand ( ˆ G ′ ) I is connected, it suffices to prove its smoothness. We already know fromLemma 4.25 that ( ˆ U ′− ) I and ( ˆ U ′ ) I are smooth over Z [ p ] so we may concentrateon the diagonalisable subgroup ( ˆ T ′ ) I . But, by the same argument as for property(1) above, this diagonalisable group has geometrically connected fibers, so this is a torus (because the base is connected and has fibers of at least two distinct residualcharacteristics), and in particular it is smooth. (cid:3) Putting the last two lemmas and Proposition 4.23 together, we get the followingresult.
Theorem 4.29. If I F preserves a pinning of ˆ G , the scheme Z ( W F /P F , ˆ G ) ¯ Z [ p ] isconnected. Corollary 4.30.
If the center of ˆ G is smooth, then Z ( W F /P F , ˆ G ) ¯ Z [ p ] is connected.Proof. Indeed, in this case there is ϕ ∈ Z ( W F /P F , ˆ G )(¯ Z [ p ]) such that Ad ϕ pre-serves a pinning (see Remark 3.9), and right multiplication by ϕ provides an iso-morphism Z ϕ ( W F /P F , ˆ G ) ¯ Z [ p ] ∼ −→ Z ( W F /P F , ˆ G ) ¯ Z [ p ] . (cid:3) Unobstructed points
As a consequence of Theorem 4.1, if L is a “good coefficient field” (i.e. either ithas finite characteristic ℓ = p , or it contains Q ℓ ), then for any L -point x : Spec L → Z ( W F /P eF , ˆ G ), the 1-cocycle ϕ x extends uniquely to an ℓ -adically continuous 1-cocycle still denoted by ϕ x : W F /P eF → ˆ G ( L ).5.1. Deformation Theory.
We fix a good coefficient field L and an L -point x of Z ( W F /P eF , ˆ G ), and we now study the tangent space T x Z ( W F /P eF , ˆ G L ). Wewill need the L -linear ℓ -adically continuous representation Ad ϕ x of W F on the Liealgebra Lie( ˆ G L ) obtained by composing L ϕ x with the adjoint representation of G L .Recall that an element of T x Z ( W F /P eF , ˆ G L ) is given by a map ˜ x : Spec L [ ǫ ] /ǫ → Z ( W F /P eF , ˆ G ) whose composition with the natural map Spec L → Spec L [ ǫ ] /ǫ isequal to x . In particular, the zero element ˜ x of T x Z ( W F /P eF , ˆ G L ) is given by thecomposition of x with the natural map Spec L [ ǫ ] /ǫ → Spec L . Note that the 1-cocycle ϕ ˜ x : W F −→ ˆ G ( L [ ǫ ] /ǫ ) defined by ˜ x extends again uniquely to an ℓ -adicallycontinuously cocycle on W F , still denoted by ϕ ˜ x .Given such a ˜ x we form a cocycle for Ad ϕ x as follows: for each w ∈ W F ,the element L ϕ ˜ x ( w ) L ϕ ˜ x ( w ) − is a tangent vector to ˆ G at the identity element ofˆ G ( L ); that is, an element of Lie( ˆ G L ). In this way one obtains a continuous 1-cocycle(crossed homomorphism) in Z ( W F , Ad ϕ x ). Moreover, this gives an isomorphismof T x Z ( W F /P eF , ˆ G L ) with Z ( W F , Ad ϕ x ). We thus obtain: Proposition 5.1.
Let L be a good coefficent field and x : Spec L → Z ( W F /P eF , ˆ G ) an L -valued point. Then the dimension of T x Z ( W F /P eF , ˆ G L ) over L is equal to dim ˆ G + dim H ( W F , (Ad ϕ x ) ⊗ ω ) , where ω is the cyclotomic character of W F .Proof. We have seen that the dimension of T x Z ( W F /P eF , ˆ G L ) is equal to that of Z ( W F , Ad ϕ x ). The latter is equal to the dimension of H ( W F , Ad ϕ x ) plus thedimension of the space of coboundaries (principal crossed homomorphisms). Theseare all of the form w wy − y , where y is an element of Ad ϕ x . The dimension ofAd ϕ x is equal to dim ˆ G , and those y that give the zero element of Z ( W F , Ad ϕ x )are precisely those fixed by W F . Thus we havedim Z ( W F , Ad ϕ x ) = dim H ( W F , Ad ϕ x ) + dim ˆ G − dim H ( W F , Ad ϕ x ) . ODULI OF LANGLANDS PARAMETERS 43
Now W F has cohomological dimension two, and it is easy to verify that everyirreducible representation of W F has Euler characteristic zero. Thus for any repre-sentation V of W F , one has dim H ( W F , V ) − dim H ( W F , V )+dim H ( W F , V ) = 0.It follows that dim T x Z ( W F /P eF , ˆ G L ) = dim G + dim H ( W F , Ad ϕ x ). The claimthen follows by Tate local duality and self-duality of Ad ϕ x . (cid:3) We can deal with the obstruction theory in a similar way. Let A be a fi-nite length local L -algebra with residue field L , ˜ x : Spec A → Z ( W F /P eF , ˆ G L )a map whose composition with the map Spec L → Spec A is equal to x , and let A ′ be a small extension of A ; that is, a finite length local L -algebra with residuefield L , and an ideal I ⊆ A ′ such that I is annihilated by the maximal ideal of A ′ , and an isomorphism A ′ /I ∼ = A . We consider the problem of lifting ˜ x to amap ˜ x ′ : Spec A ′ → Z ( W F /P eF , ˆ G L ). This is equivalent to lifting the 1-cocycle ϕ ˜ x : W F /P eF −→ ˆ G ( A ) to a 1-cocycle ϕ ˜ x ′ : W F /P eF −→ ˆ G ( A ′ ). This prob-lem is standard: let ϕ ′ be any lift of ϕ ˜ x to a continuous function (not neces-sarily a homomorphism): W F → ˆ G ( A ′ ). Then the map taking w , w ∈ W F to L ϕ ′ ( w w ) − L ϕ ′ ( w ) L ϕ ′ ( w ) is a 2-cocycle with values in (Ad ϕ x ) ⊗ I , and we canadjust our choice of ϕ ′ to yield a 1-cocycle ϕ ˜ x ′ lifting ϕ ˜ x if, and only if, this 2-cocycleis a coboundary. We thus obtain an obstruction theory for Z ( W F /P eF , ˆ G L ) in aformal neighborhood of x with values in H ( W F , Ad ϕ x ).As a formal consequence, we obtain: Proposition 5.2.
The complete local ring of Z ( W F /P eF , ˆ G L ) at x has a presen-tation with dim ˆ G + dim H ( W F , Ad ϕ x ⊗ ω ) topological generators and at most dim H ( W F , Ad ϕ x ⊗ ω ) relations. (In particular, if H ( W F , Ad ϕ x ⊗ ω ) = 0 , then x is a regular point of X G L .)Proof. This follows immediately from the above discussion once one uses Tate localduality and self-duality of Ad ϕ x to conclude that the dimensions of H ( W F , Ad ϕ x )and H ( W F , Ad ϕ x ⊗ ω ) are the same. (cid:3) Unobstructed points.
Let L be an algebraically closed good coefficient field.We have seen that an L -point x of Z ( W F /P eF , ˆ G ) is a smooth point of the fiber Z ( W F /P eF , ˆ G L ) if, and only if, H ( W F , Ad ϕ x ⊗ ω ) vanishes. We will call suchpoints of Z ( W F /P eF , ˆ G L ) unobstructed points .The primary goal of this section is to show that if the characteristic ℓ of L doesnot lie in an explicit finite set (depending only on G L ), the fiber Z ( W F /P eF , ˆ G L ) isgenerically smooth. We have already established this smoothness in characteristiczero, so we assume henceforth that L has finite characteristic ℓ . In this case, therestriction map Z ( W F /P eF , ˆ G L ) ∼ −→ Z ( W F /P eF , ˆ G L ) is an isomorphism, so thereis no need to distinguish between W F and W F . Notation 5.3.
Let ϕ : W F → ˆ G ( L ) be a continuous 1-cocycle. Denote by τ the restriction of ϕ to I F , and let ˆ G τ be the centralizer of τ in ˆ G . Then for any g ∈ ˆ G τ ( L ) there is a unique continuous 1-cocycle ϕ g , whose restriction to I F is τ ,and such that ϕ g (Fr) = gϕ (Fr).The rough version of our main result here is : Theorem 5.4.
There is a finite set of primes S depending only on G L such that,if ℓ / ∈ S , then for any continuous -cocycle ϕ : W F → ˆ G ( L ) , there exists a g ∈ ( ˆ G τ ) ◦ ( L ) such that ϕ g is unobstructed, where τ is the restriction of ϕ to I F . In order to state a more precise version, we need notations (B.2) and (B.3) ofthe appendix. In particular, h ˆ G, is the Coxeter number of the root system of ˆ G . Theorem 5.5.
Let Fr be a lift of Frobenius in W F , and denote by e the tameramification index of the finite extension of F whose Weil group is the kernel of W F −→ Out( ˆ G ) . Then the set S in Theorem 5.4 can be taken as(1) S = { primes ℓ dividing e.χ ∗ ˆ G, Fr ( q ) } , whatever ˆ G is.(2) S = { primes ℓ dividing e.χ ˆ G, Fr ( q ) . ( h ˆ G, )! } if ˆ G has no exceptional factor. Here, “exceptional” includes triality forms of D . Note also that ℓ not dividing χ ∗ ˆ G, Fr ( q ) is equivalent to q having order greater than h ˆ G, Fr in F × ℓ , which implies ℓ > h ˆ G, Fr hence also ℓ > h ˆ G, . We will also prove that, in the case where ˆ G has noexceptional factor and the action of W F is unramified and ℓ > h ˆ G, , the condition χ ˆ G, Fr ( q ) = 0 in F ℓ is also necessary to have generic smoothness.We now start the proofs of Theorems 5.4 and 5.5. Fix a ϕ and τ as in Theorem5.4; our first step will be to reduce to a setting in which the action of W F on ˆ G isunramified and stabilizes a pinning, and the image of τ is unipotent.Denote by φ ℓ the restriction of ϕ to I ℓF and by α ℓ the composition W F ϕ −→ C G L ( φ ℓ ) −→ ˜ π ( φ ℓ ), so that ϕ lies in the closed subscheme Z ( W F /P eF , ˆ G L ) φ ℓ ,α ℓ ,as defined in subsection 4.2. Then, according to Theorem 4.8, the connected com-ponent of Z ( W F /P eF , ˆ G L ) that contains ϕ has the formˆ G × C ˆ G ( φ ℓ ) ◦ Z ( W F /P eF , ˆ G L ) φ ℓ ,α ℓ . Thus we see that ϕ is a smooth point of Z ( W F /P eF , ˆ G L ) if and only if it is asmooth point of Z ( W F /P eF , ˆ G L ) φ ℓ ,α ℓ .By Theorem 4.6, there exists ϕ ′ ∈ Z ( W F /P eF , ˆ G ( L )) φ ℓ ,α ℓ such that the actionof W F on C ˆ G ( φ ℓ ) ◦ via Ad ϕ ′ preserves a Borel pair. Actually, we a have betterresult in this setting : Proposition 5.6.
We can choose ϕ ′ ∈ Z ( W F /P eF , ˆ G ( L )) φ ℓ ,α ℓ so that the actionof W F on C ˆ G ( φ ℓ ) ◦ via Ad ϕ ′ preserves a pinning.Proof. We take up the proof of Proposition 3.7, replacing P F by I ℓF and “Borel pair”by “pinning”. Since the fixator of a pinning of C ˆ G ( φ ℓ ) ◦ under conjugation is thecenter Z of C ˆ G ( φ ℓ ) ◦ , the argument of that proof shows that the obstruction to theexistence of a cocycle ϕ ′ as in this proposition lies in the group H ( W F /I ℓF , Z ( L )).On the other hand, repeating the proof of Lemma 3.8 for W F /I ℓF instead of W F /P F shows that this cohomology group vanishes if Z ( L ) can be proved to be ℓ -divisible.To prove this, recall that the group scheme Z is diagonalizable and let M be itscharacter group. Then we have a (non canonical) decomposition M ≃ M ℓ − tors × M ℓ ′ − tors × M free which induces a decomposition Z ≃ Z ℓ × Z ℓ × T Z , where T Z isa torus, Z ℓ is finite smooth and Z ℓ is finite infinitesimal. Correspondingly we get Z ( L ) ≃ Z ℓ ( L ) × T Z ( L ). Now, T Z ( L ) is clearly ℓ -divisible since L is algebraicallyclosed and Z ℓ ( L ) has prime-to- ℓ order hence is also ℓ -divisible. (cid:3) ODULI OF LANGLANDS PARAMETERS 45
Choose ϕ ′ as in this proposition and recall that the action Ad ϕ ′ on C ˆ G ( φ ℓ ) factorsover the quotient W F /I ℓF . Then we have an isomorphism η η · ϕ ′ Z ϕ ′ ( W F /I ℓF , C ˆ G ( φ ℓ ) ◦ ) ∼ −→ Z ( W F /P eF , ˆ G L ) φ ℓ ,α ℓ , The isomorphism above shows that ϕ is an unobstructed point of Z ( W F /P eF , ˆ G L )if, and only if, ϕ · ϕ ′− is an unobstructed point of Z ϕ ′ ( W F /I ℓF , C ˆ G ( φ ℓ ) ◦ ). Sowe are reduced to study unobstructedness in a much simpler case, but in order tomake this reduction step effective, we need some control on Ad ϕ ′ . Lemma 5.7.
Fix ϕ ′ as in Proposition 5.6, let w ∈ W F and denote by o w its orderin Out( ˆ G ) . Then Ad ϕ ′ ( w ) has order dividing o w | Ω ˆ G | .Proof. Put H := C ˆ G ( φ ℓ ) ◦ and let T H be a maximal torus of H that is part ofa pinning stable under Ad ϕ ′ . Pick a maximal torus T of ˆ G that contains T H .Then there is an element m of the centralizer of T H in ˆ G such that g w := mϕ ′ ( w )normalizes T . The action of ( g w ) o w on X ∗ ( T ) is the action of an element of ˆ G thatnormalizes T , so its order divides | Ω ˆ G | . Hence, Ad ϕ ′ ( w ) o w | Ω ˆ G | acts trivially on T H and therefore also on H , since it stabilizes a pinning and fixes the maximal torusof this pinning. (cid:3) As in Theorem 5.5, denote by e the tame ramification index of the finite extensionof F whose Weil group is the kernel of W F −→ Out( ˆ G ). Applying this lemma toa suitable lift of our generator s of tame inertia, we see that, if we assume that ℓ is prime to e | Ω ˆ G | , then the action Ad ϕ ′ is unramified . For this reason, we will nowfocus on the following particular setting :(5.1) (cid:26) – the action of W F on ˆ G is unramified and stabilizes a pinning,– the restriction of ϕ to I ℓF is trivial.In this setting, G L is the Langlands dual group of a uniquely determined quasi-split unramified reductive group G over F , and the restriction τ of ϕ to I F isdetermined by u := τ ( s ) which is a unipotent element of ˆ G . We will generallydenote by ˆ G u = ˆ G τ the centralizer of u in ˆ G . Our first tool to tackle this setting isthe following lemma. Lemma 5.8.
