Ramified Satake Isomorphisms for strongly parabolic characters
aa r X i v : . [ m a t h . R T ] O c t RAMIFIED SATAKE ISOMORPHISMS FOR STRONGLY PARABOLICCHARACTERS
MASOUD KAMGARPOUR AND TRAVIS SCHEDLER
Abstract.
For certain characters of the compact torus of a reductive p -adic group, which we callstrongly parabolic characters, we prove Satake-type isomorphisms. Our results generalize those ofSatake, Howe, Bushnell and Kutzko, and Roche. Contents
1. Introduction 11.1. The problem we study 11.2. History 21.3. On characters of T (O) W -invariant rational characters 72.3. Easy W -invariant characters 92.4. Extendable W -invariant characters 92.5. Comparison between easy and extendable 122.6. On parabolic characters 132.7. Proof of Theorem 3 133. Central families and Satake Isomorphisms 143.1. Recollections on decomposed subgroups 143.2. The subgroup K 153.3. Extension of ¯ µ Introduction
The problem we study.
Let F be a local non-Archimedean field with ring of integers O and residue field F q . Let G be a connected split reductive group over F with split torus T andWeyl group W = N G ( T )/ T . Let ˇ T denote the dual torus. Replacing G by an isomorphic group, we Mathematics Subject Classification.
Key words and phrases.
Satake isomorphisms, principal series, types, strongly parabolic characters, easy characters.M.K. was supported by a Hausdorff Institute Fellowship. T.S. was supported by an AIM Fellowship and the ARRA-funded NSF grant DMS-0900233. ay assume that G is defined over Z . Then G (O) is a maximal compact (open) subgroup of G ( F ) .Let H ( G ( F ) , G (O)) denote the convolution algebra of compactly supported G (O) -bi-invariantcomplex valued functions on G ( F ) . A celebrated theorem of Satake [Sat63] states that we have acanonical isomorphism of algebras(1.1) H ( G ( F ) , G (O)) ≃ C [ ˇ T / W ] . We are interested in generalizing this isomorphism to nontrivial smooth characters ¯ µ ∶ T ( O ) → C × ,as follows. Let W ¯ µ ⊆ W denote the stabilizer of ¯ µ under the action of the Weyl group. Then it isnatural to pose: Problem 1.
Construct a pair ( K, µ ) consisting of a compact open subgroup T ( O ) ⊆ K ⊆ G ( O ) and a character µ ∶ K → C × extending ¯ µ , such that we have an isomorphism of algebras (1.2) H ( G ( F ) , K, µ ) ≃ C [ ˇ T / W ¯ µ ] , where H is the convolution algebra of ( K, µ ) -bi-invariant compactly supported functions on G ( F ) . The Satake isomorphism provides a solution for the above problem for ¯ µ =
1. In this paper, wesolve the above problem for a large class of characters of T ( O ) which we call “strongly paraboliccharacters,” which are by definition characters such that W ¯ µ is the Weyl group of a Levi subgroup L < G , and moreover such that ¯ µ extends to L ( F ) . This appears to be the proper generalitywhere the problem has a positive solution. Our construction of K is tied to L . We think ofthe isomorphism H ( G ( F ) , K, µ ) ≃ C [ ˇ T / W ¯ µ ] as a Satake isomorphism for the (possibly) ramifiedcharacter ¯ µ . Therefore, we call these isomorphisms ramified Satake isomorphisms . For charactersthat are not strongly parabolic, we do not have a reason to expect a positive answer to Problem 1.1.2. History.
Following Satake, R. Howe studied Problem 1 for G = GL N [How73]. Via an iso-morphism which he called the ¯ µ -spherical Fourier transform, he completely solved the problem forthe general linear group. Howe’s paper went largely unnoticed; however, several cases of Problem1 were subsequently solved using other methods.In [Ber84], [Ber92], Bernstein constructed a decomposition of the category of representations of G ( F ) using the theory of Bernstein center. Each block admits a projective generator. In particular,for every character ¯ µ ∶ T ( O ) → C × , one has a block of representations of G ( F ) , which we denoteby R ¯ µ ( G ) . Bernstein proved that the center of R ¯ µ ( G ) is canonically isomorphic to C [ ˇ T / W ¯ µ ] ; see,for instance, [Roc09, Theorem 1.9.1.1]. Moreover, he gave an explicit description of a projectivegenerator for each of these blocks; see the RHS of (1.8). When the character ¯ µ is regular ; i.e., W ¯ µ = { } , then the center is C [ ˇ T ] , and it identifies canonically with the endomorphism ring ofBernstein’s generator.In a fundamental paper [BK98], Bushnell and Kutzko organized the study of representations of G ( F ) via compact open subgroups into the theory of types . Namely, they proposed that one shouldbe able to obtain a projective generator for every block of representations of G ( F ) by inducing afinite dimensional representation from a compact open subgroup. The pair of the compact opensubgroup and its finite dimensional representation, up to a certain equivalence, is called the type.In [BK99] and [BK93], they explicitly construct types for every block of representations of GL N .In particular, they construct projective generators for the principal series blocks R ¯ µ ( GL N ) . Whenthe character ¯ µ is regular, their construction provides a pair ( K, µ ) satisfying the requirementof Problem 1. We note, however, that Bushnell and Cuzco’s construction of types is technicallyinvolved, since they consider all blocks (not merely the principal series blocks); in particular, wewere not able to locate exactly where in their papers they construct types for the principal seriesblocks of GL N . inally, Roche [Roc98] constructed types for principal series representations of arbitrary reductivegroups in good characteristics (which excluded in particular those listed in Convention 6). In thecase that ¯ µ is regular, the type itself is a pair ( K, µ ) satisfying the conditions of Problem 1.In this paper, we build on the methods introduced by Bushnell and Kutzko and Roche, andsolve the problem for all strongly parabolic characters. We make use of Roche’s type in order toconstruct a pair ( K, µ ) satisfying the conditions of Problem 1.1.3. On characters of T ( O ) . A significant part of this paper, which may be of independentinterest, is devoted to defining and studying certain smooth characters of T ( O ) . Recall thata subgroup W ′ ⊆ W is parabolic if it is generated by simple reflections. The Levi subgroup L associated to W ′ is the subgroup generated by T and the the simple roots corresponding to thesimple reflections in W ′ along with their negatives. Definition 2.
Let ¯ µ ∶ T ( O ) → C × be a smooth character.(i) ¯ µ is parabolic if the stabilizer Stab W ( ¯ µ ) of ¯ µ in W is a parabolic subgroup.(ii) ¯ µ is strongly parabolic if it is parabolic with Levi L and extends to a character of L ( F ) .(iii) ¯ µ is easy it is parabolic and it extends to a character of L ( F ) which is trivial on [ L, L ]( F ) .It follows immediately from the definition that the trivial character and all regular charactersare easy. Moreover, it is clear that(1.3) easy Ô⇒ strongly parabolic Ô⇒ parabolic . The reverse implications can all fail; see Examples 19 and 27.To state our results regarding these characters, we need some notation. Let ∆ denote the setof roots of G . Let X , X ∨ , Q , Q ∨ denote the character, cocharacter, root and coroot lattices of G ,respectively. Below we will frequently impose the conditions that either X / Q is free or X ∨ / Q ∨ is free(or both). We remark that X ∨ / Q ∨ being free is equivalent to [ G ( C ) , G ( C )] being simply-connected,while X / Q being free is equivalent to the statement that G ( C ) has connected center. Theorem 3.