Let ˆ G and ϕ be as in (5.1). Assume further given a W F -equivariantcentral isogeny ˆ G π −→ ˆ G ′ and put ϕ ′ := π ◦ ϕ . Then there is g ∈ ( ˆ G τ ) ◦ such that ϕ g is unobstructed if, and only if, there is g ′ ∈ ( ˆ G ′ τ ′ ) ◦ such that ϕ ′ g ′ is unobstructed.Proof. Recall first that π induces an isomorphism from the unipotent subvariety ofˆ G to that of ˆ G ′ . In particular, the map ˆ G u π −→ ˆ G ′ π ( u ) is surjective for all unipotent u ∈ ˆ G ( L ), and so is the map ( ˆ G u ) ◦ π −→ ( ˆ G ′ π ( u ) ) ◦ . Therefore, it suffices to provethat ϕ is unobstructed if and only if ϕ ′ is unobstructed. If π is separable, then dπ induces a (Ad ϕ, Ad ϕ ′ )-equivariant isomorphism Lie( ˆ G ) ∼ −→ Lie( ˆ G ′ ), and thedesired property follows. If π is inseparable, dπ is still (Ad ϕ, Ad ϕ ′ )-equivariant butit is not an isomorphism. Indeed, ker π is an infinitesimal subgroup of ˆ G containedin any maximal torus ˆ T and ker dπ is its Lie algebra. However, if we choose n suchthat ker π is contained in the “ n -th kernel of Frobenius” ker(Frob n ˆ G ), then there isunique (non-central) inseparable isogeny ˆ G ′ π ∗ −→ ˆ G such that π ∗ ◦ π = Frob n ˆ G and π ◦ π ∗ = Frob n ˆ G ′ . Then we have dπ ∗ ◦ dπ = dπ ◦ dπ ∗ = 0 and it follows by dimension count that dπ ∗ induces an isomorphism coker dπ ∼ −→ ker dπ . Similarly, π restricts toan isogeny π u : ˆ G u −→ ˆ G ′ π ( u ) and Lie ˆ G ′ π ( u ) is an extension of coker dπ u ≃ ker dπ u by im dπ u ≃ coim dπ u while Lie ˆ G u is an extension in the other way around. Now, ϕ being unobstructed is equivalent to Ad ϕ (Fr) having no eigenvalue equal to q − on Lie ˆ G u which, by our discussion, is equivalent to Ad ϕ ′ (Fr) having no eigenvalueequal to q − on Lie ˆ G ′ π ( u ) , which is equivalent to ϕ ′ being unobstructed. (cid:3) For ˆ G as in (5.1), let us apply this lemma to the central isogeny ˆ G −→ ˆ G ad × ˆ G ab .Dealing with the toric part is quite easy : Lemma 5.9. If ˆ G as in (5.1) is a torus, then the following are equivalent : (1) there is an unobstructed ϕ in Z ( W F /I ℓF , ˆ G L ) , (2) any ϕ in Z ( W F /I ℓF , ˆ G L ) is unobstructed, (3) χ ˆ G, Fr ( q ) = 0 in L .Proof. For any ϕ in Z ( W F /I ℓF , ˆ G L ), we have ϕ ( s ) = 1 and ϕ (Fr) ∈ ˆ G ( L ) · Fr,so the condition for ϕ to be unobstructed is that H (Fr , ω ⊗ Lie( ˆ G L )) = 0, whichis independent of ϕ , and equivalent to q − not being an eigenvalue of Ad Fr onLie( ˆ G L ). Since Lie( ˆ G L ) = Hom Z ( X ∗ ( ˆ G ) , L ), we havedet (cid:16) q Ad Fr − id | Lie( ˆ G L ) (cid:17) = det (cid:16) q Fr − − id | X ∗ ( ˆ G ) (cid:17) = χ ˆ G, Fr ( q )so we see that q − is an eigenvalue of Ad Fr if and only if χ ˆ G, Fr ( q ) = 0 in L . (cid:3) Let us now deal with the adjoint part. We have a Fr-equivariant decompositionas a product of simple adjoint groups(5.2) ˆ G ad = ˆ G × · · · × ˆ G f × · · · × ˆ G r × · · · × ˆ G rf r where Fr permutes cyclically ˆ G i → ˆ G i → · · · → ˆ G if i and Fr f i restricts to anouter automorphism of ˆ G i . Accordingly, ϕ decomposes as a product ϕ × · · · × ϕ r and we see that ϕ is unobstructed if and only if for each i = 1 , · · · , r , the cocycle ϕ | W Ffi : W F fi −→ ˆ G i ( L ) is unobstructed, where F f i is the unramified extensionof degree f i of F . We also note the following equality, which is easily checked ondefinitions :(5.3) χ ˆ G, Fr ( T ) = χ ˆ G ab , Fr ( T ) χ ˆ G , Fr f ( T f ) · · · χ ˆ G r , Fr fr ( T f r ) . Therefore, we are reduced to study the case where ˆ G ad is simple. To this aim,we will use the following tool to construct points in Z ( W F /I ℓF , ˆ G ). Assume thatwe are given • a homomorphism λ : SL → ˆ G and • an element F ∈ ( ˆ G ⋊ Fr) λ , i.e. an element of G L that centralizes λ andprojects to Fr.Then there is a unique 1-cocycle ϕ : W F /I ℓF −→ ˆ G ( L ) such that(5.4) ϕ ( s ) = λ ( U ) and L ϕ (Fr) = λ ( S ) F , where S and U denote the matrices (cid:18) q q − (cid:19) and ( ) in SL ( L ), respectively,and where q is a choice of a square root of q in L . ODULI OF LANGLANDS PARAMETERS 47
However, we will need a condition to ensure exhaustivity of this construction.Recall that over characteristic zero fields, for any unipotent element u in ˆ G thereis a homomorphism λ : SL → ˆ G such that λ ( U ) = u and, moreover, λ is uniqueup to ˆ G -conjugacy. In finite characteristic ℓ , the situation is more subtle. Anobvious necessary condition for the existence of λ is that u have order ℓ . When ℓ is good for ˆ G , this was proven to be sufficient by Testerman in [Tes95]. In order tostudy uniqueness, Seitz [Sei00] has introduced the following notion : a morphism λ : SL → ˆ G L over L is a “good SL ” if the weights of the conjugation action of themaximal torus T ⊂ SL on Lie( ˆ G ) are bounded above by 2 ℓ − T to G m via the map z (cid:0) z z − (cid:1) ). Theorem 5.10 ([Sei00], Theorems 1.1 and 1.2) . Suppose ℓ is a good prime for ˆ G , and u is a unipotent element of ˆ G ( L ) of order ℓ . Then there is a “good SL ” λ : SL → ˆ G such that λ ( U ) = u . Moreover, any two such λ are conjugate in ˆ G ,by an L -point of the unipotent radical R u ( ˆ G u ) . Finally, the centralizer ˆ G λ of λ in ˆ G is reductive, and ˆ G u = ˆ G λ R u ( ˆ G u ) . In order to ensure that all non-trivial unipotent elements of ˆ G ( L ) have order ℓ ,we will henceforth assume that ℓ > h , where h = h ˆ G, is the Coxeter number of ˆ G . Indeed, since h is one plus the height of the highest positive root of ˆ G , it follows fromProposition 3.5 of [Sei00] and the Bala-Carter classification, that any nontrivialunipotent element of ˆ G ( L ) has order ℓ under this hypothesis. Moreover, such an ℓ is also automatically good for ˆ G (and even “very good”), so that Seitz’ theoremapplies to any u under this hypothesis. Corollary 5.11.
Let ˆ G and ϕ be as in (5.1) and suppose that ℓ > h ˆ G, . Then thereis g ∈ ( ˆ G u ) ◦ such that ϕ g is of the form (5.4) associated to a pair ( λ, F ) such that F normalizes a Borel pair (or even a pinning) of ( ˆ G λ ) ◦ .Proof. Let us choose a “good SL ” λ : SL → ˆ G with λ ( U ) = u := ϕ ( s ). Set F := λ ( S ) − ϕ (Fr). Then F ∈ ˆ G ⋊ Fr centralizes u , so F λ is a second “good SL ”that takes U to u . Since any two such are conjugate by an element centralizing u ,we have a unipotent element u ′ ∈ R u ( ˆ G u ) such that u ′ λ = F λ ; then F = u ′− F centralizes λ and, in particular, normalizes ( ˆ G λ ) ◦ . Choose a pinning ε in ( ˆ G λ ) ◦ ;then there exists h ∈ ( ˆ G λ ) ◦ such that h ε = F ε . Then F := h − F still centralizes λ , and preserves ε . Now, ϕ (Fr) = λ ( S ) u ′ h F = ( λ ( S ) u ′ hλ ( S ) − )( λ ( S ) F )with u ′ in the unipotent radical of ˆ G u and h in ( ˆ G λ ) ◦ . Thus hu ′ lies in ( ˆ G u ) ◦ , andsince λ ( S ) normalizes ( ˆ G u ) ◦ , it follows that λ ( S ) F ∈ ( ˆ G u ) ◦ ϕ (Fr). (cid:3) We now consider a particular case, which shows that the condition χ ˆ G, Fr ( q ) = 0in L is necessary for the existence of unobstructed translates. Proposition 5.12.
Let ˆ G be simple adjoint, and assume that ℓ > h ˆ G . Then thereexists ϕ as in (5.1) such that ϕ ( s ) is regular unipotent. Moreover, the followingproperties are equivalent : (1) There is an unobstructed ϕ such that ϕ ( s ) is regular unipotent. (2) Any ϕ with ϕ ( s ) regular unipotent is unobstructed. (3) χ ˆ G, Fr ( q ) = 0 in L .Proof. Fix a pinning ( ˆ
T , ˆ B, ( X α ) α ∈ ∆ ) stable under Fr. The sum E = P α ∈ ∆ X α is a regular nilpotent element of Lie( ˆ G ), which is fixed by Fr. Moreover, the sum H = P β ∈ λ + ˇ β ∈ Lie( ˆ T ) is also fixed by Fr. Then the pair ( H, E ) is part of a uniqueprincipal sl -triple, which is also fixed under Fr. Now, pick a regular unipotent u ∈ ˆ G and a good SL , say λ : SL −→ ˆ G , such that λ ( U ) = u . Then im( dλ ) isanother principal sl -triple, hence is conjugate to ( F, H, E ) by some element g ∈ ˆ G .This means that, after conjugating by some g ∈ ˆ G , we may assume that λ (andtherefore u ) is fixed by Fr, so we can construct ϕ as desired by putting ϕ ( s ) := λ ( U )and ϕ (Fr) := λ ( S ) Fr.If ϕ ′ is another L -homomorphism with ϕ ′ ( s ) regular unipotent, then we mayconjugate it so that ϕ ′ ( s ) = ϕ ( s ) = u , and this does not affect the property of beingunobstructed. Then ϕ ′ (Fr) = ϕ (Fr) g for some g ∈ ˆ G u , and ϕ ′ is unobstructed ifand only if q − is not an eigenvalue of Ad ϕ ′ (Fr) on Lie( ˆ G ) Ad u . But under ourrunning assumption ℓ > h ˆ G , which implies that ℓ is very good for ˆ G , it is knownthat Lie( ˆ G ) Ad u = Lie( ˆ G u ). Moreover ˆ G u is known to be commutative, henceAd ϕ ′ (Fr) = Ad ϕ (Fr) on Lie( ˆ G ) Ad u and we have the equivalence of (1) and (2).It remains to study when q − is an eigenvalue of Ad ϕ (Fr). Observe thatLie( ˆ G ) Ad u coincides with the centralizer Lie( ˆ G ) E of E in Lie( ˆ G ). Moreover, ourhypothesis ℓ > h ˆ G implies that ℓ does not divide the order of the Weyl group Ω ˆ G .Therefore, we can use Kostant’s section theorem as in subsection B.4. In particular,Proposition B.5 tells us thatdet (cid:16) q Ad λ ( S ) Fr − id | Lie( ˆ G ) Ad u (cid:17) = ± χ ˆ G, Fr ( q ) , which shows that ϕ is unobstructed if and only if χ ˆ G, Fr ( q ) = 0 in L . (cid:3) Remark 5.13.
Let F be any automorphism of ˆ G and suppose λ is a F -invariantgood SL such that u = λ ( U ) is regular in ˆ G . Then the same proof shows thatdet (cid:16) q Ad λ ( S ) F − id | Lie( ˆ G ) Ad u (cid:17) = χ ˆ G, F ( q ) . In order to study more general unipotent classes, the following lemma will allowus to use inductive arguments.
Lemma 5.14.
Let ˆ G and ϕ be as in (5.1), and let ˆ S be a torus in the centralizer C ˆ G ( ϕ ) of ϕ . Then : (1) ∃ h ∈ ˆ G such that ˆ M := hC ˆ G ( ˆ S ) h − is a Fr -stable Levi subgroup of ˆ G . (2) If the h -conjugate h ϕ is unobstructed in Z ( W F /I ℓF , ˆ M ) , then there is g ∈ ( ˆ G u ) ◦ such that ϕ g is unobstructed.Proof. (1) The centralizer C G L ( ˆ S ) contains L ϕ (Fr), hence it surjects onto π ( G L ).By [Bor79, Lemma 3.5], it is a “Levi subgroup” of G L in Borel’s sense. It is thusconjugate by some h ∈ ˆ G to the standard Levi subgroup of a standard parabolicsubgroup of G L . Such standard Levi subgroups are of the form L M = ˆ M ⋊ h Fr i .(2) Since unobstructedness is invariant by conjugacy, we may and will assumethat h = 1. Then observe that L ϕ factors indeed through L M , and also that Lie( ˆ M )is the weight 0 subspace of Lie( ˆ G ) in the decomposition Lie( ˆ G ) = L κ ∈ X ∗ ( ˆ S ) Lie( ˆ G ) κ ODULI OF LANGLANDS PARAMETERS 49 of Lie( ˆ G ) as a sum of weight spaces for the adjoint action of ˆ S . So, for any element s ∈ ˆ S , unobstructedness of ϕ s is equivalent to q − not being an eigenvalue of L ϕ (Fr) s on each Lie( ˆ G u ) κ . For κ = 0, this property is fulfilled by our hypothesis,since s acts trivially on Lie( ˆ G u ) . For any other κ , this property is fulfilled for s outside a proper Zariski closed subset of ˆ S , because s commutes with L ϕ (Fr).Therefore we can find s that works for all κ . (cid:3) In the case where ϕ is given by a pair ( λ, F ) as in Lemma 5.11, we will say that ϕ – or ( λ, F ) – is discrete if C ˆ G ( ϕ ), or equivalently C ˆ G ( λ ) F , contains no non-centraltorus of ˆ G . If ϕ ( s ) = λ ( U ) is a distinguished unipotent element, then ϕ is certainlydiscrete. The converse is not always true, but we note that if C ˆ G ( λ ) has positive semisimple rank, then ϕ is not discrete.In the next proposition, we include triality forms of D (i.e. any group of type D with action of Fr of order 3) in the “exceptional types”. Proposition 5.15.
Let ˆ G be as in (5.1) with no simple factor of exceptional type.Assume that ℓ > h ˆ G, , i.e. ℓ is greater than the Coxeter numbers of the simplefactors of ˆ G . Then the following are equivalent : (1) For all ϕ as in (5.1), there exists g ∈ ( ˆ G ϕ ( s ) ) ◦ such that ϕ g is unobstructed. (2) χ ˆ G, Fr ( q ) = 0 in L .Proof. The implication (1) ⇒ (2) follows from Proposition 5.12, so we focus on theother implication. By Lemma 5.8, Lemma 5.9, decomposition (5.2) and equality(5.3), we may assume that ˆ G is a classical group GL n , Sp n or SO N . In each case,we may assume that ϕ is given by a pair ( λ, F ) as in Corollary 5.11. Moreover,Lemma 5.14, Proposition B.3 (2) and an inductive argument allow us to restrictattention to discrete pairs ( λ, F ). Case ˆ G = GL N with Fr = id . Let V be an L -vector space of dimension N , and λ : SL −→ GL( V ) a morphism. Since ℓ > N , the SL -module V is semi-simpleand, for any d ≤ N , the d -dimensional representation S d = Sym d − ( L ) of SL ( L )is irreducible. Let V d be the S d -isotypic part of V . We then have decompositions V = L d ≥ V d and S d ⊗ W d ∼ −→ V d , where W d := Hom SL ( S d , V d ). In particular, weget that GL( V ) λ = Q d GL( W d ). Since this is a connected group, we may assumethat F = 1. Then we see that ( λ,
1) is discrete if and only λ is principal, i.e. u = λ ( U ) is regular. In this case we conclude thanks to Proposition 5.12.We now assume that V is endowed with a non-degenerate bilinear form of sign ε and we denote by I ( V ) the isometry group, so that I ( V ) ≃ Sp N if ε = − I ( V ) ≃ O N if ε = 1. We take up the above notations, assuming that λ factors through I ( V ). Then each V d is a non-degenerate subspace of V and thedecomposition V = L d V d is orthogonal. Further, each S d carries a natural non-degenerate bilinear form of sign ( − d − such that SL acts through I ( S d ). Then W d inherits a non-degenerate form of sign ( − d − ε such that the isomorphism S d ⊗ W d ∼ −→ V d is compatible with the tensor product form. It follows in particularthat I ( V ) λ = Q d I ( W d ). Writing r d := dim( W d ), we have(1) I ( W d ) ≃ O r d ≃ SO r d ⋊Z / Z if ( − d − ε = 1.(2) I ( W d ) ≃ Sp r d if ( − d − ε = − π ( I ( V ) λ ) admits a section into I ( V ) λ , and we may take F in theimage of such a section, so that F has order at most 2. Moreover, we see that ( I ( V ) λ ) F contains a non-trivial torus whenever there is a symplectic factor (asso-ciated to some d such that ( − d − ε = − W d = 0). Since we may restrictattention to discrete ( λ, F ), we will assume that I ( V ) λ has no symplectic factor.In particular, this fixes the parity of the d ’s such that V d = 0.We now need to investigate the eigenvalues of q Ad F Ad λ ( S ) on Lie( ˆ G ) Ad u , where u = λ ( U ). We have decompositionsEnd u ( V ) = Y d,d ′ Hom u ( V d , V d ′ ) ≃ Y d,d ′ Hom T ( S d , S d ′ ) ⊗ Hom L ( W d , W d ′ ) . The weights of λ ( T ) on V d are the weights of S d , i.e. d − , d − , · · · , − d , hencethe weights of Ad λ ( T ) on Hom u ( V d , V d ′ ) are the same as those on Hom T ( S d , S d ′ ),i.e. d + d ′ − i for 1 ≤ i ≤ min( d, d ′ ), each one occurring with multiplicity r d r d ′ .In particular, these weights are bounded above by N − d = d ′ (because then d + d ′ ≤ N ) or if d = d ′ ≤ N . On the other hand, there is at most one d > N with W d = 0 and in this case r d = 1. So any weight k > N − λ ( T ) on End u ( V ) iseven and occurs with multiplicity 1. Actually, it is easy to exhibit a weight vector.Namely, put e := d ( λ | G a )(0), which is a nilpotent endomorphism of V . We alsohave e = log( u ) since the logarithm is well defined under our hypothesis ℓ > h .Then e is a weight 2 element of End u ( V ) = End e ( V ) and for any k = 2 k ′ > N − e k ′ generates the subspace of weight k whenever it is non-zero. In otherwords, we have a decompositionEnd u ( V ) = (cid:10) e k (cid:11) k ≥ ⌊ N ⌋ ⊕ End u ( V ) ≤ N − where the last term is the sum of weight spaces of weight ≤ N − τ denote the involution ψ
7→ − ψ ∗ of End( V ) associated with the bi-linear form on V . We have Lie( I ( V )) Ad u = End u ( V ) τ and τ (Hom( V d , V d ′ )) =Hom( V d ′ , V d ). Using the fact that τ ( e k ) = ( − k +1 e k , we get :Lie( I ( V )) Ad u = End u ( V ) τ = (cid:10) e k (cid:11) k ≥ ⌊ N ⌋ , odd ⊕ End u ( V ) τ ≤ N − . Case ˆ G is symplectic or odd orthogonal. In this case, Fr acts trivially and wehave χ ˆ G, Fr ( T ) = Q ⌊ N ⌋ d =1 ( T d − q Ad ϕ (Fr) = q Ad F Ad λ ( S ) on End u ( V ) τ ≤ N − are of the form ± q k for k such that 0 < k ≤ N . For such aneigenvalue to be equal to 1, we need that q be a root of T k −
1, which is a factor of χ ˆ G, Fr ( T ). On the other hand each non-zero e k is an eigenvector of q Ad ϕ (Fr) witheigenvalue q k +1 . Of course e N = 0, so k ≤ N − k must beodd. So k + 1 is even, between 2 and N . Therefore, an eigenvalue q k +1 is 1 in L only if q is a root of χ ˆ G, Fr , as desired. Case ˆ G is even orthogonal. Here we set ˆ G = SO( V ), endowed with an outeraction of Fr of order f = 1 or 2. In these cases, setting N = 2 n , we have χ ˆ G, Fr ( T ) = ( T n + ( − f ) n − Y d =1 ( T d − . We will take advantage of the fact that, when f = 2, we have O( V ) ≃ L SO( V ).A pair ( λ, F ) thus defines a L -homomorphism ϕ for SO( V ) endowed with a trivial,resp. quadratic, action of Fr if det F = 1, resp. if det F = − ODULI OF LANGLANDS PARAMETERS 51
As above, each e k is an eigenvector of q Ad ϕ (Fr) with eigenvalue q k +1 with k + 1even. Moreover we have e N − = 0 (no Jordan matrix of rank N is orthogonal), so k + 1 ≤ N − q k +1 = 1 only if q is a root of χ ˆ G, Fr .Next, the weight spaces with weight < N − u ( V ) τN − of weight N − q Ad ϕ (Fr) on this weight space areof the form ± q n , but we need more precise information since, for example, T n + 1does not divide χ ˆ G, Fr when f = 1, and T n − χ ˆ G, Fr when f = 2.Since Hom u ( V d , V d ′ ) N − is zero unless d + d ′ = N , we have to consider two cases.(1) V = V d ⊕ V d ′ ≃ S d ⊕ S d ′ , with (necessarily) d and d ′ odd and, say d > d ′ . Inthis setting, F belongs to the center {± } × {± } of O( V d ) × O( V d ′ ). Writing F =( ε d , ε d ′ ), we see that F acts on Hom u ( V d , V d ′ ) and Hom u ( V d ′ , V d ) by multiplicationby ε d ε d ′ , and since d and d ′ are odd, we have ε d ε d ′ = det F . Now we haveEnd u ( V ) τN − = h e n − i ⊕ (Hom u ( V d , V d ′ ) ⊕ Hom u ( V d ′ , V d )) τN − . So if det F = 1, the action of F on End u ( V ) τN − is trivial, hence the eigenvalue of q Ad ϕ (Fr) is q n and we are done, since T n − χ ˆ G, Fr when f = 1. Supposenow det F = −
1. Then F acts on the second summand of End u ( V ) τN − by −
1, sothe eigenvalue of q Ad ϕ (Fr) is − q n , which is fine since T n + 1 divides χ ˆ G, Fr when f = 2. On the other hand, F acts trivially on the first summand, but the latter isnon-zero only if n is even, in which case T n − χ ˆ G, Fr .(2) V = V n . Then we may decompose V as an orthogonal sum of two λ (SL )-stable non-degenerate subspaces V = V n ⊕ V n . Moreover, since n has to be odd,( e | V in ) n − is not in End u ( V ) τ so we haveEnd u ( V ) τN − = (cid:0) Hom u ( V n , V n ) ⊕ Hom u ( V n , V n ) (cid:1) τN − . On the other hand, O( V ) λ ≃ O acts on this space through its component group {± } with the non trivial element acting as ψ ψ ∗ . So, in particular, F acts bymultiplication by det F . The eigenvalue of q Ad ϕ (Fr) is thus det F .q n and it equals1 only if q n − det F = 0, hence also only if χ ˆ G, Fr ( q ) = 0. Case ˆ G = GL N and Fr = id . In this case, we have χ ˆ G, Fr = Q Nd =1 ( T d − ( − d ).We continue with the same notations V, λ, u etc, and we assume that there is
F ∈ ( ˆ G ⋊ Fr) λ that fixes a Borel pair of ˆ G λ . Using the explicit description ˆ G λ = Q d GL( W d ), we see that ( λ, F ) is discrete if and only if r d = 1 for all d (so thatˆ G λ = G m × · · · × G m is the center of Q d GL( V d )) and ( ˆ G λ ) F = {± } × · · · × {± } .This implies that F normalizes each GL( V d ) and induces the non-trivial element α d of Out(GL( V d )). Since u | V d is regular, it follows from Proposition 5.12 and thesubsequent remark that no eigenvalue of q Ad F λ ( S ) on End u ( V d ) equals 1 unless χ GL( V d ) ,α d ( q ) = 0 in L , in which case we also have χ ˆ G, Fr ( q ) = 0 by PropositionB.3 (2). Let us now focus on the eigenvalues of q Ad F λ ( S ) on each Hom u ( V d , V d ′ )for d = d ′ . As we have already seen, the eigenvalues of q Ad λ ( S ) are of the form q ( d + d ′ ) − i with 0 ≤ i < min( d, d ′ ). So it remains to understand how F acts. Notethat F ∈ ( ˆ G λ ) F so at least we know that F = 1. We distinguish two cases.(1) Suppose that all the d ’s occuring have the same parity. Then there is a non-degenerate bilinear form on V (symplectic if the d ’s are even, orthogonal if theyare odd) such that u ∈ I ( V ), see [LS12, Cor. 3.6 (2)] for example. We may thenconjugate λ so that it factors through I ( V ). But I ( V ) is the fixed-point subgroup of an involution of the form g ⋊ Fr. So we may take this involution as our element F and we have achieved F = 1. It follows that the eigenvalues of q Ad F λ ( S ) oneach Hom u ( V d , V d ′ ) are of the form ± q k for some integer k ≤ N . Should such aneigenvalue be equal to 1, we would have q k − χ ˆ G, Fr ( q ) = 0.(2) Suppose there are both even and odd d ’s. Write ( F ) d for the component of F in GL( V d ). This is a central element of GL( V d ) equal to ±
1. We then decompose V = V + ⊕ V − where V ± = L d, ( F ) d = ± V d . We have GL( V ) F = GL( V + ) × GL( V − ),hence also GL( V ) F = GL( V + ) F × GL( V − ) F . But F acts on both GL( V − ) andGL( V + ) as an involution that induces the non trivial outer automorphism. So eachGL( V ± ) F is an orthogonal or symplectic group. This implies that all d ’s occurring inthe decomposition of V + , resp. V − , have the same parity (because all multiplicities r d are 1). As a consequence, we see that F acts on Hom u ( V d , V d ′ ) by multiplicationby ( − d + d ′ . So we now have two subcases : • if d, d ′ have the same parity, the eigenvalues of q Ad F λ ( S ) on Hom u ( V d , V d ′ )are of the form ± q k for some even integer k ≤ d + d ′ ≤ N . As before,should such an eigenvalue be equal to 1, we would have q k − χ ˆ G, Fr ( q ) = 0. • if d, d ′ have different parities, the eigenvalues of q Ad F λ ( S ) on Hom u ( V d , V d ′ )are of the form ζq k for some odd integer k ≤ d + d ′ ≤ N and a primitive4 th -root of unity ζ in L . This time, should such an eigenvalue be equal to1, we would have q k + 1 = 0 hence, again, χ ˆ G, Fr ( q ) = 0. (cid:3) It may be tempting to believe that the nice equivalence of Proposition 5.15holds in general. However, it fails in the case of triality, i.e. a group of type D with Frobenius acting with order 3. In this case, the irreducible factors of χ ˆ G, Fr ( T ) = ( T − T − T + T + 1) are Φ n ( T ) for n = 1 , , , ,
12. But toget an equivalence, we need also Φ ( T ). Lemma 5.16.
Assume that ˆ G = SO with Fr of order (triality), and ℓ > h ˆ G = 6 .Then the following are equivalent : (1) For all ϕ as in (5.1), there exists g ∈ ( ˆ G ϕ ( s ) ) ◦ such that ϕ g is unobstructed. (2) χ ′ ˆ G, Fr ( q ) = 0 in L , where χ ′ ˆ G, Fr ( T ) = T − .Proof. As in the proof of Proposition 5.15, we may focus on discrete ( λ, F ). Thereare only three possible types of decomposition associated to such a λ . Either V = V ⊕ V , or V = V ⊕ V or, V = V ⊕ V with V = S and V = S .(1) Type (7 , , λ of type (5 , F ∈ ( ˆ G ⋊ Fr) λ . Sinceˆ G λ = Z ( ˆ G ), we have (Ad F ) = id. On the other hand, the λ ( T )-weights onLie( ˆ G ) Ad u are 2, 4 and 6, so the eigenvalues of q Ad F λ ( S ) are respectively of theform ζq , q or ζq for some 3 rd -root of unity ζ . If any of these numbers equals 1 in L , then q = 1, hence χ ′ ˆ G, Fr ( q ) = 0. However, it is actually possible to prove that q is not an eigenvalue, so that the polynomial χ ˆ G, Fr is still good for this orbit.(3) Type (3 , , , G = ˆ G Fr along its non-regular distin-guished orbit. So we may pick a relevant λ that is centralized by Fr. Then π ( G L λ ) ODULI OF LANGLANDS PARAMETERS 53 is isomorphic to Z / Z and contains two elements such that ( λ, F ) is discrete : Frof order 3, and c Fr of order 6, where c is a reflection that generates π ( ˆ G λ ). Theweights are 0 , q Ad F λ ( S ) on weight 0 and 2 spacesare of the form ζq or ζq for a sixth root of unity ζ , so they are different from 1unless q = 1. On the other hand, the weight 4 space has dimension 1 and comesfrom G . So Fr acts trivially on it, and c Fr acts by ±
1. Hence the correspondingeigenvalue is ± q and is also different from 1 unless q = 1. Now, the computationin G of Remark 5.18 shows that − F = c Fr on the weight2 space, so q Ad F λ ( S ) has eigenvalue − q , and it is different from 1 if and only ifΦ ( q ) = 0. (cid:3) We now turn to the exceptional groups. Recall the polynomials χ ∗ ˆ G, Fr from (B.3). Proposition 5.17.
Suppose that ˆ G is simple of exceptional type. If χ ∗ ˆ G, Fr ( q ) = 0 in L , then for all ϕ as in (5.1), there exists g ∈ ( ˆ G ϕ ( s ) ) ◦ such that ϕ g is unobstructed. Note that χ ∗ ˆ G, Fr ( q ) = 0 is equivalent to “ q has order greater than h ˆ G in L × ”,which implies ℓ > h ˆ G . Proof.
We use the tables in Chapter 11 of [LT11]. These tables cover all the nilpo-tent classes of exceptional groups, including a description of the reductive quotient C of the centralizers (both the neutral component, denoted there by C ◦ and the π ,denoted there by C/C ◦ ), and the weights of an associated cocharacter τ on the Liealgebra centralizer (denoted by m there). Actually, they even describe the weightson each subquotient of the central series of the nilpotent part of the Lie algebracentralizer (with integer n denoting the n th step of the central series).Using a Springer isomorphism e ↔ u between the nilpotent cone and the unipo-tent variety, we get a table of unipotent classes, and we may identify the centralizersˆ G u = ˆ G e . Then for any λ associated to u , we have identifications ˆ G λ ≃ C , hencethe table provides us with descriptions of ( ˆ G λ ) ◦ = C ◦ , π ( ˆ G λ ) = C/C ◦ and theweights m of λ ( T ) on Lie( R u ( ˆ G u )).Thanks to Corollary 5.11 we may focus on ϕ associated to a pair ( λ, F ). Then,using Lemma 5.14 (together with Proposition 5.15 an inductive argument for the E series), we may restrict attention to discrete ( λ, F ). This means that, in the tablesof loc. cit. , we may restrict to classes such that C ◦ is a torus and consider all F ∈ C such that ( C ◦ ) F is finite. In particular, F has necessarily finite order dividing thesquare of the order of the image ¯ F of F in the component group C/C ◦ , and thisorder does only depend on the connected component of C that contains F . So theeigenvalues of q Ad F λ ( S ) = q Ad ϕ (Fr) on Lie( ˆ G ) Ad u are of the form ζq m +1 for someroot of unity ζ whose order t divides the order of F . Therefore, what we have tocheck is that, in all cases, we have t ( m + 1) ≤ h ˆ G, Fr when t ( m + 1) is an integer,or t ( m + 2) ≤ h ˆ G, Fr else.Below we list all “discrete” orbits except the regular ones, which are treated inProposition 5.12. The numbering is that of [LT11, § G , orbit 3. Here h = 6, m = 2 or 4, and C = S , so that t = 1 , t ( m + 1) ≤ h = 6 always hold except if t = 3 and m = 4. Butthis case doesn’t happen since the weight 4 subspace is 1 dimensional, so C = S acts on it via a character, hence via an element of order 2. F , orbit 10. Here h = 12, C = S hence t ≤
4, and the desired inequality holdstrivially for all weights except possibly for weight 6. But the weight 6 subspacehas dimension 2, so the action of C = S factors over a quotient isomorphic to S ,hence t ≤ F , orbit 14. Here h = 12 and C = S , so only the weight m = 14 space mightcontradict the desired inequality, but the column Z ♯ of the table shows that C actstrivially on this space, so t = 1 and t ( m + 1) = 8 ≤ E , orbits 17 and 19. Here again, h = 12 and C = S or { } . In each case, thedesired inequalities follow directly from the list of weights. E , orbit 11. This orbit comes from a Levi subgroup L of root system D and C ◦ is the two-dimensional center of L , while C normalizes L and a maximal torusof L and has component group C/C ◦ = S . The action of S on X ∗ ( C ◦ ) is thestandard representation or its twist by the sign character. Therefore, an elementof order 1 or 2 of C/C ◦ fixes a subtorus, and we see that if ( λ, F ) is to be discrete,then ¯ F should have order 3 in C/C ◦ . Then F itself has order 3 or 9 in C . To proveit has order 3, we embed E as a Levi subgroup of E and consider the reductivecentralizer C there. Then section 9.3.4 of loc.cit exhibits two elements c and c oforder 2 in C , whose images generate C/C ◦ = C /C ◦ , and that act on C ◦ by fixinga pinning. But C ◦ is a simple group of type D , so its center has order dividing 4.Hence the element ( c c ) , which belongs to C ◦ and fixes a pinning of C ◦ is centralin C ◦ , hence has order dividing 4. Since we have seen that c c has order 3 or 9,we conclude it has order 3. Hence F has order 3, and all desired inequalities followfrom the list of weights. E , orbit 11. Here G L λ / ( G L λ ) ◦ is an extension of Z / Z by ˆ G λ / ( ˆ G λ ) ◦ = S .Such an extension has to be split and contains a central element of order 2. Inthe present case, using notations c and g of loc. cit. , the element F := c g − Fr g belongs to ( ˆ G ⋊ Fr) λ and its image in G L λ / ( G L λ ) ◦ is the central element of order 2.It acts on ( G L λ ) ◦ by inversion, so that ( λ, F ) is discrete, as well as ( λ, c F ) for anyelement of order 3 of S . The order of F in G L λ is 2 or 4 and the desired inequalitiesfollow from the list of weights. The order of c F is 6 and the desired inequalities holdfor weights 2 and 4, while for the weight 6 space, an explicit computation showsthat, as a representation of S , it is the sum of the sign and trivial characters, sothat Ad F actually acts with order 2 and the desired inequalities hold. E , orbits 14, 16, 18. Here ˆ G λ = C = G m , and F should map to the non-trivialelement of π ( G L λ ) = π ( G L ), so F has order dividing 4. The desired inequalitiesare then straightforward for weights ≤ h = 18. For the weight 8, 10 or 14spaces, we use that fact that C ◦ acts trivially on them, so Ad F has order t ≤ E , orbits 17. Here ˆ G λ = C = {± } , hence F = ±
1, and F has a priori order 2or 4. But the representative e of the table is visibly invariant under Fr, which meansthat we can pick u and λ invariant under Fr, and set F = Fr or F = ( − . Fr. Ineach case, F has order 2 and the desired inequalities follow from the list of weights. E , orbit 19. Here ˆ G λ = C = { } , so F = 1, and the desired inequalitiesfollow from the list of weights (recall h = 18). ODULI OF LANGLANDS PARAMETERS 55 E , orbit 24. Here h = 18, C ◦ is a torus and π ( C ) = S . So F has orderdividing 4. The desired inequality 4( m + 1) ≤ h = 18 holds for all weights, exceptweight 8, but C ◦ acts trivially on this weight space, so Ad F has order t = 2 thereand the inequality holds too. E , orbits 33, 37,41,42,43. In these cases C = S or S or { } , and the inequalitiesare straightforward, except for the weight 18 space in orbit 41, where we need touse the column Z ♯ to ensure C acts trivially on this weight space. E , orbit 39. Here C ◦ = G m and C/C ◦ = S acts non trivially on C ◦ , but C isnot a semi-direct product of C ◦ by S , so F has order 4 with F = − ∈ ( C ◦ ) F .However, the explicit form of C ◦ given in the table shows that it acts with evenweights on all root subgroups (trivially on the simple roots of the Levi subsystem E and with weight 2 on the remaining simple root). Hence Ad F has order 2, andthe desired inequalities follow since all weights are even and less than 16. E , orbit 41. Here h = 30 and C = S , so t ≤
6. Hence the desired inequalitiesare at least satisfied for all weights ≤
8. This leaves us with the 4-dimensionalweight 10 space, which is stable under S , and is either a sum of (sign) charactersor the standard representation, or the twisted standard representation. In any case,the eigenvalues of elements of order 6 of S have order 1 , t ≤ E , orbits 47, 50, 52. Here h = 30 and C = G m and C/C ◦ = S . So F has orderdividing 4, which makes directly all desired inequalities hold except for weights 14and 16 subspaces, but the latter are fixed by C ◦ according to column Z , so Ad F has order t ≤ E , orbits 54, 58, 60, 62, 63, 65, 66, 67, 68. Here C = S , S or { } , and allinequalities are straightforward, except for the one dimensional weight 22 space inorbit 60 and weight 34 space in orbit 66. But the latter are fixed by C in each caseaccording to column Z ♯ , so t = 1 there and the inequalities still hold. E , orbit 55. Here C ◦ = G m and C/C ◦ = S . The table features a lift c in C of the non-trivial element of C/C ◦ . One can compute that c = 1 (e.g. by usingthe list of positive roots of E in Bourbaki). So we can take F = c and the desiredinequalities follow from the list of weights. (cid:3) Remark 5.18.