Let ¯ µ ∶ T ( O ) → C × be a smooth character. (i) ¯ µ is easy if and only if it is parabolic and can be written as a product χ ⋯ χ l , where each χ i is a character T ( O ) → C × which is a composition of a W ¯ µ -invariant rational character T ( O ) → O × and a smooth character O × → C × . (ii) The following are equivalent: (a) ¯ µ is strongly parabolic; (b) ¯ µ ○ α ∨ ∣ O × = , ∀ α ∈ ∆ L .Moreover, if q > , then these are also equivalent to: (c) ¯ µ extends to a character of L ( O ) . (iii) If X / Q is free or ∆ has no factors of type A or C n , then every parabolic character of T ( O ) is strongly parabolic. (iv) If X ∨ / Q ∨ is free, then every strongly parabolic character of T ( O ) is easy. (v) If ∆ is simply-laced and X / Q is free, then every character of T ( O ) is strongly parabolic. Section 2 is devoted to the proof of the above theorem.We now indicate what the above theorem implies for characters of various groups. By G / Z wemean G / Z ( G ) . The letter N denotes a positive integer. We let E n , n = , , and This follows from the fact that if G is a (connected split) semisimple group, then X / Q equals the dual of Z ( G ( C )) and X ∨ / Q ∨ equals the dual of π ( G ( C )) ; see, for example [Con12, Example 6.7]. For example, for SL , we have ( X, Q, X ∨ , Q ∨ ) = ( Z , Z , Z , Z ) . ) denote the split reductive group whose associated complex group is the connected, simply-connected, simple group of type E n (resp. F and G ).(1.4) Reductive group Properties Characters GL N , E simply-laced, X / Q and X ∨ / Q ∨ free all characters are easyPGL N , GO N , simply-laced and all characters areSO N / Z , E / Z, E / Z X/Q free strongly parabolicSL N ( N ≥ N , X ∨ / Q ∨ free, and hypothesis of (iii) all parabolic characters are easySpin N , E N ( N ≥ , G Sp N / Z , GO N , SO N hypothesis of (iii) all parabolic charactersare strongly parabolic Remark . Let G be a (connected) algebraic group over a field k . Let ¯ k denote an algebraic closureof k . Then G is said to be easy if every g ∈ G ( ¯ k ) is in the neutral connected component of itscentralizer in G ⊗ k ¯ k . This definition is due to V. Drinfeld. Based on the discussion in, e.g.,[Boy10, § k has characteristic zero. Namely, here we show that, if [ G, G ] is simply connected and Z ( G ) isconnected, then every parabolic character is easy (and the parabolic assumption is not needed inthe simply-laced case); in [Boy10, § G being easy in Drinfeld’s sense. Remark . To every character ¯ µ ∶ T ( O ) → C × , Roche [Roc98, §
8] associated a possibly disconnectedsplit reductive group ˜ H = ˜ H ¯ µ over F . The connected component of ˜ H is an endoscopy group for G .It follows from Theorem 3.(ii) that strongly parabolic characters are exactly those characters forwhich ˜ H is the Levi of a parabolic of G (and in particular connected). In more detail, by [Roc98,Definition 6.1], the coroots α ∨ of the connected component H of the identity of ˜ H (as a complexreductive group) are exactly those for which ¯ µ ○ α ∨ ∣ O × =
1, and by [Roc98, Lemma 8.1.(i)], thestabilizer of ¯ µ equals the Weyl group of H (and is not bigger) if and only if ˜ H = H . Then, weconclude because the Weyl group of H is a parabolic subgroup of the Weyl group of G if and onlyif H is a Levi subgroup of G (i.e., its roots form a closed root subsystem of those of G ).1.4. Satake isomorphisms.
In this section, we let G be a connected split reductive group over alocal field F . We impose the following restrictions on the residue characteristic of F . Convention . For every irreducible direct factor of the root system of G , we assume that theresidue characteristic of F is not one of the following primes:(1.5) Root system Excluded primes B n , C n , D n { } F , G , E , E { } E { , , } Theorem 7.
Let G be a connected split reductive group over a local field F whose residue charac-teristic satisfies the above restrictions. Then for every strongly parabolic character ¯ µ ∶ T ( O ) → C × ,there exists a compact open subgroup K < G ( O ) and an extension µ ∶ K → C × such that (1.6) H ( G ( F ) , K, µ ) ≃ C [ ˇ T / W ¯ µ ] As mentioned above, in the case of G = GL N , the above theorem is due to Howe [How73], andif ¯ µ is regular, then the above theorem follows by combining results of Bernstein [Ber84], [Ber92],Bushnell-Kutzko [BK98], [BK99] and Roche [Roc98]. As far as we know, the generalization tostrongly parabolic characters is new. xample . Let G = GL and let T ( O ) ≃ ( O × ) denote the group of diagonal matrices. Write¯ µ = ( ¯ µ , ¯ µ , ¯ µ ) where each ¯ µ i is a smooth character O × → C × . Suppose ¯ µ = ¯ µ and that theconductor cond ( ¯ µ / ¯ µ ) equals n ≥
2. (The conductor of a character χ ∶ O × → C × is the smallestpositive integer c for which χ ( + p c ) = { } .) If we follow Howe’s approach, we would take K = ⎛⎜⎝ O O OO O O p n p n O ⎞⎟⎠ ∩ G ( O ) . On the other hand, in the present article, following more closely the types of [Roc98], we takeinstead K = ⎛⎜⎜⎝ O O p [ n ] O O p [ n ] p [ n + ] p [ n + ] O ⎞⎟⎟⎠ ∩ G ( O ) . In both cases, ¯ µ extends to a character µ ∶ K → C × and one has an isomorphism H ( G ( F ) , K, µ ) ≃ C [ ˇ T ] . This example shows that the subgroup K of Theorem 7 is not necessarily unique.To prove Theorem 7, we use Roche’s result on types for principal series representations. Given anarbitrary smooth character ¯ µ ∶ T ( O ) → C × , Roche [Roc98] constructed a compact open subgroup J ⊂ G ( F ) (which depends on the choice of B ) and an extension µ J ∶ J → C × such that the compactlyinduced representation(1.7) W ∶= ind G ( F ) J µ J is a progenerator for the principal series Bernstein block of G defined by ¯ µ . More precisely, acombination of results of Bushnell and Kutzko, Dat, and Roche implies that in this situation, onehas an explicit isomorphism of G ( F ) -modules(1.8) Φ ∶ W ≃ Ð→ Π ∶= ι G ( F ) B ( F ) ( ind T ( F ) T (O) ¯ µ ) . Here, ι denotes the functor of parabolic induction. See § W is canonically isomorphic with H ( G ( F ) , J, µ J ) .Now suppose the character ¯ µ is strongly parabolic. Let L denote the corresponding Levi and let µ L ( F ) ∶ L ( F ) → C × denote an extension of ¯ µ to L ( F ) . Let µ L = µ L (O) ∶= µ L ( F ) ∣ L (O) denote its re-striction to L ( O ) . We prove that K = J L ( O ) is a subgroup of G ( F ) . Moreover, we show that thereexists a canonical character µ ∶ K → C × which extends µ J and µ L . Theorem 7 states that the Heckealgebra H ( G ( F ) , K, µ ) , consisting of compactly supported ( K, µ ) -bi-invariant functions on G ( F ) ,is isomorphic to C [ ˇ T / W ¯ µ ] . To prove this result, we realize H ( G ( F ) , K, µ ) as an endomorphismring of a family of principal series representations, which we call a central family .1.5. Central families.Definition 9.
Let ¯ µ be a strongly parabolic character with the corresponding Levi L . Let K = J L ( O ) denote the corresponding compact open subgroup. The central family of principal seriesrepresentations of G attached to ¯ µ is defined by(1.9) V ∶= ind G ( F ) K µ. Note that V is a submodule of W and the latter is a progenerator for the principal seriesblock corresponding to ¯ µ . According to Theorem 7, the endomorphism ring H of this familyidentifies with the center of the corresponding Bernstein block (which is isomorphic to the centerof H ( G ( F ) , J, µ J ) , and hence isomorphic to C [ ˇ T / W ¯ µ ] ; cf. § Note that here and in (1.10), it does not matter if we use normalized or unnormalized parabolic induction since therepresentation being induced is isomorphic to its twist by any unramified character. or generic maximal ideals m ⊂ H , the G ( F ) -module V / m V is an irreducible principal seriesrepresentation. (We will neither prove nor use the last statement.) We will now give an alternativedescription of V . Let P ⊇ B be a parabolic subgroup whose Levi is isomorphic to L . Let(1.10) Θ ∶= ι G ( F ) P ( F ) ( ind L ( F ) L (O) µ L ) . Theorem 10.
Under the assumptions of Theorem 7, we have a canonical isomorphism of G ( F ) -modules V ≃ Ð→ Θ . We prove the above theorem by identifying V and Θ with submodules of W and Π, respectively.Then, using the explicit description of Φ in (1.8), we show that Φ ∣ V ∶ V → Π defines an isomorphismonto Θ. On the other hand, the endomorphism ring V identifies with H ( G ( F ) , K, µ ) . Thus, toprove Theorem 7, we need to compute the endomorphism algebra of Θ. To this end, we will use atheorem of Roche [Roc02] on parabolic induction of Bernstein blocks. Remark . (i) As mentioned above, in this paper, we construct the pair ( K, µ ) satisfyingrequirement of Problem 1 by using Roche’s pair ( J, µ J ) . In this case, the subgroup K depends only on the kernel of ¯ µ ; that is, if ker ( ¯ µ ) = ker ( ¯ µ ′ ) then K ¯ µ = K ¯ µ ′ . In fact, it onlydepends on the conductors of the restrictions of ¯ µ to the coroot subgroups (i.e., the minimal c α ≥ µ ∣ α ∨ ( + p cα ) is trivial) together with the collection of roots α such that theentire restriction ¯ µ ∣ α ∨ (O × ) is trivial. This follows immediately from the construction of J ;see § ( J, µ J ) is a type for the Bernstein block R ¯ µ ( G ) . Types for Bernstein blocks arenot, however, necessarily unique. Therefore, it is natural to wonder if our constructioncould work using a different type ( J ′ , µ J ′ ) . In the case G = GL N , this is true in view of theresults of [How73], as we observed (for N =
3) in Example 8. We do not, however, pursuethis question in the current text.1.6.
Further directions.
The proof of Theorem 7 given in this paper is rather indirect; moreover,it relies on nontrivial results of Bernstein, Bushnell and Kutzko, Roche, and Dat. In a forthcomingpaper [KS], we hope to give a direct proof of this theorem by writing an explicit support preservingisomorphism H ( L ( F ) , L ( O ) , µ L ) ≃ Ð→ H ( G ( F ) , K, µ ) . In other words, we hope to prove Theorem7 using combinatorics and the classical Satake isomorphism. This proof should also make clear thegeometric nature of the group K and some of its double cosets in G ( F ) ; in particular, we expectthat it will help with the geometrization program (see below).In Definition 9, for strongly parabolic characters of the compact torus, we constructed “centralfamilies”. The endomorphism ring of the central family identifies canonically with the center of theblock defined by the character; moreover, generic irreducible representations in the block appearwith multiplicity one in the central family. It would be interesting to find analogous central familiesfor other Bernstein blocks.It is well-known that the Satake isomorphism allows one to realize the local unramified Langlandscorrespondence. In more detail, let ˇ G denote the complex reductive group which is the Langlandsdual of G . Using the classical Satake isomorphism (1.1), one can show that we have a bijectionunramified irreducible representations of G ( F ) ↔ characters of H Combining this with the bijectionscharacters of H ↔ points of ˇ T / W ↔ semisimple conjugacy classes in ˇ G e obtain a bijection between unramified representations of G ( F ) and semisimple conjugacy classesin ˇ G . It would be interesting to study the role of the ramified Satake isomorphisms (i.e., the onesgiven by Theorem 7) in the local Langlands program.In [HR10], a version of the Satake isomorphism for non-split groups is proved. On the otherhand, there is also now a Satake isomorphism in characteristic p ; see [Her11]. We expect that thereis also a version of Theorem 7 for non-split groups and one in characteristic p .Finally, we expect that there is a geometric version of Theorem 7. The geometric version ofthe usual Satake isomorphism is proved by Mirkovic and Vilonen [MV07], completing a projectinitiated by Lusztig, Beilinson and Drinfeld, and Ginzburg. In the case of regular characters; i.e.,in the case that the stabilizer of the character in the Weyl group is trivial, a geometric version ofTheorem 7 is proved in [KS11]. In [KS11, § Acknowledgements.
We would like to thank Alan Roche for very helpful email correspon-dence. He sketched proofs of several technical results for us; moreover, he brought to our attentionthe references [Roc02], [Dat99], and [Blo05]. We would like to thank J. Adler for reading an earlierdraft and several useful discussions. We also thank Loren Spice for helpful conversations. Wethank V. Drinfeld for pointing out to us Proposition 45 which plays a crucial role in this paper andsharing with us his notes on easy algebraic groups. Finally, we thank the Max Planck Institute forMathematics in Bonn for its hospitality.2.
Parabolic, strongly parabolic, and easy characters
Conventions.
Let F be a local field with ring of integers O , unique maximal ideal p , residuefield F q , and uniformizer t . Let G be a connected split reductive group over F with F -split torus T . Replacing G if necessary by an F -isomorphic group, we may (and do) assume that G and T aredefined over Z . Let W = N G ( T )/ T denote the Weyl group.Let ∆ = ∆ G denote the roots of G (with respect to T ). For α a root in ∆, we write α ∨ for thecorresponding coroot. Let X = Hom ( T, G m ) and X ∨ = Hom ( G m , T ) denote the the character andcocharacter lattices, respectively. Let Q ⊆ X be the root lattice, and let Q ∨ ⊆ X ∨ denote the corootlattice. Let ( Q ∨ ) sat be the saturation of Q ∨ in X ∨ , i.e., ( Q ∨ ) sat = { λ ∈ X ∨ ∣ m ⋅ λ ∈ Q ∨ , some m ∈ Z } . By definition X ∨ /( Q ∨ ) sat is a torsion free abelian group. To an element λ ∈ X ∨ , we associate t λ = λ ( t ) ∈ T ( F ) .For every α ∈ ∆, let u α ∶ G a → G be the one-parameter root subgroup, where G a is the additivegroup. We assume these root subgroups satisfy the conditions specified in [Roc98, § U α < G be the image of u α . For all i ∈ Z , let U α,i = u α ( p i ) < G ( F ) . In particular, U α, = u α ( O ) .Let H and K be topological groups and suppose H < K . Let χ ∶ H → C × be a character of H . We write ind KH χ for the space of left ( H, χ ) -invariant relatively compactly supported functionson K ; that is, those functions f ∶ K → C whose support has compact image in K / H and satisfy f ( hk ) = χ ( h ) f ( k ) for all h ∈ H and k ∈ K . The group K acts on this space by right translation.2.2. W -invariant rational characters. We start this section with a general lemma which we willrepeatedly use below.
Lemma 12.
Let H be a group and K < H a subgroup. Then a character χ ∶ K → C × extends to acharacter of H if and only if χ ∣ K ∩[ H,H ] is trivial. The same is true if H is an l-group (i.e., a locally ompact totally disconnected Hausdorff topological group), and K is a closed subgroup. Finally,the same is true if H is a connected split reductive algebraic group, K is a closed subgroup, and χ ∶ K → G m is a rational character.Proof. It is clear that the assumption that χ be trivial on K ∩ [ H, H ] is necessary. Conversely, if thisis true, extending the character is the same as extending the induced character of K /( K ∩ [ H, H ]) to H /[ H, H ] . Therefore, all the statements of the lemma reduce to the case that H is commutative.Then, the statement that any character of a subgroup of an abstract (discrete) abelian groupextends to the entire group follows from the fact that C × is divisible, and hence injective.For the locally compact analogue, write C × ≅ S × R > . For characters to S , the statement followsfrom Pontryagin duality. For R > , note first that, if H is compact, then there are no nontrivialcontinuous characters to R > . As an l-group always contains a compact open subgroup, this reducesthe problem to the case H is discrete, where it follows as in the previous paragraph, since R > isdivisible, and hence injective, as a discrete abelian group.For the algebraic analogue, i.e., where H and K are connected split tori, the statement followsbecause applying Hom ( − , G m ) to a short exact sequence 1 → K → H → H / K → H to characters of K is surjective. (cid:3) Lemma 13.