Consider the orbit 3 of G , on page 73 of [LT11]. The reflection c exchanges the two root vectors e and e , which have both weight 2. So − F := c on the space generated by these vectors, hence − q is aneigenvalue of q Ad F λ ( S ) . But Φ ( T ) does not divide χ ˆ G, Fr ( T ) = ( T − T − χ ˆ G, Fr really fails, just as for D . It fails also for F due to the weight 8 spaceof orbit 13, which requires Φ and Φ (depending on F ) although none of thesepolynomials divides χ F , . The same orbit and the same weight space viewed in E and E (orbit 17) again requires Φ and Φ , although Φ does not divide χ E , and Φ does not divide χ E , Fr . In orbit 33 of E , taking F = c , the weight 8space requires Φ , which does not divide χ E , . Finally, in orbit 66 of E , taking F = c , the weight 26 space requires Φ , which does not divide χ E , . Note it iscertainly possible in each case to compute explicitly a polynomial χ ′ dividing χ ∗ for which equivalence between χ ′ ( q ) = 0 and generic smoothness holds. Corollary 5.19.
Let G be as in (5.1). If χ ∗ ˆ G, Fr ( q ) = 0 in L , then for any ϕ as in(5.1), there is g ∈ ˆ G ◦ ϕ ( s ) such that ϕ g is unobstructed.Proof. This follows from Lemma 5.8, Lemma 5.9, decomposition (5.2), Proposition5.15, Lemma 5.16 and Proposition 5.17. Note again that χ ∗ ˆ G, Fr ( q ) = 0 is equivalentto q having order greater than h ˆ G, Fr , which implies ℓ > h ˆ G, Fr hence also ℓ > h ˆ G . (cid:3) Proof of Theorem 5.5. (1) We assume that ℓ does not divide eχ ∗ ˆ G, Fr ( q ). Fix ϕ ∈ Z ( W F , ˆ G ( L )) and choose ϕ ′ as in Proposition 5.6. By Lemma 5.7, the action Ad ϕ ′ of W F on ˆ H := C ˆ G ( ϕ ( I ℓF )) ◦ is unramified and η := ϕ · ( ϕ ′ ) − ∈ Z ( W F /I ℓF , ˆ H ( L )).By Proposition B.3, we have χ ∗ ˆ H, Ad ϕ (Fr) ( q ) = 0 in L , so the last Corollary gives us anelement h ∈ ( ˆ H η ( s ) ) ◦ such that η h is unobstructed in Z ( W F /I ℓF , ˆ H ( L )). We haveexplained above Proposition 5.6 that η h · ϕ ′ is then unobstructed in Z ( W F , ˆ G ( L )),but we have ( ˆ H η ( s ) ) ◦ = ( ˆ G τ ) ◦ and η h · ϕ ′ = ( η · ϕ ′ ) h = ϕ h .(2) We assume here that ˆ G has no exceptional factor, that ℓ > h ˆ G, , and that ℓ does not divide χ ˆ G, Fr ( q ). Then we repeat the above argument, observing thatˆ H = C ˆ G ( ϕ ( I ℓF )) is again a group with no exceptional component. Indeed, it sufficesto check this in a classical group where it is fairly standard. However, the action of ϕ ′ (Fr) on ˆ H may feature instances of triality. Fortunately, this is harmless becausethe modified polynomial χ ′ ˆ H, Fr still divides χ ˆ G, Fr . Indeed, if Φ ( T ) divides χ ˆ G, Fr for ˆ G a classical group, then so does Φ ( T ). (cid:3) G L -banal primes. We keep the general setup of this section.
Proposition 5.20.
Let ℓ = p be a prime. Then the following are equivalent :(1) For every algebraically closed field of characteristic ℓ , and every continuous L -homomorphism ϕ : W F → G L ( L ) , there is g ∈ C ˆ G ( ϕ ( I F )) ◦ such that ϕ g isunobstructed.(2) For any e ∈ N and any finite place v of O e [ p ] such that the residue field k v has characteristic ℓ , the fiber Z ( W F /P eF , ˆ G k v ) of Z ( W F /P eF , ˆ G ) is reduced.Proof. Assume (1). It suffices to prove reducedness for k v replaced by its alge-braic closure L . Let x be an L -point of Z ( W F /P eF , ˆ G L ) contained on exactly oneirreducible component of Z ( W F /P eF , ˆ G L ). Let τ be the restriction of ϕ x to I F ;there then exists a g ∈ ˆ G τ ( L ) such that ϕ gx is unobstructed. The L -point y of Z ( W F /P eF , ˆ G L ) lies in the same irreducible component of Z ( W F /P eF , ˆ G L ) as x ,so that irreducible component is generically reduced. Since x was arbitrary, wededuce that Z ( W F /P eF , ˆ G L ) is generically reduced; since it is also a local completeintersection, it must be reduced.Now assume (2). Following the same reduction process as above Proposition 5.6,we may assume that G L and ϕ are as in (5.1). Recall from the discussion aboveLemma 2.2 the mapˆ G × ˆ G ◦ ϕ ( s ) −→ Z ( W F /I ℓF , ˆ G ) L , ( h, g ) h ( ϕ g ) . We have shown there that Z ( W F /I ℓF , ˆ G ) L is covered by the images of finitely manyof these maps, and that these images all have the same dimension. This implies thatthe closure of these images are the irreducible components of Z ( W F /I ℓF , ˆ G ). In ODULI OF LANGLANDS PARAMETERS 57 particular, the image of the above map is dense in one of the components that con-tain ϕ . Therefore, since reducedness implies generic smoothness of all components,we get (1). (cid:3) Definition 5.21.
A prime ℓ = p is G L -banal if the properties of the last propositionhold for ℓ .One way to view this reducedness of fibers from a philosophical standpoint isto say that there are no nontrivial “congruences” between Langlands parametersmodulo an G L -banal prime ℓ : the closures of distinct irreducible components incharacteristic zero remain distinct modulo ℓ . One expects that this should corre-spond, on the other side of the local Langlands correpondence, to a lack of nontrivialcongruences between admissible smooth representations of the reductive group G over F whose L -group is G L . So this should be related to the representation theo-retic notion of “banal”. Recall indeed that, for a reductive group G over F , a prime ℓ = p is called banal if it does not divide the order of a torsion element of G ( F ).For the sake of precision, we will reterm this as “ G -banal”. Lemma 5.22.
Suppose that G is a reductive group over O F , and denote by G L =ˆ G ⋊ h Fr i its Langlands dual group. (1) A prime ℓ is G -banal if and only if it divides the order of G ( k F ) . (2) The set of G -banal primes only depends on the isogeny class of G . (3) We have | G ( k F ) | = q N · χ ˆ G, Fr ( q ) .Proof. (1) Let g ∈ G ( F ) have finite order prime to p . Then it stabilizes a facetof the Bruhat-Tits building of G ( F ), and fixes its barycenter. This barycenterbecomes a hyperspecial point in the building of G ( F ′ ) for some totally ramifiedextension of F . So the order of g divides | G ( k F ′ ) | , but k F ′ = k F .(3) This is the Chevalley-Steinberg formula, see Theorem B.4.(2) This follows from (1), see the proof of Theorem B.4. (cid:3) Corollary 5.23.
Suppose G is an unramified group over F with no exceptionalfactor, and let ℓ be a prime greater than the Coxeter number of G . Then ℓ is G L -banal if and only if it is G -banal.Proof. This follows from the above lemma together with Proposition 5.15. (cid:3)
It is a bit surprising that our results in Lemma 5.16 and Remark 5.18 show thatthis equivalence does not hold for exceptional groups.6.
The GIT quotient in the banal case
Our aim in this section is to get a complete description of the affine quotient Z ( W F /P eF , ˆ G ) (cid:12) ˆ G after base change to ¯ Z [ N ] for some sufficiently well controlledinteger N . Our strategy rests on the universal homeomorphism (4.10) Z ( W F /I eF , ˆ G ) (cid:12) ˆ G −→ Z ( W F /P eF , ˆ G ) (cid:12) ˆ G. We have already singled out the so-called G L -banal primes, which are particu-larly well behaved for the RHS. On the other hand, the integer N ˆ G defined aboveCorollary 4.16 plays a particular role regarding the LHS: Lemma 6.1.
The structural morphism Z ( W F /I eF , ˆ G ) −→ Spec( Z [ p ]) is smoothover Spec( Z [ pN ˆ G ]) . Proof.
Since the finite group I F /I eF has invertible order in Z [ pN ˆ G ], Lemma A.1 tellsus that Z ( I F /I eF , ˆ G ) is smooth over Spec( Z [ pN ˆ G ]). On the other hand, the map ϕ ϕ (Fr) identifies the restriction morphism Z ( W F /I eF , ˆ G ) −→ Z ( I F /I eF , ˆ G )with the ˆ G -transporter from Fr φ univ to φ univ , where φ univ denotes the universal 1-cocycle I F /I eF −→ ˆ G ( O Z ( I F /I eF , ˆ G ) ). Hence, by Lemma A.1 again, this restrictionmorphism is also smooth, and the lemma follows. (cid:3) Recall that the universal homeomorphism (4.10) becomes an isomorphism aftertensoring by Q . The next result gives a bound on the set of integers that actuallyneed to be inverted. Proposition 6.2.
The morphism Z ( W F /I eF , ˆ G ) (cid:12) ˆ G −→ Z ( W F /P eF , ˆ G ) (cid:12) ˆ G of(4.10) is an isomorphism after inverting N ˆ G and the non G L -banal primes.Proof. Consider the dual map (4.9) on rings of functions ( R e G L ) ˆ G −→ ( S eG L ) ˆ G . Wealready know it is injective and its cokernel is a torsion abelian group. Let ℓ be anassociated prime of this cokernel. If ℓ does not divide N ˆ G , there is no ℓ -torsion in S eG L (by the last lemma), hence the reduced map ( R e G L ) ˆ G ⊗ F ℓ −→ ( S eG L ) ˆ G ⊗ F ℓ is not injective. But this map induces a bijection on ¯ F ℓ -points, so its kernel liesin the Jacobson radical, and we deduce that ( R e G L ) ˆ G ⊗ F ℓ is not reduced. On theother hand, ( R e G L ) ˆ G is an ℓ -adically saturated submodule of R e G L , so that the map( R e G L ) ˆ G ⊗ F ℓ −→ ( R e G L ⊗ F ℓ ) ˆ G is actually injective. So we infer that R e G L ⊗ F ℓ isnot reduced, hence ℓ is not G L -banal. (cid:3) Remark 6.3.
When G L is the Langlands dual group of an unramified group,Proposition 5.12 and the estimate of Proposition 4.14 show that the prime divisorsof N ˆ G are non G L -banal. We believe this is true in general.In view of the last proposition, we focus in the next subsection on the explicitdescription of Z ( W F /I eF , ˆ G ) (cid:12) ˆ G , over ¯ Z [ pN ˆ G ]. The description that we obtain inTheorem 6.7 bears a striking analogy with the usual description of the Bernsteincenter. Actually, in Subsection 6.3, we extend scalars to C and we show that ourdescription gives back Haines’ definition of a structure of algebraic variety on theset of semisimple complex Langlands parameters.6.1. Description of Z ( W F /I eF , ˆ G ) (cid:12) ˆ G over ¯ Z [ pN ˆ G ] . Since the order of I F /I eF isinvertible in Z [ pN ˆ G ], we can obtain decompositions of Z ( W F /I eF , ˆ G ) ¯ Z [ pN ˆ G ] similarto (4.2) and (4.4) by restricting cocycles to I F instead of restricting to P F . Indeed,we first infer the following results from Theorems A.7, A.9, A.12 and A.13 in theappendix. Proposition 6.4.
There is a finite extension ˜ K e of K e and a set ˜Φ e ⊂ Z (cid:16) I F /I eF , ˆ G (cid:0) O ˜ K e (cid:2) /pN ˆ G (cid:3)(cid:1)(cid:17) , such that • for each φ ∈ ˜Φ e , the group scheme C ˆ G ( φ ) ◦ is split reductive and π ( φ ) := π ( C ˆ G ( φ )) is constant over O ˜ K e [ pN ˆ G ] and ODULI OF LANGLANDS PARAMETERS 59 • we have an orbit decomposition Z ( I F /I eF , ˆ G ) O ˜ Ke [ pN ˆ G ] = a φ ∈ ˜Φ e ˆ G · φ ≃ a φ ∈ ˜Φ e ˆ G/C ˆ G ( φ ) where each summand represents the corresponding ´etale sheaf quotient. The above decomposition induces in turn the following ones : Z ( W F /I eF , ˆ G ) O ˜ Ke [ pN ˆ G ] = a φ ∈ ˜Φ adm e ˆ G × C ˆ G ( φ ) Z ( W F , ˆ G ) φ . (6.1) ( Z ( W F /I eF , ˆ G ) (cid:12) ˆ G ) O ˜ Ke [ pN ˆ G ] = a φ ∈ ˜Φ adm e Z ( W F , ˆ G ) φ (cid:12) C ˆ G ( φ ) . Here, Z ( W F , ˆ G ) φ is the affine scheme over O ˜ K e [ pN ˆ G ] that classifies all 1-cocycles ϕ : W F −→ ˆ G such that ϕ | I F = φ and, as usual, we say that φ is admissible if thisscheme is not empty.Define the Fr-twist of φ by Fr φ ( i ) := Fr( φ (Fr − i Fr)). Then we have an isomor-phism ϕ ϕ (Fr) Z ( W F , ˆ G ) φ ∼ −→ T ˆ G ( Fr φ, φ )where the RHS denotes the transporter in ˆ G from Fr φ to φ for the natural actionof ˆ G on Z ( I F , ˆ G ). This isomorphism is C ˆ G ( φ )-equivariant if we let C ˆ G ( φ ) act onthe transporter by Fr-twisted conjugation c · t := c ◦ t ◦ Fr( c ) − . On the other hand, T ˆ G ( Fr φ, φ ) is also a left pseudo-torsor over C ˆ G ( φ ) under composition ( c, t ) c ◦ t .When φ is admissible, T ˆ G ( Fr φ, φ ) is actually a C ˆ G ( φ )-torsor for the ´etale topology,and the ´etale sheaf quotient π ( Fr φ, φ ) := T ˆ G ( Fr φ, φ ) /C ˆ G ( φ ) ◦ is a π ( φ )-torsor.Therefore π ( Fr φ, φ ) is representable by a finite ´etale O ˜ K e [ pN ˆ G ]-scheme and, aftermaybe enlarging ˜ K e , we may and will assume that it is constant . Then we get afurther decomposition T ˆ G ( Fr φ, φ ) = a β ∈ π ( Fr φ,φ ) T ˆ G ( Fr φ, φ ) β which is nothing but the decomposition into connected components, and where eachcomponent is a left C ˆ G ( φ ) ◦ -torsor. Moreover, the Fr-twisted conjugation action of C ˆ G ( φ ) on T ˆ G ( Fr φ, φ ) induces an action of π ( φ ) on π ( Fr φ, φ ). Denote by π ( φ ) β the stabilizer of β for this action, and by π ( Fr φ, φ ) a set of representatives oforbits. Then we get(6.2) T ˆ G ( Fr φ, φ ) (cid:12) C ˆ G ( φ ) = a β ∈ π ( Fr φ,φ ) (cid:0) T ˆ G ( Fr φ, φ ) β (cid:12) C ˆ G ( φ ) ◦ (cid:1) /π ( φ ) β . Our next result will allow us to compute each term of this decomposition. Beforewe can state it, note that if R is an O ˜ K e [ pN ˆ G ]-algebra and ˜ β ∈ T ˆ G ( R ) ( Fr φ, φ ),then conjugation by ˜ β ⋊ Fr in ˆ G ( R ) ⋊ W F normalizes C ˆ G ( R ) ( φ ). We denote theautomorphism thus induced by Ad ˜ β . Theorem 6.5.
Fix a pinning ε φ = ( B φ , T φ , ( X α ) α ) of C ˆ G ( φ ) ◦ over O ˜ K e [ pN ˆ G ] .Then, after maybe enlarging the finite extension ˜ K e , we can find for each β ∈ π ( Fr φ, φ ) , an element ˜ β ∈ T ˆ G ( O ˜ Ke [ pN ˆ G ]) ( Fr φ, φ ) such that Ad ˜ β normalizes ε φ . Proof.