Let G be a connected split reductive algebraic group over a field k with split torus T .Let χ ∶ T → G m be a rational character. The following are equivalent: (1) χ is trivial on T ∩ [ G, G ] . (2) χ extends to a character G → G m ; (3) χ is W -invariant; (4) χ ○ α ∨ ∶ G m → G m is trivial, for every α ∈ ∆ .Proof. Lemma 12 implies immediately that ( ) Ô⇒ ( ) . Next, it is clear that [ N G ( T ) , T ] ⊆ [ G, G ] ∩ T ; therefore, if we restrict a character of G to T , we obtain a character which is invariantunder the conjugation action of N T ( G ) . This proves (2) Ô⇒ (3). Next, suppose χ is W -invariant.Then(2.1) χ ○ α ∨ = ( s α .χ ) ○ α ∨ = χ ○ ( − α ∨ ) = ( χ ○ α ) − It follows that ( χ ○ α ∨ ) =
1. Since G m has no nontrivial character of order 2, it follows that χ ○ α ∨ =
1. Hence, (3) Ô⇒ (4). For the final implication, we use the canonical identification(2.2) T ∩ [ G, G ] = G m ⊗ Z ( Q ∨ ) sat . By the notation on the RHS we mean the group subscheme of T whose R points equals R × ⊗ Z ( Q ∨ ) sat ,where R is a ring over k . Now if χ ○ α ∨ is trivial for every α ∈ ∆, then χ is trivial on T ∩ [ G, G ] .This proves (4) Ô⇒ (1). (cid:3) Remark . It follows from the above lemma that the group of characters of T which satisfy theabove equivalent conditions is canonically isomorphic to(2.3) Hom ( T /( T ∩ [ G, G ]) , G m ) ≃ X W ≃ Hom ( X ∨ / Q ∨ , Z ) ≃ Hom ( X ∨ /( Q ∨ ) sat , Z ) . The last isomorphism follows from the following: the quotient X ∨ / Q ∨ ↠ X ∨ /( Q ∨ ) sat splits, since X ∨ /( Q ∨ ) sat is free, and the resulting pullback maps Hom ( X ∨ / Q ∨ , Z ) ↔ Hom ( X ∨ /( Q ∨ ) sat , Z ) are We don’t need to assume that H is totally disconnected, if we use the fact from [HM06, Corollary 7.54] thatevery locally compact Hausdorff topological group contains a compact subgroup H ′ such that the quotient H / H ′ isisomorphic to R n × D for a discrete group D . For a proof of this statement over an algebraically closed field see, for instance, [DM91], § ( Q ∨ ) sat ; more precisely, we have ¯ k × ⊗ Z Q ∨ = T ( ¯ k ) ∩ [ G, G ]( ¯ k ) = ( T ∩ [ G, G ])( ¯ k ) . nverse to each other since the quotient X ∨ / Q ∨ ↠ X ∨ /( Q ∨ ) sat has finite kernel and Z is torsion-free.(More generally, for any finite-kernel quotient of finitely-generated abelian groups, the pullback mapon Hom ( − , Z ) is an isomorphism.)2.3. Easy W -invariant characters. Let G be a reductive group defined over Z . Let Hom sm ( O × , C × ) denote the group of smooth characters O × → C × . Proposition 15.
The following conditions are equivalent for a smooth character ¯ µ ∶ T ( O ) → C × : (i) The restriction ¯ µ ∣ ([ G,G ]∩ T )(O) is trivial; (ii) The character ¯ µ is a product of compositions of W -invariant rational characters T ( O ) → O × with smooth characters O × → C × .Remark . The same statement and proof holds when O is replaced by any (topological) ring. Definition 17.
A smooth W -invariant character ¯ µ ∶ T ( O ) → C × is easy (with respect to G ) if theequivalent conditions of Proposition 15 are satisfied. Remark . By Lemma 13 and Proposition 15, the group of easy characters T ( O ) identifies canon-ically withHom sm (( T /( T ∩ [ G, G ]))( O ) , C × ) ≃ Hom ( X ∨ /( Q ∨ ) sat , Z ) ⊗ Z Hom sm ( O × , C × ) ≃ Hom sm ( X ∨ /( Q ∨ ) sat ⊗ Z O × , C × ) . The last isomorphism follows from the fact that X ∨ /( Q ∨ ) sat is free. Proof of Proposition 15.
The implication (ii) ⇒ (i) is immediate from Lemma 13. For the reverseimplication, note that by assumption ¯ µ is a character of T ( O ) = ( G m ⊗ Z X ∨ )( O ) which is trivialon ([ G, G ] ∩ T )( O ) = ( G m ⊗ Z ( Q ∨ ) sat )( O ) . Therefore ¯ µ is canonically a character of ( G m ⊗ Z X ∨ )( O )/( G m ⊗ Z ( Q ∨ ) sat )( O ) = (( G m ⊗ Z X ∨ )/( G m ⊗ Z ( Q ∨ ) sat ))( O ) = ( G m ⊗ Z X ∨ /( Q ∨ ) sat )( O ) = ( X ∨ /( Q ∨ ) sat ) ⊗ Z O × . We conclude that ¯ µ is a product of compositions of (smooth) characters of O × with rational char-acters Hom ( X ∨ /( Q ∨ ) sat , Z ) . By Remark 14, the group of such rational characters is canonicallyisomorphic to the sublattice X W of W -invariant rational characters. Therefore, ¯ µ has the formclaimed in Part (ii). (cid:3) Note that, if ¯ µ is easy, then Lemma 13 implies that ¯ µ extends to a character of G ( F ) , and henceof G ( O ) . As the following example illustrates, the converse is not, in general, true. Example . Let G = P GL . Then the determinant map GL ( F ) → F × descends to a map G ( F ) → F × /( F × ) ≅ { ± } × t X ∨ / t X ∨ . Take the composition and the further quotient by the second factor,and view it as a character G ( F ) → C × (which is trivial on t X ∨ ). The restriction of this character to T ( O ) is nontrivial, even though there are no nonzero W -invariant rational characters (and hencenon nontrivial easy characters).Nonetheless, in the next subsection, we give a combinatorial description of all characters of T ( O ) which extend to characters of G ( F ) , similar to the description of easy characters above.2.4. Extendable W -invariant characters.Proposition 20. The following conditions are equivalent for a smooth W -invariant character ¯ µ ∶ T ( O ) → C × : (a) ¯ µ extends to a character of G ( F ) ; b) For all α ∈ ∆ , we have (2.4) ¯ µ ○ α ∨ ∣ O × = . If in addition q > , then these are also equivalent to (c) ¯ µ extends to a character of G ( O ) . Definition 21.
A smooth character ¯ µ ∶ T ( O ) → C × is said to be extendable (to G ( F ) ) if it satisfiesthe equivalent conditions of the above proposition. Remark . Note that α ∨ generate the coroot lattice Q ∨ ⊂ X ∨ ; hence, ⟨ α ∨ ( O × )⟩ = Q ∨ ⊗ Z O × . Alsonote that T = X ∨ ⊗ Z G m ; therefore, T ( O ) = X ∨ ⊗ Z O × . It follows from the above proposition thatthe group of extendable (to G ( F ) ) characters of T ( O ) identifies with(2.5)Hom sm ( T ( O )/⟨ α ∨ ( O × )⟩ α ∈ ∆ , C × ) = Hom sm (( X ∨ ⊗ Z O × )/( Q ∨ ⊗ Z O × ) , C × ) ≃ Hom sm ( X ∨ / Q ∨ ⊗ Z O × , C × ) . The last isomorphism follows because ( X ∨ ⊗ Z O × )/( Q ∨ ⊗ Z O × ) = X ∨ / Q ∨ ⊗ Z O × (by definition of ⊗ ,or by its right-exactness). Note that X ∨ / Q ∨ ⊗ Z O × has a topology which is induced by the topologyon O × . Therefore, we can speak of its smooth characters.The rest of this subsection is devoted to the proof of Proposition 20. To this end, we will provethe following:(2.6) [ G ( O ) , G ( O )] ∩ T ( O ) = ⟨ ¯ µ ○ α ∨ ( O × )⟩ α ∈ ∆ if q > , [ G ( F ) , G ( F )] ∩ T ( F ) = ⟨ ¯ µ ○ α ∨ ( F )⟩ α ∈ ∆ , ∀ q. Let us explain how this implies the proposition. First, note that, given any character of T ( O ) , thereis a unique extension to a character of T ( F ) trivial on t X ∨ . Moreover, if the original character wastrivial on α ∨ ( O × ) , then the extension is trivial on α ∨ ( F × ) , for all α ∈ ∆. Applying this observation,together with the second identity in (2.6) and Lemma 12, we conclude that parts (a) and (b) ofthe proposition are equivalent. The first identity in (2.6) similarly implies, for q >
2, that parts (b)and (c) of the proposition are equivalent.2.4.1.