The proof goes along the same argument as for Theorem 3.4. Let us first dothe translation to the notation of Subsection 3.2. To this aim, choose an L -groupsuch that I F /I eF embeds into π ( G L ). Then we can form the subgroup scheme C G L ( φ ) as in Subsection 3.2 and, denoting by Fr the image of Fr in π ( G L ), we seethat the map ϕ ϕ L (Fr) defines an isomorphism Z ( W F , ˆ G ) φ ∼ −→ C G L ( φ ) ∩ ( ˆ G ⋊ Fr) , and that the map ˜ β ˜ β ⋊ Fr defines a second isomorphism T ˆ G ( Fr φ, φ ) ∼ −→ C G L ( φ ) ∩ ( ˆ G ⋊ Fr)whose composition with the previous one is the isomorphism introduced just above.Now, define ˜ π ( φ ) and Σ( φ ) as above Definition 3.3, so that we have for eachadmissible φ a further decomposition Z ( W F , ˆ G ) φ = a α ∈ Σ( φ ) Z ( W F , ˆ G ) φ,α . Then we have a bijection α α (Fr) between Σ( φ ) and the fiber of the map ˜ π ( φ ) → π ( G L ) over Fr. On the other hand, we have a natural injection π ( Fr φ, φ ) ֒ → ˜ π ( φ )whose image is precisely the said fiber. So we get a bijection β ↔ α between π ( Fr φ, φ ) and Σ( φ ), and it is easily checked that Z ( W F , ˆ G ) φ,α = T ˆ G ( Fr φ, φ ) β .Now the same proof as that of Theorem 3.4 applies, and actually the strongervariant of Remark 3.9 applies too, because what is needed from Lemma 3.8 in theproof of this variant is now trivial : since W F /I F ≃ Z , we have H ( W F /I F , A ) = { } for any abelian group A with action of W F /I F .So we get the existence of a finite extension ˜ K e and, for each α , a cocycle(6.3) ϕ α : W F −→ ˆ G ( O ˜ K e [1 /pN ˆ G ])that restricts to φ , induces α , normalizes ε φ and has finite image. Writing ϕ α (Fr) =˜ β ⋊ Fr provides us with the desired element ˜ β . (cid:3) With the notation of this theorem we now have an identification c c ◦ ˜ βC ˆ G ( φ ) ◦ ∼ −→ T ˆ G ( Fr φ, φ ) β and the Fr-twisted conjugation action of C ˆ G ( φ ) ◦ on T ˆ G ( Fr φ, φ ) β corresponds to theAd ˜ β -twisted conjugation action of C ˆ G ( φ ) ◦ on itself. We thus get T ˆ G ( Fr φ, φ ) β (cid:12) C ˆ G ( φ ) ◦ = ( C ˆ G ( φ ) ◦ ⋊ Ad ˜ β ) (cid:12) C ˆ G ( φ ) ◦ . Now, denote by Ω ◦ φ the Weyl group of the maximal torus T φ of C ˆ G ( φ ) ◦ , anddenote by Ω φ := N C ˆ G ( φ ) ( T φ ) /T φ its “Weyl group” in C ˆ G ( φ ). The natural map N C ˆ G ( φ ) ( T φ , B φ ) −→ π ( φ ) induces an isomorphism N C ˆ G ( φ ) ( T φ , B φ ) /T φ ≃ π ( φ ),hence Ω φ = Ω ◦ φ ⋊ π ( φ ) is a split extension of π ( φ ) by Ω ◦ φ . Since the automorphismAd ˜ β of C ˆ G ( φ ) stabilizes T φ and B φ , it acts on Ω φ and preserves the semi-directproduct decomposition. Note that the actions of Ad ˜ β on T φ and Ω φ only dependon β and not on the choice of ˜ β as in the theorem. We will thus denote these actionssimply by Ad β . Observe that the invariant subgroup (Ω φ ) Ad β of Ω φ decomposes as(Ω φ ) Ad β = (Ω ◦ φ ) Ad β ⋊ π ( φ ) β and acts naturally on the coinvariant torus ( T φ ) Ad β . ODULI OF LANGLANDS PARAMETERS 61
Proposition 6.6.
The inclusion T φ ֒ → C ˆ G ( φ ) ◦ induces an isomorphism ( T φ ) Ad β (cid:12) (Ω ◦ φ ) Ad β ∼ −→ (cid:16) C ˆ G ( φ ) ◦ ⋊ Ad ˜ β (cid:17) (cid:12) C ˆ G ( φ ) ◦ Proof.
Consider the inclusion T φ ⋊ Ad ˜ β ֒ → C ˆ G ( φ ) ◦ ⋊ Ad ˜ β . Under the conjuga-tion action of C ˆ G ( φ ) ◦ on the RHS, the LHS is stable by the subgroup scheme N C ˆ G ( φ ) ◦ ( T φ ) β of N C ˆ G ( φ ) ◦ ( T φ ) given as the inverse image of (Ω ◦ φ ) Ad β . Whence amorphism (cid:16) T φ ⋊ Ad ˜ β (cid:17) (cid:12) N C ˆ G ( φ ) ◦ ( T φ ) β −→ (cid:16) C ˆ G ( φ ) ◦ ⋊ Ad ˜ β (cid:17) (cid:12) C ˆ G ( φ ) ◦ . Now observe that ( T φ ) Ad β = (cid:16) T φ ⋊ Ad ˜ β (cid:17) (cid:12) T φ , so that the above morphism inducesin turn a morphism( T φ ) Ad β (cid:12) (Ω ◦ φ ) Ad β −→ ( C ˆ G ( φ ) ◦ ⋊ Ad ˜ β ) (cid:12) C ˆ G ( φ ) ◦ . Now, for any algebraically closed field L over O ˜ K e [1 /pN ˆ G ], Lemma 6.5 of [Bor79]tells us that this morphism induces a bijection on L -points. In particular thecorresponding map on rings of functions is injective since the source is reduced.Its surjectivity can be proved as in [Bor79, Prop. 6.7], which deals with complexcoefficients. Namely, put R := O ˜ K e [1 /pN ˆ G ] and let X denote the character groupof T φ . Then the ring of functions of ( T φ ) Ad β (cid:12) (Ω ◦ φ ) Ad β is R [ X Ad β ] (Ω ◦ φ ) Ad β , hencehas a natural R -basis given by (Ω ◦ φ ) Ad β -orbits in X Ad β . Any such orbit has aunique representative in the dominant cone of X with respect to B φ . So let λ ∈ X Ad β be dominant in X and let L λ be the corresponding invertible sheaf on theflag variety C ˆ G ( φ ) ◦ /B φ . Then M λ := H ( C ˆ G ( φ ) ◦ /B φ , L λ ) is a free R -module offinite rank with an algebraic action of C ˆ G ( φ ) ◦ . Actually, since λ is Ad β -invariant,it defines a character of the group scheme T φ ⋊ h Ad β i , and since C ˆ G ( φ ) ◦ /B φ =( C ˆ G ( φ ) ◦ ⋊ h Ad β i ) / ( B φ ⋊ h Ad β i ), we see that M λ is actually a C ˆ G ( φ ) ◦ ⋊ h Ad β i -module. In particular, the map g tr( g ⋊ Ad β | M λ ) is in the ring of functions of( C ˆ G ( φ ) ◦ ⋊ Ad ˜ β ) (cid:12) C ˆ G ( φ ) ◦ . Its restriction to T φ factors over ( T φ ) Ad β and is of theform X λ ′ ∈ (Ω ◦ φ ) Ad β λ ′ + X µ<λ a µ µ, a µ ∈ N . So we deduce inductively the desired surjectivity. (cid:3)
Eventually, after choosing a pinning ε φ for each φ ∈ ˜Φ e and inserting the resultof the above proposition inside decompositions (6.1) and (6.2), we get our desireddescription of the affine quotient over O ˜ K e [ pN ˆ G ]. Theorem 6.7.
The collection of embeddings T φ ֒ → C ˆ G ( φ ) induce an isomorphismof O ˜ K e [ pN ˆ G ] -schemes a φ ∈ ˜Φ adm e a β ∈ π ( Fr φ,φ ) ( T φ ) Ad β (cid:12) (Ω φ ) Ad β ∼ −→ ( Z ( W F /I eF , ˆ G ) (cid:12) ˆ G ) O ˜ Ke [ pN ˆ G ] . We note that the LHS does not depend on the choices of elements ˜ β as in Theorem6.5. But the maps from the LHS to the RHS a priori depend on these choices. The GIT quotient over a banal algebraically closed field.
With The-orem 6.7 and Proposition 6.2, we now have a description of the affine quotient Z ( W F /P F , ˆ G ) (cid:12) ˆ G after inverting N ˆ G and the non G L -banal primes. Let us nowconsider affine quotients over algebraically closed fields. Theorem 6.8.
Let L be an algebraically closed field over O ˜ K e [ pN ˆ G ] and of G L -banal characteristic. Then the natural maps induce isomorphisms a φ ∈ ˜Φ adm e a β ∈ π ( Fr φ,φ ) ( T φ,L ) Ad β (cid:12) (Ω φ ) Ad β ∼ −→ Z ( W F /I eF , ˆ G L ) (cid:12) ˆ G L Z ( W F /I eF , ˆ G L ) (cid:12) ˆ G L ∼ −→ Z ( W F /P eF , ˆ G L ) (cid:12) ˆ G L Z ( W F /P eF , ˆ G L ) (cid:12) ˆ G L ∼ −→ ( Z ( W F /P eF , ˆ G ) (cid:12) ˆ G ) L . Proof.
The first isomorphism holds without the G L -banal hypothesis. It is provedexactly as the isomorphism of Theorem 6.7, and is part of an obvious commutativesquare involving the latter isomorphism base changed to L .If, in addition, the order of each (Ω φ ) Ad β is invertible in L , which is certainly thecase if char( L ) is G L -banal, then ( T φ,L ) Ad β (cid:12) (Ω φ ) Ad β ∼ −→ (( T φ ) Ad β (cid:12) (Ω φ ) Ad β ) L .Then, the fourth map of this commutative square has also to be an isomorphism : Z ( W F /I eF , ˆ G L ) (cid:12) ˆ G L ∼ −→ ( Z ( W F /I eF , ˆ G ) (cid:12) ˆ G ) L . So we now have a commutativesquare involving the analogous map for W F /P eF , which, in terms of rings, reads( R e G L ) ˆ G ⊗ L / / ∼ (cid:15) (cid:15) ( R e G L ⊗ L ) ˆ G (cid:15) (cid:15) ( S eG L ) ˆ G ⊗ L ∼ / / ( S eG L ⊗ L ) ˆ G The right vertical map is surjective since the bottom map is surjective, and it isalso injective since it induces a bijection on L -points and ( R e G L ⊗ L ) ˆ G is reduced.Therefore it is an isomorphism, and so is the upper map. (cid:3) Remark 6.9.
Let L be an algebraically closed field as in the theorem.i) The index set in the first isomorphism can be replaced by any set Ψ e ( L )of representatives of ˆ G ( L )-conjugacy classes of pairs ( φ, β ) consisting of a cocycle φ : I F /I eF −→ G L ( L ) and an element β ∈ π ( Fr φ, φ ). Since any φ as above isautomatically semisimple, this is the same set as in Corollary 4.22.ii) As usual, the set of L -points of Z ( W F /I eF , ˆ G L ) (cid:12) ˆ G L is the set of closedˆ G L -orbits in Z ( W F /I eF , ˆ G ( L )). For a cocycle ϕ : W F /I eF −→ ˆ G ( L ) in some Z ( W F /I eF , ˆ G L ) φ , we claim that the following statements are equivalent, provided L has characteristic G L -orbit is closed in Z ( W F /I eF , ˆ G L ),(2) its C ˆ G ( φ ) L -orbit is closed in Z ( W F /I eF , ˆ G L ) φ ,(3) ϕ L (Fr) is a semisimple element of G L ( L ).(4) ϕ L ( W F ) consists of semisimple elements.Indeed, (1) ⇒ (2) since the small orbit is the intersection of the big one with theclosed subset Z ( W F /I eF , ˆ G L ) φ . Moreover, (2) is equivalent to the orbit of ϕ L (Fr)being closed in C G L ( φ )( L ), which in turn is equivalent to ϕ L (Fr) being a semisimpleelement of C G L ( φ )( L ), hence also of G L ( L ). Further, (3), being equivalent to (2), ODULI OF LANGLANDS PARAMETERS 63 applies to any lift of Frobenius, so implies (4). Eventually, (4) implies that theˆ G ( L )-orbits of a finite set of generators of ϕ L ( W F ) are closed, which implies (1).The cocycles that satisfy property (3) are often called “Frobenius semi-simple”in the literature. When L has positive characteristic, a cocycle with closed orbitmay not be Frobenius semi-simple. For example suppose q = q F has prime order ℓ = p in some ( Z /n Z ) × with n prime to both p and ℓ , and consider the character θ : I F ։ µ n ֒ → ¯ F × ℓ . Extend this character to I F · Fr ℓ Z by setting θ (Fr ℓ ) = 1 andinduce to W F . We obtain an irreducible representation ϕ : W F −→ GL ℓ (¯ F ℓ ) suchthat ϕ (Fr) has order ℓ .6.3. Comparison with the Haines variety.
In this subsection, we assume thatthe action of W F stabilizes a pinning of ˆ G , so that G L is an L -group associated tosome reductive group G over F . As noted in point ii) of the last remark, the set of C -points of the affine categorical quotient Z ( W F /I eF , ˆ G ) C (cid:12) ˆ G C is the set of ˆ G ( C )-conjugacy classes of Frobenius semisimple L -homomorphisms W F /I eF −→ G L ( C ). In [Hai14], Haines endows this set with the structure of a com-plex affine variety that mimics Bernstein’s description of the center of the categoryof complex representations of G ( F ). We will denote by Ω e ( ˆ G ) the Haines varietyand we wish to compare his construction to ours. In a rather abstract form, themain result of this section is the following. Theorem 6.10.
The set-theoretic identification between ( Z ( W F /I eF , ˆ G ) (cid:12) ˆ G )( C ) and Ω e ( ˆ G ) is induced by an isomorphism of varieties Ω e ( ˆ G ) ≃ Z ( W F /I eF , ˆ G ) C (cid:12) ˆ G C . We need to recall some features of Haines’ construction in Section 5 of [Hai14].Let ϕ L : W F /I eF → G L ( C ) be a Frobenius-semisimple L -morphism (called an“infinitesimal character” by Haines and Vogan), and choose a Levi subgroup M of G L that contains ϕ L ( W F ) and is minimal for this property. Here we consider Levisubgroups in the sense of Borel [Bor79, § M ◦ = M ∩ ˆ G is a Levisubgroup of ˆ G and π ( M ) ∼ −→ π ( G L ) is a quotient of W F . As a consequence, theaction of M by conjugation on the center Z ( M ◦ ) of M ◦ factors through an actionof π ( G L ), and provides thus a canonical action of W F on Z ( M ◦ ). We may thenconsider the torus ( Z ( M ◦ ) I F ) ◦ given by the neutral component of the I F -invariants,and which still carries an action of W F /I F = h Fr i . To any z ∈ ( Z ( M ◦ ) I F ) ◦ , Hainesassociates a new parameter z · ϕ L defined by( z · ϕ L )( w ) := z ν ( w ) ϕ L ( w ) , with ν : W F ։ Fr Z . The conjugacy class ( z · ϕ L ) ˆ G only depends on the image of z in the Fr-coinvariants( Z ( M ◦ ) I F ) ◦ ) Fr , hence we get a map(6.4) ( Z ( M ◦ ) I F ) ◦ Fr −→ Ω e ( ˆ G ) = ( Z ( W F /I eF , G L ) (cid:12) ˆ G )( C )By Haines’ definition of the variety structure on Ω e ( ˆ G ), this map is a morphismof algebraic varieties ( Z ( M ◦ ) I F ) ◦ Fr −→ Ω e ( ˆ G ). Even better, there is a finitegroup W M ,ϕ of algebraic automorphisms of ( Z ( M ◦ ) I F ) ◦ Fr , whose precise defini-tion is not needed here, such that the map (6.4) factors over an injective map( Z ( M ◦ ) I F ) ◦ Fr /W M ,ϕ ֒ → Ω e ( ˆ G ). Then, the corresponding morphism of varieties ( Z ( M ◦ ) I F ) ◦ Fr (cid:12) W M ,ϕ −→ Ω e ( ˆ G ) is an isomorphism onto a connected componentof Ω e ( ˆ G ), by Haines’ construction. Moreover, all connected components are ob-tained in this way.At this point, we have recalled enough to prove one direction. Lemma 6.11.
The set-theoretic identification between Ω e ( ˆ G ) and ( Z ( W F /I eF , G L ) (cid:12) ˆ G )( C ) is induced by a morphism of varieties Ω e ( ˆ G ) −→ Z ( W F /I eF , G L ) C (cid:12) ˆ G C . Proof.
By the foregoing discussion, it now suffices to prove that each map (6.4) isinduced by a morphism of schemes( Z ( M ◦ ) I F ) ◦ Fr −→ Z ( W F /I eF , G L ) C (cid:12) ˆ G C . By construction, the map (6.4) is part of a commutative diagram( Z ( M ◦ ) I F ) ◦ (cid:15) (cid:15) / / Z ( W F /I eF , ˆ G )( C ) (cid:15) (cid:15) ( Z ( M ◦ ) I F ) ◦ Fr / / ( Z ( W F /I eF , ˆ G ) (cid:12) ˆ G )( C )where the top map is given by z z · ϕ L . Denote by ζ ∈ G L ( C [( Z ( M ◦ ) I F ) ◦ ])the element corresponding to the closed immersion ( Z ( M ◦ ) I F ) ◦ ֒ → G L . Then ζ · ϕ L is an element of Z ( W F /I eF , ˆ G ( C [( Z ( M ◦ ) I F ) ◦ ])), hence corresponds to amorphism ( Z ( M ◦ ) I F ) ◦ −→ Z ( W F /I eF , ˆ G ). By definition, this morphism inducesthe top map of the above diagram on the respective sets of C -points. Moreover,the composition of this morphism with the morphism underlying the right verticalmap of the diagram is Fr-equivariant for the trivial action of Fr on the target, so ithas to factor over a morphism which induces the bottom map of the diagram, asdesired. (cid:3) We now go in the other direction.
Lemma 6.12.
The set-theoretic identification between ( Z ( W F /I eF , G L ) (cid:12) ˆ G )( C ) and Ω e ( ˆ G ) is induced by a morphism of varieties Z ( W F /I eF , G L ) C (cid:12) ˆ G C −→ Ω e ( ˆ G ) . Proof.