Proof of (2.6) . Lemma 23.
Fix an arbitrary ring R . Then, (2.7) [ G ( R ) , G ( R )] ∩ T ( R ) ⊆ ⟨ ¯ µ ○ α ∨ ( R )⟩ α ∈ ∆ . The opposite inclusion, [ G ( R ) , G ( R )] ⊇ ⟨ ¯ µ ○ α ∨ ( R )⟩ α ∈ ∆ holds for all G if and only if it holds when G ( C ) is semisimple and simply connected. In the latter situation, it is equivalent to (2.8) [ G ( R ) , G ( R )] ⊇ T ( R ) . Proof.
More generally, suppose we have an arbitrary morphism of groups ˜ G → G and a subgroup T < G such that G is generated by T and the image of ˜ G . Then, it is obvious that [ G, G ] is thenormal subgroup generated by [ π ( ˜ G ) , π ( ˜ G )] = π [ ˜ G, ˜ G ] , [ π ( ˜ G ) , T ] , and [ T, T ] .Returning to the situation of the lemma, let ˜ G be the connected reductive algebraic group suchthat ˜ G ( C ) is the universal cover of [ G, G ]( C ) . Let π denote the canonical morphism ˜ G → G .Abusively, we will use π also to denote the induced morphism ˜ G ( R ) → G ( R ) . Note that π is anisomorphism on root subgroups; therefore, G ( R ) is generated by π ( ˜ G ( R )) and T ( R ) . Applying theconsiderations of the previous paragraph, we conclude that [ G ( R ) , G ( R )] is the normal subgroupgenerated by the image π ([ ˜ G ( R ) , ˜ G ( R )]) , along with [ T ( R ) , π ( ˜ G ( R ))] and [ T ( R ) , T ( R )] . Notethat these are all contained in the normal subgroup generated by π ( ˜ G ( R )) .We claim, however, that π ( ˜ G ( R )) is normal, so that [ G ( R ) , G ( R )] ⊆ π ( ˜ G ( R )) . Since G ( R ) isgenerated by π ( ˜ G ( R )) and T ( R ) , it suffices to show that π ( ˜ G ( R )) is closed under conjugation by T ( R ) . This follows because π ( ˜ G ( R )) is generated by the root subgroups and π ( ˜ T ( R )) , and each f these are closed under conjugation by T ( R ) . Therefore, for ˜ T < ˜ G the maximal torus of ˜ G , weconclude that(2.9) [ G ( R ) , G ( R )] ∩ T ( R ) ⊆ π ( ˜ T ( R )) = ⟨ ¯ µ ○ α ∨ ( R )⟩ α ∈ ∆ . The final equality holds because, for a simply connected semisimple root system, the maximal torusis generated by the coroot subgroups.The second assertion follows since [ G ( R ) , G ( R )] ⊇ π [ ˜ G ( R ) , ˜ G ( R )] and the set ∆ of roots is thesame for G and ˜ G . For the final assertion (the case G = ˜ G ), we only need to note that, in this case, T is generated by the coroot subgroups, so T ( R ) = ⟨ ¯ µ ○ α ∨ ( R )⟩ α ∈ ∆ . (cid:3) Lemma 24.
For a fixed ring R , the inclusion (2.8) holds for all G such that G ( C ) is semisimpleand simply connected if and only if it holds for G = SL .Proof. Suppose G ( C ) is semisimple and simply connected. Then T ( O ) is generated by its corootsubgroups, so it is enough to show (2.8) for the image of all subgroups SL → G corresponding toroots of G . (cid:3) Lemma 25.
The inclusion (2.8) holds for G = SL in the case that either R = O for q > or R isa field.Proof. Let f ∈ R × and let α be the positive simple root. Then one can verify that(2.10) ( − ( f − ) / f ) [( f − ) , ( −
10 1 )] ( ( − f )/ f ) = ( f f − ) = α ∨ ( f ) . We now consider the question of when the first and last matrices on the LHS are in [ G ( R ) , G ( R )] .Generally, for g ∈ R × ,(2.11) [( x ) , ( g g − )] = ( x ( − g ) ) . Let I R ∶ = ( − g ∣ g ∈ R ) = { x ( − g ) ∣ x, g ∈ R } be the ideal of elements appearing in the lower-leftentry of the final matrix. If this ideal is the unit ideal, then the LHS of (2.10) is in [ G ( R ) , G ( R )] ,as desired. This is clearly true if R is a field such that ∣ R ∣ > I R is also the unit ideal when R = O and q >
3. First, more generally for q > I O ⊇ p . Indeed, the squaring operation is bijective on 1 + p , for all z ∈ p . So, we cantake g ∈ O such that g = + z , and hence z ∈ I R .So for q >
3, to show I O is the unit ideal reduces to showing that I O/ p = I F q is the unit ideal,which as we pointed out is true in this case.It remains to show that, for R = O for q =
3, or R = F q for q ≤
3, that (2.8) holds. We have toshow this slightly differently, since now I R is not the unit ideal.First, if R = F q and q =
2, there is nothing to show because now T ( F q ) is trivial. If R = F q for q = [ G ( F q ) , G ( F q )] has index three in G ( F q ) ; since T ( F q ) has order two, itfollows that T ( F q ) must be in the kernel of the abelianization map G ( F q ) → G ( F q )/[ G ( F q ) , G ( F q )] ,i.e., that [ G ( F q ) , G ( F q )] ⊇ T ( F q ) . This completes the proof of the lemma for R equal to a field.Next, suppose that R = O and q =
3. Then, it suffices to show that [ G ( O ) , G ( O )] ⊇ α ∨ ( + p ) .Indeed, if we show this, the inclusion reduces to [ G ( F q ) , G ( F q )] ⊇ α ∨ ( F × q ) , which we have nowestablished.To show this, note that, by the above argument, I O ⊇ p . Hence, we can apply (2.11) to the case f ∈ + p , and we conclude that α ∨ ( + p ) ⊇ [ G ( O ) , G ( O )] , as desired. (cid:3) .5. Comparison between easy and extendable.Corollary 26.
Let ¯ µ ∶ T ( O ) → C × be a smooth character of G . Then ¯ µ is easy for G Ô⇒ ¯ µ is extendable to G ( F ) Ô⇒ ¯ µ is W -invariant Ô⇒ ( ¯ µ ○ α ∨ ∣ O × ) = , ∀ α ∈ ∆ . Proof.