The description of Theorem 6.7 shows that it suffices to prove that for anypair ( φ, β ) as in Theorem 6.7 and any choice of ˜ β ∈ T ˆ G ( Fr φ, φ ) β as in Theorem 6.5,the map(6.5) ( T φ ) Ad β ( C ) −→ ( Z ( W F /I eF , G L ) (cid:12) ˆ G )( C ) = Ω e ( ˆ G )is induced by a morphism of algebraic varieties ( T φ ) Ad β −→ Ω e ( ˆ G ).To prove this, we will identify the maps (6.5) to instances of maps (6.4). Letus thus fix a pair ( φ, β ) and ˜ β as in the statement, so that Ad ˜ β fixes a pinning ε φ of C ˆ G ( φ ) with maximal torus T φ . We denote by ϕ ˜ β the unique extension of φ such that ϕ ˜ β (Fr) = ˜ β . Consider the torus ( T φ ) Ad ˜ β , ◦ . Its centralizer M in G L contains ϕ L ˜ β ( W F ), hence maps onto π ( G L ) and is thus a Levi subgroup in thesense of Borel. Moreover, the canonical action of W F on Z ( M ◦ ) is induced byAd ϕ ˜ β . We claim that M is minimal among Levi subgroups of G L that contain ODULI OF LANGLANDS PARAMETERS 65 ϕ L ˜ β ( W F ). Indeed, if ϕ L ˜ β ( W F ) ⊂ M ′ ⊂ M , then ( T φ ) Ad ϕ ˜ β (Fr) , ◦ ⊂ Z ( M ) ◦ ⊂ Z ( M ′ ) ◦ ⊂ C ˆ G ( ϕ ˜ β ) ◦ . But ( T φ ) Ad ˜ β , ◦ is a maximal torus of C ˆ G ( ϕ ˜ β ) ◦ = C ˆ G ( φ ) Ad ˜ β , ◦ by[DM94, Thm 1.8 iii)], so all inclusions above have to be equalities and in particular M ′ = M since a Levi subgroup of G L is the centralizer in G L of its connectedcenter by [Bor79, Lem. 3.5]. Now, observe that( Z ( M ◦ ) I F ) ◦ = Z ( M ◦ ) Ad φ ( IF ) , ◦ = ( Z ( M ◦ ) ∩ C ˆ G ( φ )) ◦ = T φ . Indeed, the last equality comes from [DM94, Thm 1.8 iv)] which implies that thecentralizer of ( T φ ) Ad ˜ β , ◦ in C ˆ G ( φ ) ◦ is T φ . It follows in particular that( Z ( M ◦ ) I F ) ◦ Fr = ( T φ ) Ad ˜ β . Now, by construction, the map (6.5) is part of a commutative diagram T φ ( C ) (cid:15) (cid:15) / / Z ( W F /I eF , ˆ G )( C ) (cid:15) (cid:15) ( T φ ) Ad ˜ β ( C ) / / ( Z ( W F /I eF , ˆ G ) (cid:12) ˆ G )( C )where the top map is given by t (cid:16) w t ν ( w ) ϕ ˜ β ( w ) (cid:17) , where ν is the projection W F −→ W F /I F = Fr Z . This proves that the bottommap of the diagram, i.e. the map of the lemma, is an instance of (6.4), hence is amorphism of algebraic varieties. (cid:3) Proof of Theorem 6.10.
It follows from the two above lemmas. (cid:3)
Remark 6.13.
The isomorphism of Theorem 6.10 induces of course a bijectionbetween the sets of connected components on both sides. This bijection is easilydescribed as follows : • π (Ω e ( ˆ G )) is the set of “inertial classes” of Frobenius-semisimple cocycles ϕ , as defined in [Hai14, Def 5.5.3] • π ( Z ( W F /I eF , ˆ G ) C (cid:12) ˆ G C ) is the set of conjugacy classes of pairs ( φ, β ) asin iii) of Remark 6.9. • The bijection takes ϕ to ( ϕ | I F , p ( ϕ (Fr)) with p the projection T ˆ G ( Fr φ, φ ) −→ π ( Fr φ, φ ).Now, we wish to compare more explictly Haines’ construction with our description.This can be done component-wise, so let us fix data ( φ, β, ε φ , ˜ β ) and put ϕ = ϕ ˜ β asin the last proof. Recall also the Levi subgroup M = C G L (( T φ ) Ad ˜ β , ◦ ) of G L thatappeared in the last proof. So we have ( T φ ) Ad ˜ β = ( Z ( M ◦ ) I F , ◦ ) Fr . The associatedconnected component in Theorem 6.7 is ( T φ ) Ad ˜ β (cid:12) (Ω φ ) Ad ˜ β while this connectedcomponent is described as a two-step quotient (cid:0) ( Z ( M ◦ ) I F , ◦ ) Fr / Stab( ϕ ) (cid:1) /W ϕ, M ◦ . in Lemmas 5.3.7 and 5.3.8 of Haines’ paper [Hai14]. In order to compare explicitlythese two descriptions, put[ ϕ ] := (cid:8) z · ϕ, z ∈ Z ( M ◦ ) I F , ◦ (cid:9) ⊂ Z ( W F /I eF , ˆ G ( C )) and denote by N ˆ G ([ ϕ ]) its stabilizer in ˆ G . Then we have T φ = Z ( M ◦ ) I F , ◦ ⊂ N ˆ G ([ ϕ ]) ⊂ N C ˆ G ( φ ) ( T φ ) . Note that all these groups are stable under Ad ˜ β (which is induced by conjuga-tion under ˜ β ⋊ Fr = ϕ L (Fr) in G L ). Through the bijection Z ( M ◦ ) I F , ◦ ∼ −→ [ ϕ ],the natural action of N ˆ G ([ ϕ ]) on the RHS corresponds to the Ad ˜ β -twisted con-jugation on the LHS. As a consequence, this action descends to an action of thefinite group N ˆ G ([ ϕ ]) /T φ on ( T φ ) Ad ˜ β = ( Z ( M ◦ ) I F , ◦ ) Fr . This finite subgroup of Ω φ is fixed by Ad ˜ β since N ˆ G ([ ϕ ]) stabilizes T φ for the Ad ˜ β -twisted conjugacy. Con-versely, any element of N C ˆ G ( φ ) ( T φ ) that maps into (Ω φ ) Ad ˜ β stabilizes [ ϕ ], so we seethat N ˆ G ([ ϕ ]) /T φ = (Ω φ ) Ad ˜ β . To compare with Haines’ two-step quotient, let usintroduce the subgroup N M ◦ ([ ϕ ]) of N ˆ G ([ ϕ ]). It is distinguished and contains T φ .The map m m. Ad ˜ β ( m ) − induces a morphism from N M ◦ ([ ϕ ]) /T φ onto the sub-group Stab( ϕ ) ⊂ ( T φ ) Ad ˜ β of [Hai14, Lemma 3.5.7] and the action of N M ◦ ([ ϕ ]) /T φ on ( T φ ) Ad ˜ β factors over this map. In addition, the quotient N ˆ G ([ ϕ ]) /N M ◦ ([ ϕ ]) isnothing but the group W ˆ Gϕ, M ◦ of [Hai14, Lemma 3.5.8]. Appendix A. Moduli of cocycles
A.1.
Schemes of cocycles.
Let H be an affine group scheme over a noetherianring R and let Γ be a finite group. Consider the functor Hom(Γ , H ), which to any R -algebra R ′ associates the set of homomorphisms Hom(Γ , H ( R ′ )). It is representedby a closed and finitely presented R -subscheme of the affine R -scheme H (Γ) , sinceit is the inverse image of the closed subscheme { H } (Γ × Γ) of H (Γ × Γ) by the R -morphism H (Γ) −→ H (Γ × Γ) defined by ( h γ ) γ ∈ Γ ( h γ h γ ′ h − γγ ′ ) ( γ,γ ′ ) ∈ Γ × Γ .The group scheme H acts by conjugation on Hom(Γ , H ). Given an R -algebra R ′ and an homomorphism φ ∈ Hom(Γ , H ( R ′ )), the orbit maps g Ad g ◦ φ , H ( R ′′ ) −→ Hom(Γ , H ( R ′′ )) define an R ′ -morphism H R ′ −→ Hom(Γ , H ) R ′ of finitepresentation, that we call an orbit morphism (here R ′′ runs over R ′ -algebras). Thefiber over any other homomorphism φ ′ ∈ Hom(Γ , H ( R ′ )) of this morphism is thetransporter T H ( φ, φ ′ ) of φ , which to any R ′′ over R ′ associates the set-theoretictransporter from φ to φ ′ in H ( R ′′ ). Lemma A.1.
Assume that H is smooth and that Γ has order invertible in R .Then Hom(Γ , H ) is smooth over R , all the orbit morphisms are smooth and alltransporters are smooth.Proof. By finite presentation, to prove smoothness it suffices to prove formal smooth-ness. Let R ′ be an R -algebra and let I be an ideal of R ′ of square 0. We needto show that the map Hom(Γ , H ( R ′ )) −→ Hom(Γ , H ( R ′ /I )) is surjective. So let φ : Γ −→ H ( R ′ /I ) be a group homomorphism. By smoothness of H we may lift φ to a map h : Γ −→ H ( R ′ ). Consider the map Γ × Γ −→ ker( H ( R ′ ) −→ H ( R ′ /I ))that takes ( γ, γ ′ ) ∈ Γ × Γ to h ( γ ) h ( γ ′ ) h ( γγ ′ ) − . Note that conjugation by h ( γ )endowes the abelian group ker( H ( R ′ ) −→ H ( R ′ /I )) with an action of Γ that ac-tually only depends on φ . In fact, if we identify ker( H ( R ′ ) −→ H ( R ′ /I )) withthe R ′ /I -module Lie( H ) ⊗ R I then this action is induced by the R ′ /I -linear actionof Γ on Lie( H ) ⊗ R R ′ /I given by the adjoint representation composed with thehomomorphism φ . Now the map defined above is a 2-cocycle, hence since | Γ | isinvertible in R , it has to be cohomologically trivial, so there is a map k : Γ −→ ODULI OF LANGLANDS PARAMETERS 67 ker( H ( R ′ ) −→ H ( R ′ /I )) such that h ( γ ) h ( γ ′ ) h ( γγ ′ ) − = k ( γ )( h ( γ ) k ( γ ′ )) k ( γγ ′ ) − .Then the map γ φ ( γ ) := k ( γ ) − h ( γ ) is a group homomorphism φ : Γ −→ H ( R ′ )that lifts φ , and the smoothness of Hom(Γ , H ) follows.Now fix a homomorphism φ : Γ −→ H ( R ) and let us show that the correspondingorbit morphism is smooth, by using the infinitesimal criterion. Let again R ′ be an R -algebra together with an ideal I of square 0, and let φ ′ be another homomorphismΓ −→ H ( R ′ ) whose image φ ′ in Hom(Γ , H ( R ′ /I )) is conjugate to φ by an element h in H ( R ′ /I ). We must find an element h ∈ H ( R ′ ) that conjugates φ ′ to φ . Bysmoothness of H we can pick an element h ′ ∈ H ( R ) that maps to h . Then themap γ φ ( γ )( h ′ φ ′ ( γ ) − h ′− ) defines a 1-cocycle of Γ in ker( H ( R ′ ) −→ H ( R ′ /I ))endowed with the action associated with φ as above. By the same argument asabove, this cocycle is a coboundary, so there is some k ∈ ker( H ( R ′ ) −→ H ( R ′ /I ))such that φ ( γ ) h ′ φ ′ ( γ ) − h ′− = ( φ ( γ ) kφ ( γ ) − ) k − , from which we get an element h = k − h ′ as desired. Hence the orbit morphism is smooth. By base change, thecentralizers and the transporters are therefore smooth too. (cid:3) Suppose now that we are given an action of Γ on H by automorphisms of groupschemes over R . Identifying 1-cocycles Γ −→ H ( R ′ ) with cross-section homomor-phisms Γ −→ H ⋊ Γ, we see that the functor R ′ Z (Γ , H ( R ′ )) is represented byan R -scheme that is a direct summand of Hom(Γ , H ⋊ Γ). We denote this schemeby Z (Γ , H ). It is stable under the conjugation action of H ⋊ Γ restricted to H .When H is smooth, so is H ⋊ Γ, hence the above lemma implies :
Corollary A.2.
Assume that H is smooth and that Γ has order invertible in R .Then Z (Γ , H ) is smooth over R , all the H -orbit morphisms are smooth and alltransporters are smooth. A.2.
The sheafy quotient.
We henceforth assume that H is smooth and Γ hasorder invertible in R , and we are now interested in the quotient object H (Γ , H ) of Z (Γ , H ) by the conjugation action of H . As for now, we define it as the quotientsheaf, say for the ´etale topology, that is, the sheaf associated to R H (Γ , H ( R )). Corollary A.3.
Assume that R is a local Henselian G -ring, and denote by k itsresidue field. Then the map H (Γ , H ( R )) −→ H (Γ , H ( k )) is a bijection.Proof. By smoothness and Artin approximation, any k -point of Z (Γ , H ) extends toa section over R , that is, the map Z (Γ , H ( R )) −→ Z (Γ , H ( k )) is surjective. Hencethe map of the lemma is surjective too. To prove injectivity, let φ , φ ′ : Γ −→ H ( R )be two 1-cocycles whose images φ , φ ′ are H -conjugate in Hom(Γ , H ( k ) ⋊ Γ) by some h ∈ H ( k ). By the previous lemma, the transporter scheme T H ( φ, φ ′ ) is smoothover R . Hence by Artin approximation its k -point h extends to an R -section h that conjugates φ to φ ′ . (cid:3) Lemma A.4.
Assume that H is reductive, that R is a strictly Henselian local G -ring, and denote by R ′ any non-zero R -algebra.(i) the map H (Γ , H ( R )) −→ H (Γ , H ( R ′ )) is injective.(ii) it is surjective if R is a d.v.r. and R ′ is a strictly Henselian local G -ring.Proof. We adapt the proof of Thm 4.8 of [BHKT19].(i) We need to prove that if two cocycles φ , φ ′ in Z (Γ , H ( R )) get H ( R ′ )-conjugate in Hom(Γ , H ( R ′ ) ⋊ Γ), then they are H ( R )-conjugate. By the last corol-lary, it suffices to prove that their images φ , φ ′ ∈ Z (Γ , H ( k )) are H ( k )-conjugate. We will need V. Lafforgue’s theory of pseudocharacters for the group H ⋊ Γ. Thisnotion is introduced without name nor formal definition in the preamble of Propo-sition 11.7 of [Laf18]. A formal definition is given in [BHKT19, Def 4.1] wherethe name “pseudocharacter” is also introduced. Unfortunately, unlike Lafforgue,these authors restrict attention to connected (split reductive) groups. However, onehas merely to replace Z [ ˆ G n ] ˆ G by Z [( H ⋊ Γ) n ] H in [BHKT19, Def 4.1] to get thecorrect definition for the non-connected group H ⋊ Γ (note that H is a split reduc-tive group over R , since R is strictly Henselian). Then, as in [BHKT19, Lemma4.3], it follows from the definition that any homomorphism φ : Γ −→ H ( R ) ⋊ Γdefines a “ H ⋊ Γ-pseudocharacter of Γ over R ” denoted by Θ φ . Moreover, if φ , φ ′ in Z (Γ , H ( R )) become H ( R ′ )-conjugate in Hom(Γ , H ( R ′ ) ⋊ Γ), then Θ φ ≡ Θ φ ′ [mod I ]where I = ker( R −→ R ′ ) (as in lemmas 4.3 and 4.4.i of [BHKT19]). Thereforewe get Θ φ = Θ φ ′ . Then, the main result on pseudocharacters asserts that thesemi-simplifications of φ and φ ′ are conjugate under H ( k ). Here, the notion ofsemi-simplicity is the notion of H ⋊ Γ-complete reducibility of [BMR05, § k has character-istic 0 and in [BHKT19, Thm 4.5] in any characteristic, but in the connected case.We leave the reader convince himself that their argument can be adapted to thenon-connected case in any characteristic.It suffices now to show that φ and φ ′ are actually H ⋊ Γ-completely reducible.Choose an R -parabolic subgroup P of H ⋊ Γ containing φ (Γ) and minimal for thisproperty. Let P π −→ L P be its Levi quotient and let L P ι −→ P be a Levi section of P . Then φ ss := ι ◦ π ◦ φ is by definition a semisimplification of φ . If we denote by U P the unipotent radical of P , the map Γ −→ U P , γ φ ss ( γ ) φ ( γ ) − is a 1-cocyclefor the action of Γ by conjugation on U P through φ . But U P has a d´evissage bycopies of the additive group k , and H (Γ , k ) is trivial since | Γ | is invertible in k .Therefore the above 1-cocycle is a coboundary, and we can find some u ∈ U P suchthat φ ss ( γ ) φ ( γ ) − = u − φ ( γ ) uφ ( γ ) − . So u − conjugates φ to φ ss and φ issemisimple (ie H ⋊ Γ-completely reducible) as claimed.(ii) The general case follows from combinations of the two following special cases.(a) The case where R ′ = ¯ K is an algebraic closure of the fraction field K of R .This case will follow from the following facts of Bruhat-Tits theory : any vertex ofthe semi-simple building B ( H, K ) becomes hyperspecial in B ( H, K ′ ) for a suitablefinite extension K ′ of K and, moreover, two hyperspecial points in B ( H, K ′ ) become H ( K ′′ )-conjugate in B ( H, K ′′ ) for some further finite extension K ′′ . Note thatthese facts hold even though K is not necessarily complete. Observe also that Γacts on the building B ( H, K ) and fixes the hyperspecial point o fixed by H ( R ). Nowlet φ ∈ Z (Γ , H ( ¯ K )). Then φ belongs to Z (Γ , H ( K )) for some finite extension K of K . Pick a point x of B ( H, K ) fixed by L φ (Γ) ⊂ H ( K ) ⋊ Γ. It becomeshyperspecial over some finite extension K of K and we may even assume thatthere is some h ∈ H ( K ) such that hx = o . Then h ( L φ )(Γ) fixes o so, writing h ( L φ )( γ ) = ( h φ ( γ ) , γ ) ∈ H ( K ) ⋊ Γ, we see that h φ ( γ ) fixes o hence belongs to H ( K ) o = H ( R ) Z H ( K ) for all γ ∈ Γ, i.e. h φ ∈ Z (Γ , H ( R ) Z H ( K )), where R is the normalization of R in K . Now, note that H (Γ , Z H ( K ) /Z H ( R )) maynot be trivial, but maps trivially in H (Γ , Z H ( K ) /Z H ( R )) for any further finiteextension K such that | Γ | divides Z H ( K ) /Z H ( R ) Z H ( K ). This means that thereis z ∈ Z H ( K ) such that zh φ ∈ Z (Γ , H ( R )). But R is an Henselian local R -algebra with the same residue field as R , so by the previous corollary there is ODULI OF LANGLANDS PARAMETERS 69 h ′ ∈ H ( R ) such that h ′ zh φ ∈ Z (Γ , H ( R )). So the class [ φ ] in H (Γ , H ( ¯ K )) is theimage of [ h ′ zh φ ] ∈ H (Γ , H ( R )), as desired.(b) The case where R is an algebraically closed field. Here, using the last corollarywe may assume that R ′ is also a (by assumption algebraically closed) field. Thiscase can certainly be handled via pseudocharacters. Namely, using [BHKT19, Thm4.5] and the fact that all morphisms Γ −→ H ( R ) ⋊ Γ are H ⋊ Γ-semisimple (asproved above), we see that it suffices to prove that any H ⋊ Γ-pseudocharacter ofΓ over R ′ is actually R -valued. However, the result is true under the much moregeneral assumption that H is smooth over R . Indeed, since the orbit morphismsare smooth, the H ( R )-orbits in Z (Γ , H ( R )) are open for the Zariski topology.Since two orbits are either equal or disjoint, there are only finitely many of them.Let φ , · · · , φ n be representatives. The orbit morphisms yield a smooth surjectivemorphism ( ⊔ ni =1 H ) −→ Z (Γ , H ) which induces in turn a surjection on R ′ -points( ⊔ ni =1 H ( R ′ )) −→ Z (Γ , H )( R ′ ) since R ′ is algebraically closed. So we see that each H ( R ′ )-orbit in Z (Γ , H ( R ′ )) comes from an H ( R )-orbit in Z (Γ , H ( R )). (cid:3) Recall now the ´etale sheafification H (Γ , H ) of the functor R ′ H (Γ , H ( R ′ ))on R -algebras. Here we consider the “big” site of affine schemes of finite presen-tation over R with the ´etale topology. The maps H (Γ , H ( R )) −→ H (Γ , H ( R ′ ))define a morphism from the constant presheaf associated to the set H (Γ , H ( R ))to the presheaf R ′ H (Γ , H ( R ′ )). It induces in turn a morphism of sheaves H (Γ , H ( R )) −→ H (Γ , H )where the left hand side is a “constant” sheaf. Proposition A.5.