The first implication is immediate from Propositions 15 and 20. The second implicationfollows from the facts W ≃ N G (O) ( T ( O ))/ T ( O ) and [ N G (O) ( T ( O )) , T ( O )] ⊆ [ G ( O ) , G ( O )] ∩ T ( O ) . For the last implication, note that ( ¯ µ ○ α ∨ ) ( x ) = ¯ µ ([ α ∨ ( x ) , s α ]) , where s α is any lift to N G (O) ( T ( O )) of the simple reflection s α . (cid:3) The reverse implications can all fail. For the first implication, see Example 19. For the remainingtwo, we have the following:
Example . (i) Let G = SL . Let ¯ µ ∶ T ( O ) → C × denote the composition T ( O ) ≃ O × → O × /( O × ) θ Ð→ C × , where θ is a nontrivial character. Then ¯ µ ∶ T ( O ) = O × → C × is W -invariant; however, itdoes not extend to G ( F ) by Proposition 20.(ii) [Roc98, Example 8.4] Let G = Sp n , n ≥
2. Identify T ( O ) with ( O × ) n , and let ¯ µ = ( θ, ⋯ , θ ) .Then ¯ µ is W -invariant; however, it does not extend to G ( F ) . This is because, as observedin [Roc98, Example 8.4], the composition ¯ µ ○ α ∨ is not trivial for all α (and in fact, theroot subsystem whose coroots have trivial composition produces an endoscopic group SO n ,which is not a subgroup of G ). Example . Let G = SL . Define¯ µ ( diag ( a, b, a − b − )) = θ ( a ) θ ( b ) , a, b ∈ O × , where θ is a nontrivial quadratic character of O × . By assumption, ( ¯ µ ○ α ∨ ) = G ; however, ¯ µ is not invariant under the transformation ( a, b, a − b − ) ↦ ( a − b − , b, a ) ; in particular,it is not W -invariant.In certain situations, either (or both) of the first two implications in the above corollary becomebiconditionals. Lemma 29. (i)
Suppose that Q ∨ = ⟨ λ − w ( λ ) ∣ λ ∈ X ∨ , w ∈ W ⟩ . Then every W -invariantcharacter of T ( O ) is extendable to G ( F ) . (i’) The hypothesis of (i) is equivalent to the statement that, for some choice of simple roots α i ,there exist cocharacters λ i ∈ X ∨ such that ⟨ λ i , α i ⟩ = . Moreover, this condition is impliedby either of the following: (a) X / Q is free (b) The root system of G has no factors of type A or C n . (ii) Suppose X ∨ / Q ∨ is torsion-free. Then every extendable character of T ( O ) (to G ( F ) ) iseasy.Proof. (i) If the coroot lattice equals the span of the elements λ − w ( λ ) for w ∈ W and λ ∈ X ∨ , then(2.4) is satisfied. This is because W -invariance implies ¯ µ ( λ ( x )) = ¯ µ ( w ( λ )( x )) for all x ∈ G m ( O ) ,and hence ¯ µ (( λ − w ( λ ))( x )) = x ∈ G m ( O ) .(i’) First, we claim that Q ∨ ⊇ ⟨ λ − w ( λ ) ∣ λ ∈ X ∨ , w ∈ W ⟩ . Let α i , i ∈ I be a choice of simple roots.Since W is generated by the s α i , ⟨ λ − w ( λ ) ∣ λ ∈ X ∨ , w ∈ W ⟩ = ⟨ λ − s α i λ ∣ λ ∈ X ∨ , i ∈ I ⟩ = ⟨⟨ λ, α i ⟩ α ∨ i ∣ λ ∈ X ∨ , i ∈ I ⟩ . This proves the desired containment. So, we need to show that the opposite inclusion is equivalentto the condition stated in (i’). iven λ i such that ⟨ λ i , α i ⟩ =
1, we obviously get α ∨ i in the RHS of the above equation. Conversely,if α ∨ i ∈ ⟨⟨ λ, α i ⟩ α ∨ i ∣ λ ∈ X ∨ , i ∈ I ⟩ , then there must exist λ i ∈ X ∨ such that ⟨ λ i , α i ⟩ =
1. Applying thisto all i yields the desired equivalence (since Q ∨ is spanned by the α ∨ i ).(a) If X / Q is torsion-free, then Q must be saturated in X , so the condition (i’) is satisfied.(b) For the root system A , with simple roots α and α , one has s α ( α ∨ ) − α ∨ = α ∨ , andsimilarly with indices 1 and 2 swapped, so that one concludes that α ∨ , α ∨ ∈ ⟨ λ − w ( λ )⟩ and hence Q ∨ = ⟨ λ − w ( λ )⟩ . The same argument shows that, for every root system in which every simple rootis contained in a root subsystem of type A , then every coroot is contained in ⟨ λ − w ( λ )⟩ and hence(i) is also satisfied.This takes care of all root systems except for types A , B n , C n , and G . For type B n with n ≥ α , . . . , α n where α n is theshort simple root, then α ∨ i ∈ ⟨ λ − w ( λ )⟩ for i < n , since these are incident to a subdiagram of type A ; for α ∨ n , it is still true that s α n ( α ∨ n − ) − α ∨ n − = α ∨ n , so also α ∨ n ∈ ⟨ λ − w ( λ )⟩ . For type G , if thesimple roots are α and α , we see that s α ( α ∨ + α ∨ ) − ( α ∨ + α ∨ ) = ± α ∨ , so α ∨ ∈ ⟨ λ − w ( λ )⟩ , and thesame fact holds (with opposite sign) when indices 1 and 2 are swapped. Note also that B = C , sowe do not need to separately exclude B .(ii) The hypothesis is equivalent to the condition that Q ∨ is saturated in X ∨ ; i.e., Q ∨ = ( Q ∨ ) sat .The result then follows from Remarks 18 and 22. (cid:3) On parabolic characters.
Recall that a smooth character ¯ µ ∶ T ( O ) → C × is parabolic if itsstabilizer in W is a parabolic subgroup. Here is an example of a character which is not parabolic. Example . (cf. [Roc98, Example 8.3], due to Sanje-Mpacko) Let N ≥ G = SL N . Define¯ µ ( diag ( a , a , . . . , a N − , a − ⋯ a − N − )) = χ ( a ) χ ( a ) ⋯ χ N − ( a N − ) , where χ ∶ O × → C × is a character of order N . Then the stabilizer of ¯ µ in W is the subgroup Z / N of cyclic permutations, which is not parabolic (and in particular is not all of W ).On the other hand, as the following proposition illustrates, in certain situations all charactersare parabolic. Proposition 31.
Let G be a connected simply laced split reductive group. If X / Q is free, then everysmooth character of T ( O ) is strongly parabolic. If, moreover, X ∨ / Q ∨ is free, then every smoothcharacter of T ( O ) is easy.Proof. Let ∆ ¯ µ denote the collection of roots α ∈ ∆ such that ¯ µ ○ α ∨ =
1. We claim that ∆ ¯ µ is a closedroot subsystem. Indeed, if α, β ∈ ∆ ¯ µ and ⟨ α, β ⟩ = −
1, then ( α + β ) ∨ = α ∨ + β ∨ , and so α + β ∈ ∆ ¯ µ aswell. Let L denote the Levi subgroup corresponding to ∆ ¯ µ . It follows from Proposition 20 alongwith [Roc98], Lemma 8.1.(i) and the comment at the end of p. 395, that ¯ µ is strongly parabolicwith Levi L (cf. Remark 5). Alternatively, if we use only from [Roc98] that ¯ µ is parabolic, thenwe can apply Lemma 29 to deduce strong parabolicity. For the final statement, we again applyLemma 29. (cid:3) Proof of Theorem 3.
Parts (i) and (ii) follow from Propositions 15 and 20, respectively.Next, we need a basic fact from the theory of reductive groups.
Lemma 32.
Let G be a connected split reductive group over Z with split torus T . Let L < G be aLevi containing T . (i) If X ∨ / Q ∨ is torsion free, so is X ∨ / Q ∨ L . (ii) If X / Q is torsion free, then so is X / Q L . (iii) If the equivalent conditions (i) or (i’) of Lemma 29 are satisfied for G (i.e., Q ∨ = ⟨ λ − w ( λ ) ∣ λ ∈ X ∨ ⟩ or, for some choice of simple roots α i , there exist cocharacters λ i ∈ X ∨ such that ⟨ λ i , α i ⟩ = ), then they are also satisfied when G is replaced by L . roof. Parts (i) and (ii) follow from the fact that Q / Q L and Q ∨ / Q ∨ L are always torsion free. Forpart (iii), we consider the condition (i’), i.e., the second condition. Note that, if this condition issatisfied for some choice of simple roots, it must be satisfied for all choices of simple roots, since twodifferent choices are related by an element of the Weyl group. But, by definition, one can choosesimple roots of L which form a subset of a choice of simple roots of G (and note that the (co)weightlattices are the same for L as for G ). Hence condition (i’) is satisfied for L . (cid:3) Then, parts (iii) and (iv) both follow from Lemmas 32 and 29. Finally, part (v) follows fromProposition 31 and Lemma 32.3.