Suppose that H is reductive over a strictly Henselian discretevaluation G -ring R in which the order of Γ is invertible. Then the above morphismof sheaves is an isomorphism. In particular, H (Γ , H ) is representable by a productof finitely many copies of R .Proof. We first note that the functor R ′ H (Γ , H ( R ′ )) defined over all R -algebrascommutes with filtered colimits. Indeed, this property is certainly true for the func-tors R ′ Z (Γ , H ( R ′ )) and R ′ H ( R ′ ) since both these functors are representedby finitely presented R -algebras. Elementary formal nonsense shows that this prop-erty holds in turn for the quotient functor R ′ H (Γ , H ( R ′ )).Therefore, if A is any R -algebra and x is a geometric point of Spec( A ) then,writing A shx for the strict henselization of A at x , the set H (Γ , H ( A shx )) is thestalk of the sheaf H (Γ , H ) at x . When A is finitely presented over R , the ring A shx is a G -ring. So by the last lemma, the map H (Γ , H ( R )) −→ H (Γ , H ( A shx ))is bijective. This means that the morphism of sheaves under consideration is anisomorphism on stalks. Thus it is an isomorphism. (cid:3) Remark A.6.
Here is a concrete paraphrase of the proposition. First note thatthe map Z (Γ , H ( R )) −→ H (Γ , H )( R ) is surjective since R is strictly Henselian,so that we can pick a finite subset Φ ⊂ Z (Γ , H ( R )) mapping bijectively to H (Γ , H )( R ). Now, suppose that A is an integral finitely generated R -algebraand let φ be a 1-cocycle Γ −→ H ( A ). Then there is a unique cocycle φ ∈ Φ anda faithfully ´etale map A −→ A ′ such that φ is H ( A ′ )-conjugate to the “constant”cocycle φ .We now globalize a bit the previous proposition. Theorem A.7.
Suppose that H is reductive over a Dedekind G -ring R in which theorder of Γ is invertible. Then H (Γ , H ) is representable by a finite ´etale R -algebra.Proof. Let ¯ K be an algebraic closure of the fraction field K of R . For a closedpoint s of Spec( R ), denote by R shs a strict henselization of R at s and by ¯ K s an algebraic closure of its fraction field. Let us choose a set of representativesΦ s ⊂ Z (Γ , H ( R shs )) of H (Γ , H ( R shs )). Since Z (Γ , H ) is finitely presented, theserepresentatives are defined over some ´etale R -domain R ′ , so that Φ s comes froma subset Φ R ′ ⊂ Z (Γ , H ( R ′ )). Now if s ′ is another closed point of Spec( R ) in theimage of Spec( R ′ ) and if we choose an R -morphism R ′ −→ R shs ′ , then we claim thatthe natural map Φ R ′ → H (Γ , H ( R shs ′ )) is also a bijection. Indeed, this follows fromthe following commutative diagram H (Γ , H ( R shs )) ∼ / / H (Γ , H ( ¯ K s ))Φ R ′ qqqqqqqqqqq & & ▼▼▼▼▼▼▼▼▼▼▼ / / H (Γ , H ( ¯ K )) ∼ h h ◗◗◗◗◗◗◗◗◗◗◗◗ ∼ v v ♠♠♠♠♠♠♠♠♠♠♠♠ H (Γ , H ( R shs ′ )) ∼ / / H (Γ , H ( ¯ K s ′ ))where we have chosen an R -embedding R ′ ֒ → ¯ K and two R ′ -embeddings ¯ K ֒ → ¯ K s and ¯ K ֒ → ¯ K s ′ , and where the ∼ denote bijections granted by Lemma A.4. Asa consequence, denoting by Φ R ′ the constant sheaf on R ′ -algebras associated tothe set Φ R ′ , we see as in the last proof that the natural morphism of sheavesΦ R ′ −→ H (Γ , H ) R ′ is an isomorphism.Now, varying the point s and using the quasicompacity of Spec( R ) we get afaithfully ´etale morphism R ֒ → R ′′ = R ′ × · · · × R ′ n and a set Φ R ′′ = Φ R ′ ×· · · × Φ R ′ n such that the natural morphism of sheaves on R ′′ -algebras Φ R ′′ −→ H (Γ , H ) R ′′ is an isomorphism. In particular, the sheaf H (Γ , H ) is representableafter base change to R ′′ by a sum of copies of R ′′ . Since the map of (Spec R ) ´et -sheaves H (Γ , H ) × Spec R Spec R ′′ −→ H (Γ , H ) is visibly representable, ´etale andsurjective, it follows that H (Γ , H ) is an algebraic space over (Spec R ) ´et . Thisalgebraic space has to be finite ´etale (and in particular separated) over R since itis so after base change to R ′′ . Hence by Corollary II.6.17 of [Knu71], this algebraicspace is actually a scheme, and it is finite ´etale over R . (cid:3) A.3.
Relation with the affine GIT quotient.
Let us investigate the relationshipbetween H (Γ , H ) and another natural quotient of Z (Γ , H ) by H . Namely, denoteby O the R -algebra such that Z (Γ , H ) = Spec( O ). The action of H on Z (Γ , H )translates into a comodule structure O ρ −→ O ⊗ R R [ H ] on O under the Hopf R -algebra R [ H ] corresponding to H . As usual, put O H := ker( ρ − id ⊗ ε )where ε is the counit of R [ H ]. Then the morphism Spec( O ) −→ Spec( O H ) is acategorical quotient of Z (Γ , H ) by H in the category of affine R -schemes.Note that H (Γ , H ) is a categorical quotient in the much larger category ofsheaves on the big ´etale site of Spec( R ). However, under suitable assumptions,Theorem A.7 shows that it is actually represented by an affine R -scheme. So, by ODULI OF LANGLANDS PARAMETERS 71 uniqueness of categorical quotients, we conclude that up to a unique isomorphism,we have H (Γ , H ) = Spec( O H ) , which we summarize in the following corollary. Corollary A.8.
Suppose that H is reductive over a Dedekind G -ring R in whichthe order of Γ is invertible. Then O H is a finite ´etale R -algebra and represents thesheaf H (Γ , H ) . In particular, its formation commutes with any change of rings R −→ R ′ . A.4.
Representatives.
Suppose that H is reductive over a Dedekind G -ring R in which the order of Γ is invertible. Theorem A.7 ensures that after replacing R by a finite ´etale extension, H (Γ , H ) is a constant sheaf (associated to the set H (Γ , H )( R )). The map Z (Γ , H ( R )) −→ H (Γ , H )( R ) need not be surjective,but if R is any R -algebra such that H (Γ , H )( R ) is in the image of the map Z (Γ , H ( R )) −→ H (Γ , H )( R ), then for any finite set Φ ⊂ Z (Γ , H ( R )) map-ping bijectively to H (Γ , H )( R ), the constant sheaf property ensures that : for anyconnected R -algebra A and any φ ∈ Hom(Γ , H ( A )) , there is a unique φ ∈ Φ suchthat φ and φ become H ( A ′ ) -conjugate in Hom(Γ , H ( A ′ )) for some faithfully ´etale A -algebra A ′ . By definition of H (Γ , H ), we certainly can find a R as above that is faithfully´etale over R . However in general, it is not clear whether we can find R finite ´etaleover R . The following result uses the strong approximation property to prove that,if R is a localization of a ring of integers in a number field, then we can at leastfind R finite (not necessarily ´etale) over R . Theorem A.9.
Assume that H is reductive over a normal subring R of somenumber field K , and that Γ has invertible order in R . Then there is a finite extension K of K and a finite set Φ ⊂ Z (Γ , H ( R )) (with R the normalization of R in K ) such that for any connected R -algebra A and any φ ∈ Z (Γ , H ( A )) , there is aunique φ ∈ Φ such that φ and φ becomes H ( A ′ ) -conjugate in Z (Γ , H ( A ′ )) forsome faithfully ´etale A -algebra A ′ .Proof. As we have just argued, we may assume that H (Γ , H ) is a constant sheaf,and the problem boils down to finding K such that the map Z (Γ , H ( R )) −→ H (Γ , H )( R ) = H (Γ , H )( R )is surjective. We certainly can find a faithfully ´etale R ′ over R such that any[ φ ] ∈ H (Γ , H )( R ) has a representative φ ′ ∈ Z (Γ , H ( R ′ )). Let us choose such data,and assume further that H is split over R ′ . Let R ′ = Q ni =1 R ′ i be the decomposi-tion of R ′ in connected components and let φ = ( φ i ) i =1 , ··· ,n be the correspondingdecomposition of φ . Replacing R by its normalization in the residue field at somegeneric point of R ′ ⊗ R · · · ⊗ R R ′ n , we may assume that each R ′ i is a localization of R (i.e. Spec( R ′ ) −→ Spec( R ) is a Zariski cover). Then we simplify the notation andwrite R i := R ′ i . Since all φ i map to the same element [ φ ] ∈ H (Γ , K ), they becomepairwise H -conjugate over some finite extension of K . Replacing R by its normal-ization in this finite extension, we may thus assume that they are H ( K )-conjugatein Z (Γ , H ( K )). Actually we may, and we will, even assume that they are pairwise Z ( H ) ◦ ( K ) × H sc ( K )-conjugate through the canonical isogeny Z ( H ) ◦ × H sc −→ H ,where H sc denotes the simply connected covering group of the adjoint group H ad . We now try to construct a φ ∈ Z (Γ , H ( K )) that is H ( K )-conjugate to each φ i ,and such that φ (Γ) ⊂ H ( R ).If n = 1, we are obviously done. Otherwise, start with φ and pick elements( z i , h i ) ∈ Z ( H ) ◦ ( K ) × H sc ( K ) such that z i h i φ = φ i in Z (Γ , H ( K )), for all i =2 , · · · , n . For any prime p ∈ S := Spec( R ) \ Spec( R ) there is some i ≥ p ∈ Spec( R i ). Pick such an i and put ( z p , h p ) := ( z i , h i ). Since H sc is a split simplyconnected semisimple group over K , the strong approximation theorem with thefinite set S ensures the existence of an element h ∈ H sc ( R ) such that h ∈ H sc ( R p ) h p for all p ∈ S . Then we have ( h φ )(Γ) ⊂ H ( R ) and ( z p h φ )(Γ) ⊂ H ( R p ) for all p ∈ S . Now, since Z ( H ) ◦ is a split torus, say of dimension d , the obstruction tofinding z ∈ Z ( H ) ◦ ( R ) ∩ T p Z ( H ) ◦ ( R p ) z p lies in the d th power of the ideal classgroup C ℓ ( K ) d . Hence it vanishes over the Hilbert class field K h of K and we canat least find z ∈ Z ( H ) ◦ ( R h ) ∩ T p Z ( H ) ◦ ( R h p ) z p , where the superscript h denotesnormalization in K h . Then we see that zh φ (Γ) ⊂ H ( R h ) and zh φ (Γ) ⊂ H ( R h p )for all p . Therefore we have ( zh φ )(Γ) ⊂ H ( R h ) as desired. (cid:3) Remark A.10 (Orbits) . With the notation of the theorem, the morphism Z (Γ , H ) R π −→ H (Γ , H ) R = { π ( φ ) , φ ∈ Φ } provides a decomposition as a disjoint union of affine R -schemes Z (Γ , H ) R = G φ ∈ Φ π − ( π ( φ ))Moreover, the action h h · φ of H R provides a surjective morphism of R -schemes H R −→ π − ( π ( φ )), which at the level of ´etale sheaves identifies π − ( π ( φ )) withthe quotient H R /C H ( φ ) with C H ( φ ) denoting the centralizer of φ . In particular,we see that this quotient sheaf is representable by an affine scheme which identifieswith the orbit H · φ := π − ( π ( φ )) of φ . To put it in different words, the naturalmap H −→ H · φ , h h · φ is a C H ( φ )-torsor for the ´etale topology.A.5. Centralizers.
Our next task is to study the centralizer C H ( φ ) of a cocycle φ ∈ Z (Γ , H ( R )). We have seen in Lemma A.1 that this is a smooth group schemeover R . Moreover, by [PY02, Thm 2.1], its geometric fibers have reductive neutralcomponents. In other words, the “neutral” component C H ( φ ) ◦ is a reductive groupscheme over R . Thus it follows from Prop 3.1.3 of [Con14] that the quotient sheaf π ( C H ( φ )) := C H ( φ ) /C H ( φ ) ◦ is representable by a separated ´etale group schemeover R . Our aim here is to prove that π ( C H ( φ )) is actually finite over R , at leastwhen R is a Dedekind G -ring and Γ is a solvable group.Note that C H ( φ ) is also the subgroup of Γ-fixed points in H for the Ad φ -twistedaction of Γ on H . So, up to changing the action of Γ on H , it suffices to study thefiniteness of π ( H Γ ) as an R -scheme. Lemma A.11. i) Let H ′ −→ H be a Γ -equivariant central isogeny of reductivegroup schemes over R . If π ( H ′ Γ ) is finite over R , then so is π ( H Γ ) .ii) Let Γ ′ be a normal subgroup of Γ . If π ( H Γ ′ ) and π (( H Γ ′ , ◦ ) Γ / Γ ′ ) are finite,then so is π ( H Γ ) .Proof. i) Let Z be the kernel of the isogeny, which is a finite central subgroupscheme of H ′ of multiplicative type over R . We claim that the sheaf H (Γ , Z ) isrepresentable by a finite ´etale group scheme over R . Indeed, since the category offinite group schemes of multiplicative type over R is abelian ([SGA3-II, IX.2.8]), the ODULI OF LANGLANDS PARAMETERS 73 sheaves Z (Γ , Z ), B (Γ , Z ) and, consequently, H (Γ , Z ) are finite group schemesof multiplicative type over R . Let us decompose Z = Q p Z p into a finite productof its p -primary components. Then H (Γ , Z ) decomposes accordingly as a productof H (Γ , Z p ). But H (Γ , Z p ) is trivial unless p divides the order of Γ. Since thisorder is invertible in R , so is the rank of H (Γ , Z ), which is therefore ´etale over R .Let us now look at the following exact sequence of sheaves of groups on the big´etale site of Spec( R ).1 −→ Z Γ −→ H ′ Γ −→ H Γ −→ H (Γ , Z ) −→ H (Γ , H ′ ) . In this sequence, we now know that all terms are R -schemes. Since H (Γ , Z ) isfinite ´etale, the morphism H Γ −→ H (Γ , Z ) has to be trivial on the reductivesubgroups ( H Γ ) ◦ , so that we deduce the following exact sequence : Z Γ −→ π ( H ′ Γ ) −→ π ( H Γ ) −→ H (Γ , Z ) −→ H (Γ , H ′ ) . Now assume that π ( H ′ Γ ) is finite over R , and therefore finite ´etale. Since Z Γ isfinite, its image in π ( H ′ Γ ) is closed, hence is finite ´etale. Therefore π ( H Γ ) appearsas the middle term of a five terms exact sequence in which all the four remainingterms are finite ´etale group schemes (the last one is only a pointed scheme and is´etale by theorem A.7). Going to a finite ´etale covering R ′ of R over which all these´etale groups become constant, we see that π ( H Γ ) also becomes constant and finiteover R ′ , hence is already finite over R .ii) Put H ′ := ( H Γ ′ ) ◦ . Applying the Γ / Γ ′ -invariants functors to the exact se-quence H ′ ֒ → H Γ ′ ։ π ( H Γ ′ ), we get an exact sequence1 −→ ( H ′ ) Γ / Γ ′ −→ H Γ −→ π ( H Γ ′ ) Γ / Γ ′ −→ H (Γ / Γ ′ , H ′ ) . By assumption, π ( H Γ ′ ) is finite ´etale, so the invariant subgroup π ( H Γ ′ ) Γ / Γ ′ isalso ´etale and finite since it is closed. Therefore the map from H Γ factors over π ( H Γ ). Since (( H ′ ) Γ ) ◦ = ( H Γ ) ◦ , we thus get an exact sequence1 −→ π (( H ′ ) Γ / Γ ′ ) −→ π ( H Γ ) −→ π ( H Γ ′ ) Γ / Γ ′ −→ H (Γ / Γ ′ , H ′ ) . All terms but possibly the middle one are finite ´etale (by Theorem A.7 for the lastone). Therefore, the middle one is also finite ´etale, as desired. (cid:3)
Theorem A.12.