Central families and Satake Isomorphisms
Recollections on decomposed subgroups.
We begin this section with some general re-marks on compact open subgroups of G ( F ) . Let P be a parabolic subgroup of G with Levidecomposition LU + P . Let P − = LU − P denote the opposite of P relative to L . (According to [Bor91,Proposition 14.21], the opposite parabolic is unique up to conjugation by a unique element of U + P .)Let J ⊂ G be a compact open subgroup. Let J + P = J ∩ U + P ( F ) , J P = J ∩ L ( F ) , J − P = J ∩ U − P ( F ) . For a parabolic P = LU + P , we let ∆ + P denote the set of roots of U + P . Similarly, we let ∆ − P denotethe set of roots of U − P . Note that ∆ = ∆ L ⊔ ∆ + P ⊔ ∆ − P . Definition 33. (1) The subgroup J is decomposed with respect to P if the product J + P × J P × J − P → J is surjective (and hence bijective).(2) The group J is totally decomposed with respect to P if it is decomposed, and in addition,the product maps ∏ α ∈ ∆ ± P U α ( F ) → J ± P are surjective (and hence bijective) for any ordering of the factors on the left hand side.(3) We say that J is absolutely totally decomposed if it is totally decomposed with respect toall parabolic subgroups P .The above definitions are closely related to the ones given in [BK98, §
6] and [Bus01, § § § Lemma 34.
Let J be totally decomposed in G with respect to a Borel subgroup B . Then J is totallydecomposed with respect to every parabolic P containing B . In particular, if J is totally decomposed with respect to all Borels, then it is absolutely totallydecomposed. The following lemma is also immediate from the definitions. Lemma 35.
Let J be a compact open subgroup of G which is decomposed with respect to a parabolic P . Suppose L ( O ) normalizes J + P and J − P . Then the subset K = J L ( O ) is a subgroup of G ( F ) ;moreover, it is decomposed with respect to P ; that is, K = K + P K P K − P where K ± P = J ± P and K P = L ( O ) . .2. The subgroup K.
Let f ∶ ∆ → Z be a function satisfying the properties(a) f ( α ) + f ( β ) ≥ f ( α + β ) , whenever α, β, α + β ∈ ∆;(b) f ( α ) + f ( − α ) ≥ f is concave in the sense of Bruhat and Tits (see [BT72], § § J = J f ∶ = ⟨ U α,f ( α ) , T ( O ) ∣ α ∈ ∆ ⟩ . Using the results of Bruhat and Tits, specifically [BT72, Proposition 6.4.9], Roche proved thefollowing lemma.
Lemma 36. [Roc98, Lemma 3.2]
The group J is absolutely totally decomposed in G . Moreover, J ∩ U α ( F ) = U α,f ( α ) for all α ∈ ∆ . Next, let L be a Levi subgroup of G (by which we always mean a Levi for a parabolic subgroupcontaining T ). We are interested to know when K = J L ( O ) is a group. In view of Lemma 37, it isenough to check that L ( O ) normalizes J ± P for some choice of parabolic P with Levi component L . Lemma 37.
Let P be a parabolic with Levi component L . Suppose that (3.2) f ( β ) = f ( α + β ) , ∀ α ∈ ∆ L , β ∈ ∆ ∖ ∆ L such that α + β ∈ ∆ . Then L ( O ) normalizes J ± P . To prove the above, we will make use of the following lemma, which will also be useful later:
Lemma 38.
Assume that g ∶ ∆ ∖ ∆ L → Z satisfies (3.2) , in addition to conditions (a) and (b)(restricting α, β , and α + β to lie in ∆ ∖ ∆ L ). Let ∆ + L ⊆ ∆ L be any choice of positive roots, and let f ∶ ∆ → Z be the function defined by (3.3) f ∣ ∆ ∖ ∆ L = g, f ∣ ∆ + L = , f ∣ ∆ − L = . Then f satisfies conditions (a) and (b). Note that J f ∣ ∆ L = I L is the Iwahori subgroup of L ( O ) corresponding to ∆ + L ⊆ ∆ L . Proof.
It is clear (and standard) that f ∣ ∆ L satisfies conditions (a) and (b) (where we require in (a)that α, β , and α + β lie in ∆ L ). By hypothesis, f ∣ ∆ ∖ ∆ L satisfies conditions (a) and (b) (requiring α, β , and α + β to be in ∆ ∖ ∆ L in (a)). So we only need to check that, if α ∈ ∆ L and β ∈ ∆ ∖ ∆ L ,then condition (a) is satisfied in the case that α + β ∈ ∆. This is immediate from (3.2). (cid:3) Proof of Lemma 37.
Choose a subset ∆ + L ⊆ ∆ L of positive roots for L . Let g = f ∣ ∆ ∖ ∆ L , and let f ′ ∶ ∆ → Z be as in Lemma 38 (i.e., f ′ ∣ ∆ ∖ ∆ L = f ∣ ∆ ∖ ∆ L , f ′ ∣ ∆ + L = f ′ ∣ ∆ − L = I L = J ∣ f ′ ∣ ∆ L < L ( O ) be the corresponding Iwahori subgroup containing T ( O ) . Then I L ≤ J f ′ , and hence I L normalizes J f ′ . It also normalizes U ± P (since L normalizes the unipotent radical U + P ), so I L normalizes J f ′ ∩ U ± P = J f ∩ U ± P = J ± P . On the other hand, L ( O ) is generated by all its Iwahorisubgroups, so L ( O ) also normalizes J ± P . (Note that we could have also used the decomposition L ( O ) = I L W L I L , for W L the Weyl group of L , and the fact that W L normalizes J + P under hypothesis(3.2).) (cid:3) Proposition 39.
Let L be a Levi subgroup of G . Assume that the function f ∶ ∆ → Z satisfiesconditions (a) and (b) as well as (3.2) , and set J = J f . Then K = J L ( O ) is a group; moreover, K is decomposed with respect to every parabolic P with Levi L .Proof. By Lemma 37, L ( O ) normalizes J + P and J − P . The result follows then from Lemma 35. (cid:3) The following corollary gives an alternative definition of K . Corollary 40.
Let L be a Levi subgroup of G . Let g ∶ ∆ → Z be a function satisfying the followingproperties: i) g ( α ) = for all α ∈ ∆ L ; (ii) g ( α ) + g ( − α ) ≥ for all α ∈ ∆ ∖ ∆ L ; (iii) g ( α ) + g ( β ) ≥ g ( α + β ) , whenever α, β, α + β ∈ ∆ .Then K = ⟨ U α,g ( α ) , T ( O )⟩ is a compact open subgroup of G . Moreover, K ∩ U α = U α,g ( α ) . Finally, K = L ( O ) J f , where f ∶ ∆ → C × is defined from g ∣ ∆ ∖ ∆ L by Lemma 38 (for any choice of positiveroots ∆ + L ⊆ ∆ L ).Proof. The inclusion K ⊆ J f L ( O ) is clear. For the reverse inclusion, note that it is obvious that J ± f ⊂ K , so we only need to show that L ( O ) ⊂ K . This follows from the fact that L is generated by T and the root subgroups. Thus, K = J f L ( O ) . In particular, by the above proposition, we havea direct product decomposition K = K + P K P K − P for every parabolic with Levi L . This implies thatfor α ∈ ∆ ± P , K ∩ U α = U α,g ( α ) . On the other hand it is clear that for α ∈ ∆ L , we have K ∩ U α = U α, since U α, ⊂ L ( O ) . (cid:3) Extension of ¯ µ . Let ¯ µ ∶ T ( O ) → C × be a smooth character. Following Roche [Roc98], wedefine a compact open subgroup J associated to ¯ µ . To this end, we have to choose a partition∆ = ∆ + ⊔ ∆ − . (Note that this amounts to choosing a Borel B ⊂ G .) For every α ∈ ∆, let(3.4) c α ∶ = cond ( ¯ µ ○ α ∨ ) denote the conductor of ¯ µ ○ α ∨ ; that is, the smallest positive integer c for which ¯ µ ( α ∨ ( + p c )) = { } .Let(3.5) f ¯ µ ( α ) = ⎧⎪⎪⎨⎪⎪⎩[ c α / ] , if α > [( c α + )/ ] , if α < Lemma 41. [Roc98, § Suppose that characteristic of F q satisfies the conditions in (1.5) . Then f ¯ µ satisfies conditions (a) and (b) of (3.2) . In particular, in view of Lemma 36, we obtain an associated compact open subgroup J = J ¯ µ = J f ¯ µ .Note that the function f ¯ µ and the corresponding group J ¯ µ depend on the partition of ∆ into positiveand negative roots (or equivalently, on the chosen Borel B ). While we ignore this in the notation,the reader should keep in mind that the Borel B is present. In particular, we have a decomposition J = J + J J − , where J ± = J ± B . Let J ● = ⟨ J + , J − ⟩ . Lemma 42. [Roc98, § There exists a unique character µ J ∶ J → C × whose restriction to J = T ( O ) equals ¯ µ and whose restriction to J ● is trivial. Let ¯ µ be a strongly parabolic character of T ( O ) and let L denote the corresponding Levi. Let P be a parabolic for L , and B the Borel subgroup of P . In terms of B , let f = f ¯ µ denote the functionassociated by Roche, and let J = J ¯ µ denote the corresponding compact open subgroup of G ( F ) . Lemma 43.