Assume that H is reductive over a Dedeking G -ring R and isacted upon by a solvable finite group Γ with invertible order in R . Then π ( H Γ ) isa finite ´etale group scheme over R .Proof. As already mentioned in the beginning of this subsection, the problem isto prove finiteness. Thanks to item ii) of the last lemma, we can use induction toreduce the case of a solvable Γ to the case of an abelian Γ, and then further reduceto the case of a cyclic Γ. So let us assume that Γ is cyclic.By Theorem 5.3.1 of [Con14], there is a unique closed semi-simple subgroupscheme H der of H over R that represents the sheafification of the set-theoreticalderived subgroup and such that the quotient H/H der is a torus. Then the naturalmorphism Z ( H ) ◦ × H der −→ H is a central isogeny by the fibrewise criterion,and moreover is Γ-equivariant (here Z ( H ) ◦ denotes the maximal central torus of H ). Further, by Exercise 6.5.2 of [Con14], there is a canonical central isogeny H sc −→ H der over R , such that all the geometric fibers of H sc are simply connectedsemi-simple groups. Being canonical, the action of Γ on H der lifts uniquely to H sc .Let us now consider the Γ-equivariant central isogeny Z ( H ) ◦ × H sc −→ H . By item i) of the previous lemma, it suffices to prove the finiteness of π (( Z ( H ) ◦ ) Γ ) and thatof π (( H sc ) Γ ). The first one is clear since ( Z ( H ) ◦ ) Γ is smooth and of multiplicativetype. For the second one, we use Steinberg’s theorem [Ste68, Thm 8.2], which canbe applied here since a generator of Γ induces a semisimple automorphism of eachgeometric fiber of H sc , and which ensures that ( H sc ) Γ has connected fibers, so that π (( H sc ) Γ ) is even the trivial group. (cid:3) A.6.
Splitting a reductive group scheme over a finite flat extension.
Areductive group scheme over any ring R is known to split over a faithfully ´etaleextension of R . However, in general it won’t split over a finite ´etale extension.Already over R = Z , there are examples where a non-trivial Zariski localization isneeded. Here we use a similar argument as in the proof of Theorem A.9 in order toprove that if R is a localization of a ring of integers, then a reductive group schemeover R splits over a suitable finite flat extension of R . Proposition A.13.
Assume that H is reductive over a normal subring R of anumber field K . Then there is a finite extension K of K such that H splits overthe normalization R of R in K .Proof. Pick a faithfully ´etale R ′ over R such that H splits over R ′ . Let R ′ = Q ni =1 R ′ i be the decomposition of R ′ in connected components. Of course, if n = 1 we aredone, so we assume n >
1. Replacing R by its normalization in the residue field atsome generic point of R ′ ⊗ R · · ·⊗ R R ′ n , we may assume that each R ′ i is a localizationof R (i.e. Spec( R ′ ) −→ Spec( R ) is a Zariski cover). Let T i ⊂ H R ′ i be a split maximaltorus defined over R ′ i . The generic fibers T i,K are split maximal tori in H K , henceare conjugate under H ( K ). After replacing K by a finite extension, we may assumethat they are conjugate under H sc ( K ). So there are elements h i ∈ H sc ( K ), i > h i T ,K = T i,K . Put S := Spec( R ) \ Spec( R ′ ) (a finite set) and for p ∈ S , pick a i > p ∈ Spec( R ′ i ) and put h p = h i . Then by the strongapproximation theorem, there is some h ∈ H sc ( R ′ ) such that h ∈ H sc ( R p ) h p for all p ∈ S . We claim that the K -torus T K := h T ,K of H K extends (canonically) to aan R -subtorus of H . Indeed, recall that the functor Tor H/R which to any R -algebra R ′ associates the set of maximal subtori of H R ′ is known to be representable by asmooth quasi-affine, hence in particular separated, scheme over R , see e.g. [Con14,Thm 3.2.6]. By construction T comes from a R ′ -torus T ′ of H R ′ , which is uniqueby separateness of Tor H/R . Similarly for each p ∈ S , there is a unique extensionof T K to a R p -torus T p of H R p . This means that the K -section of Tor H/R givenby T K extends uniquely to a Zariski covering of Spec R , hence extends to Spec R itself, whence a maximal torus T R of H extending T K . Since T K is split and sincetori are known to split over finite ´etale coverings of the base, T R is split too.Now, the root subspaces of T R in Lie( H ) are rank 1 locally free R -modules.Replacing K by its Hilbert class field, we may assume that they are actually free.Since R is connected, this is enough for H to split over R . (cid:3) Remark A.14.
Exercise 7.3.9 of [Con14] provides another proof that does notuse strong approximation. Namely, start by enlarging R so that H K splits. So H K contains a Borel subgroup B K , which extends uniquely to a Borel subgroupscheme B of H by the properness of the scheme of Borel subgroups. Let ( H ′ , B ′ )be the constant split pair over R that extends ( H K , B K ). Then the functor I of isomorphisms between ( H ′ , B ′ ) and ( H, B ) is a torsor over the automorphism
ODULI OF LANGLANDS PARAMETERS 75 group A = B ′ ad ⋊ Out( H ) of the pair ( H ′ , B ′ ). Its class in H (Spec R, A ) hastrivial image in H (Spec R, Out( H )) since H is split over K . On the other hand H (Spec R, B ′ ad ) is isomorphic to a sum of copies of Pic( R ) = H (Spec R, G m ). Solet K be the Hilbert class field of K . Since Pic( R ) −→ Pic( R ) has trivial image, I becomes a trivial A -torsor over R , hence H splits over R . Appendix B. Twisted Poincar´e polynomials
B.1.
Some characteristic polynomials attached to root data.
Let Σ =( X , X ∨ , ∆ , ∆ ∨ ) be a based root datum with Weyl group Ω and group of auto-morphisms Aut(Σ). Both Ω and Aut(Σ) embed as groups of linear automorphismsof X and X ∨ , and Aut(Σ) normalizes Ω. In particular Aut(Σ) acts on the ringof Ω-invariant polynomials Sym • ( X ) Ω on X ∨ and on the conormal module M • of X ∨ / Ω along the zero section M • := Sym • > ( X ) Ω / (Sym • > ( X ) Ω ) . For any α ∈ Aut(Σ) we consider its weighted characteristic polynomial on M • Q (B.1) χ α | M • ( T ) := Y d> det (cid:0) T d − α | M d Q (cid:1) ∈ Z [ T ] . A priori M • may have torsion, but a result of Demazure [Dem73, Thm 3] showsthat M • ⊗ Z [ | Ω | ] is torsion free, so we deduce the following Remark B.1. If ℓ does not divide the order of Ω, the image of χ α | M • in F ℓ [ T ] isthe weighted characteristic polynomial of α on M • F ℓ .Since Ω is a reflection subgroup of Aut( X ∨ ), the ¯ Q -algebra Sym • ( X ) Ω¯ Q is knownto be a weighted polynomial algebra. More precisely, any graded section M • ¯ Q ֒ → Sym • > ( X ) Ω¯ Q induces a graded isomorphism Sym( M • ¯ Q ) ∼ −→ Sym • ( X ) Ω¯ Q . In particular,for α = 1, we have χ | M • ( T ) = Q ri =1 ( T d i −
1) where r = rk Z ( X ) and d ≤ · · · ≤ d r are the so-called fundamental degrees of Ω acting on X ∨ . Here d r is known as the Coxeter number of Σ and is the maximal n ∈ N such that Φ n ( T ) divides χ | M • ( T ).More generally, using an Aut(Σ)-equivariant section M • ¯ Q ֒ → Sym • > ( X ) Ω¯ Q , we seethat, at least when α has finite order, χ α | M • = ( T d − ε ,α ) · · · ( T d r − ε r,α ) wherethe ε i,α are as in Lemma 6.1 of [Spr74]. Note that in this case, χ α | M • is a productof cyclotomic polynomials, hence it lies in Z [ T, T − ] and we have χ α | M • = χ α − | M • .The maximal n ∈ N such that Φ n ( T ) divides χ α | M • ( T ) has been known in theliterature as the twisted Coxeter number associated to α . Now, a fundamentalconsequence of Springer’s work in this setup is the following result. Proposition B.2. χ α | M • ( T ) is the lowest common multiple in ¯ Q [ T ] of the charac-teristic polynomials χ ωα | X ( T ) of ωα on X ¯ Q , where ω runs over Ω .Proof. When α has finite order, this is a reformulation of Theorem 6.2 (i) of [Spr74].In general, this follows from the decompositions X Q = X Ω Q ⊕ Q h ∆ i and M • Q = X Ω Q ⊕ Sym • > ( Q h ∆ i ) Ω / (Sym • > ( Q h ∆ i ) Ω ) and the fact that α | Q h ∆ i has finite order,while ωα | X Ω Q = α | X Ω Q for all ω ∈ Ω. (cid:3) B.2.
Application to reductive groups.
Let ˆ G be a reductive group over analgebraically closed field L of characteristic ℓ . Attached to ˆ G is a root datum Σ asbefore, with an identification Aut(Σ) = Out( ˆ G ). Now, let β be an automorphismof ˆ G with image α in Out( ˆ G ). Using the notation of the last subsection, we put(B.2) χ ˆ G,β ( T ) := χ α | M • ( T ) ∈ Z [ T ] . Further, we denote by h ˆ G,β the twisted Coxeter number of Σ associated to α andwe put(B.3) χ ∗ ˆ G,β ( T ) := Y n ≤ h ˆ G,β Φ n ( T ) ∈ Z [ T ] . The following result is crucial to track the “banal” primes in this paper.
Proposition B.3.
Let β be an automorphism of ˆ G .(1) If ˆ H is a reductive subgroup of ˆ G stable under β , then χ ˆ H,β divides χ ˆ G,β , h ˆ H,β ≤ h ˆ G,β and χ ∗ ˆ H,β divides χ ∗ ˆ G,β .(2) Let t be a semi-simple element of ˆ G ( L ) such that β ( t ) = t q . Then t has finiteorder, and this order divides χ ˆ G,β ( q ) .Proof. For (1), let T ˆ H be a maximal torus of ˆ H , and pick h ∈ ˆ H such that Ad h ◦ β stabilizes a pinning of ˆ H with torus T ˆ H . Using this pinning to identify X ∗ ( T ˆ H )with X ˆ H , the action of Ad h ◦ β on X ∗ ( T ˆ H ) corresponds to the action of the image α ˆ H of β in Out( ˆ H ). More generally, for all n ∈ N ˆ H ( T ˆ H ) with image ω ˆ H ∈ Ω ˆ H , theaction of Ad nh ◦ β on X ∗ ( T ˆ H ) is given by ω ˆ H α ˆ H . Now, let T be a maximal torusof ˆ G containing T ˆ H , and pick m ∈ C ˆ G ( T ˆ H ) such that Ad mnh ◦ β stabilizes T . Usingany pinning of ˆ G with torus T in order to identify X ∗ ( T ) and X ˆ G = X , the actionof Ad mnh ◦ β on X ∗ ( T ) is of the form ωα for some ω ∈ Ω ˆ G . Since m centralizes T ˆ H ,this action induces ω ˆ H α ˆ H on X ∗ ( T ˆ H ). It follows that the characteristic polynomial χ ω ˆ H α ˆ H | X ˆ H ( T ) divides the characteristic polynomial χ ωα | X ( T ). By Proposition B.2,we deduce that χ α ˆ H | M • ˆ H divides χ α | M • , as desired.(2) The connected centralizer ˆ H := C ˆ G ( t ) ◦ contains t and is stable under β ,so by (1) it suffices to prove the statement when t is central in ˆ G . Then we maycompose β with some Ad g so that it fixes a pinning of ˆ G , with maximal torus ˆ T .Now, consider t as a homomorphism X ∗ ( ˆ T ) −→ L × . Since β ( t ) = t q , we see thatthis homomorphism factors over the cokernel of the endomorphism β − q of X ∗ ( ˆ T ).But this cokernel is finite of order χ ˆ T ,β ( q ) = det( q − β ). So t has order dividing χ ˆ T ,β ( q ), hence also dividing χ ˆ G,β ( q ). (cid:3) B.3.
The Chevalley-Steinberg formula.
Let now G be a reductive group over F q . Let G ∗ be a split form of G ¯ F q over F q , so that there is an isomorphim ψ : G ¯ F q ∼ −→ G ∗ ¯ F q . Then Fr := Frob ψ − ◦ ψ is an automorphism of G ¯ F q , and we have thefollowing Chevalley-Steinberg formula for the number of F q -rational points of G . Theorem B.4 (Chevalley-Steinberg) . | G ( F q ) | = q N .χ G, Fr ( q ) , where N is the di-mension of a maximal unipotent subgroup of G ¯ F q . ODULI OF LANGLANDS PARAMETERS 77
Proof.
This formula is stated for absolutely simple adjoint groups in Theorems25 and 35 of [Ste16]. It is also true for a torus S , since we have an isomor-phism X ∗ ( S ) / ( q Fr − X ∗ ( S ) ∼ −→ S ( F q ) [DL76, (5.2.3)], from which it follows that | S ( F q ) | = | det( q Fr − | = | χ S, Fr − ( q ) | = χ S, Fr ( q ).To prove the formula in general, we first observe that if G π −→ G ′ is an isogeny,then | G ( F q ) | = | G ′ ( F q ) | . Indeed, the kernel H := ker( π )(¯ F q ) is a finite group withan action of Frobenius Fr and we have an exact sequence1 −→ H Fr −→ G ( F q ) −→ G ′ ( F q ) −→ H (Fr , H ) = H Fr −→ H ( F q , G ) = 1. But we also have an exactsequence H Fr ֒ → H Fr − id −→ H ։ H Fr which shows that | H Fr | = | H Fr | , so we get | G ( F q ) | = | G ′ ( F q ) | .Now we deduce the formula for general G by applying this observation to theisogeny G −→ G ab × G ad and decomposing G ad as a product of restriction of scalarsof absolutely simple groups. (cid:3) B.4.
Kostant’s section theorem.
We return to the setting of a reductive groupˆ G over an algebraically closed field L and, for simplicity, we assume that ˆ G is simpleadjoint. We also assume that the characteristic ℓ of L does not divide the order ofthe Weyl group Ω ˆ G .Let us fix a pinning ε = ( ˆ T , ˆ B, ( X α ) α ∈ ∆ ) of ˆ G . The sum E = P α ∈ ∆ X α is thena regular nilpotent element of Lie( ˆ G ). The sum H = P β ∈ Φ + ˇ β ⊗ ∈ X ∗ ( ˆ T ) ⊗ L =Lie( ˆ T ) is a regular semisimple element of Lie( ˆ G ) and the pair ( H, E ) is part of aunique “principal” sl -triple ( F, H, E ). Denote by Lie( ˆ G ) E the centralizer of E inLie( ˆ G ). Under our assumption on ℓ , Veldkamp has proved that Kostant’ sectiontheorem still holds, [Vel72, Prop 6.3]. This states that the mapLie( ˆ G ) E −→ Lie( ˆ G ) (cid:12) ˆ G, X ( F + X ) mod ˆ G is an isomorphism of varieties. Moreover, seeing λ := P β ∈ Φ + β ∨ as a cocharacter ofˆ G , this map is G m -equivariant for the action ( t, y ) t · y := t Ad λ ( t ) ( y ) on the LHSand the action ( t, x ) t x on the RHS. Composing with the Chevalley isomorphism(which also holds in this context) yields an isomorphism of G m -varieties π : Lie( ˆ G ) E ∼ −→ X ∨ L / Ω ˆ G . Now let Aut( ˆ G ) ε be the group of automorphisms of ˆ G that preserve the pinning ε . This group fixes E , so it acts on Lie( ˆ G ) E . It also acts on Lie( ˆ G ) (cid:12) ˆ G and X ∨ L / Ω ˆ G ,and both the Chevalley map and the Kostant map are equivariant for these actions.Identifying Out( ˆ G ) with Aut( ˆ G ) ε , we thus get on conormal modules at the originan isomorphism M • L ∼ −→ (Lie( ˆ G ) E ) ∗ which is Out( ˆ G )-equivariant, as well as G m -equivariant for the (dual) action de-scribed above on the RHS and the action associated with “twice the • -grading” onthe LHS. So we deduce the following result. Proposition B.5.
For t ∈ L × and β ∈ Aut( ˆ G ) ε of finite order, we have det (cid:16) t Ad λ ( t ) Ad β − id | Lie( ˆ G ) E (cid:17) = ± χ ˆ G,β ( t ) . Proof.
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Jean-Franc¸ois Dat, Institut de Math´ematiques de Jussieu, Sorbonne Universit´e, Uni-versit´e de Paris, CNRS 4, place Jussieu, 75252, Paris, France.
E-mail address : [email protected] David Helm, Department of Mathematics, Imperial College, London, SW7 2AZ, UnitedKingdom.
E-mail address : [email protected] Robert Kurinczuk, Department of Mathematics, Imperial College, London, SW7 2AZ,United Kingdom.
E-mail address : [email protected] Gil Moss, Department of Mathematics, The University of Utah, Salt Lake City, UT84112, USA.
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