The set K = J L ( O ) is a compact open subgroup of G ( F ) . Moreover, for everyparabolic subgroup P containing L , we have a decomposition K = K + P K P K − P where K ± P = J ± P and K P = L ( O ) .Proof. If α ∈ ∆ L , then ¯ µ ○ α ∨ is trivial by (2.4). Now, if β ∈ ∆ is such that α + β ∈ ∆, then ( α + β ) ∨ = aα ∨ + bβ ∨ where a and b are relatively prime to q (by our assumption on the characteristicof F q ; see Conventions 6). Therefore, for every β ∈ ∆ such that α + β ∈ ∆, the conductor of ¯ µ ○ ( α + β ) ∨ equals the conductor of ¯ µ ○ ( β ∨ ) ; i.e.,(3.6) f ¯ µ ( α + β ) = f ¯ µ ( β ) . The result then follows from Proposition 39. (cid:3) et B L = B ∩ L denote the corresponding Borel subgroup of L . Let I L denote the correspondingIwahori subgroup of L . Note that by (3.5), we have J P = J ∩ L ( O ) = I L . Let µ L ( F ) denote anextension of µ to L ( F ) . Let µ L = µ L (O) ∶ = µ L ( F ) ∣ L (O) denote its restriction to L ( O ) . Note that µ L is automatically trivial on I + L and I − L , since these groups are in [ L ( F ) , L ( F )] . Set K ● P = ⟨ K + P , K − P ⟩ . Proposition 44.
There exists a unique character µ = µ K ∶ K → C × whose restrictions to K ● P , J and L ( O ) equal , µ J , and µ L , respectively.Proof. We need the following elementary fact: let H + , H , H − be subgroups of a group H whichgenerate the group. Suppose that H normalizes H ± . Let χ be a character of H which is trivial on ⟨ H + , H − ⟩ ∩ H . Then the map ˜ χ ∶ H → C × defined by ˜ χ ( h + h h − ) = χ ( h ) is a well-defined extensionof χ to H .By assumption the characters µ J and µ L agree on J ∩ L ( O ) = I L ; in particular, µ L is trivialon K ● P ∩ L ( O ) (since µ is trivial on J ● ). Applying the above fact, we conclude that there exists acharacter µ ∶ K → C × whose restriction to K ± P is trivial and whose restriction to L ( O ) equals µ L .The latter statement implies that the restriction of µ to I ± L is trivial; hence, the restriction of µ to J ± = K ± P I ± L is also trivial. Moreover, the restriction of µ to T ( O ) equals ¯ µ . By Lemma 42, therestriction of µ to J equals µ J . (cid:3) Proof of Theorem 10.
Let ¯ µ ∶ T ( O ) → C × be a strongly parabolic character of T ( O ) withLevi L , and extensions µ L ( F ) and µ L (O) = µ L as above. Pick a parabolic P containing L and aBorel B < P . Let J = J ¯ µ denote the compact open subgroup associated by Roche to B and ¯ µ . Let µ J ∶ J → C × denote the canonical extension of ¯ µ to J . Let(3.7) W ∶ = ind G ( F ) J µ J . By definition, W is realized on the space of left ( J, µ J ) -invariant compactly supported functions on G ( F ) . The group G ( F ) acts on this space by right translation. Let f be the function supportedon J which there equals µ J . Then f ∈ W ; moreover, every element of W can be written as a finitelinear combination of elements of the form f .g , g ∈ G ( F ) .Note that J ∩ L ( O ) = I L is the Iwahori subgroup of J . Let µ I denote the restriction of µ J to I L . Let P ○ I ∶ = I L U + P ( O ) . The character µ I extends uniquely to a character of P ○ I which is trivial on U + P ( O ) . By an abuse of notation, we denote this character of P ○ I by µ I as well. Let(3.8) Π ∶ = ind G ( F ) P ○ I µ I . Then Π is realized as the space of left ( P ○ I , µ I ) -invariant functions on G ( F ) . The group G ( F ) actsby right translation. Define a ∶ G ( F ) → C to be the function supported on P ○ I J such that a ( pj ) = µ I ( p ) µ J ( j ) , p ∈ P ○ I , j ∈ J. One can check that a is a well-defined right ( J, µ J ) -invariant function in Π. It follows that theassignment f ↦ a defines a morphism of G ( F ) -modules Φ ∶ W → Π. Proposition 45. Φ is an isomorphism.Proof. According to [Roc98, Theorem 7.5], ( J, µ J ) is a cover of ( T ( O ) , ¯ µ ) , in the sense of [BK98,Definition 8.1] (in [Roc98], the residue characteristic is further restricted so as to obtain a nonde-generate bilinear form on the Lie algebra, but this restriction can be relaxed using the dual Liealgebra as in [Yu01]: see [KS11, § § A.2]). It follows from [BK98, Proposition 8.5] that ( J, µ ) is also a cover of ( I L , µ L ) . The explicit isomorphism above is constructed (in the general settingof types) in [Dat99, §
2] (see also [Blo05, Theorem 2], where an even more general statement aboutcovers is proved). (cid:3) It is easy to check that Π, thus defined, is isomorphic to the Π defined in (1.8). ecall from (1.9) that V ∶ = ind G ( F ) K µ . By definition, this is a submodule of W . On the otherhand, let P ○ = L ( O ) U + P ( F ) . The character µ L extends uniquely to a character of P ○ which is trivialon U + P ( F ) . By an abuse of notation, we denote this character by µ L as well. Then, recalling thedefinition of Θ in (1.10), we have an isomorphismΘ ∶ = ι G ( F ) P ( F ) ind L ( F ) L (O) µ L ≃ ind G ( F ) P ○ µ L . We identify Θ with the G ( F ) -module on the RHS of the above isomorphism. With this convention,it is clear that we have an inclusion Θ ↪ Π. To establish Theorem 10, we prove that the restrictionof Φ to V defines an isomorphism G ( F ) -modules V ≃ Ð→ Θ. To this end, we define averaging (orsymmetrization) maps W → V and Π → Θ and show that they are compatible with Φ.Recall that W is realized on the space of left ( J, µ J ) -invariant functions on G ( F ) . Under thisidentification, the subspace V ⊂ W is identified with the space of left ( L ( O ) , µ L ) -invariant functionsin W . On the other hand, Π is identified with the space of left ( P ○ I , µ I ) -invariant functions on G ( F ) ,and Θ is the subspace of left ( L ( O ) , µ L ) -invariant functions in Π.Choose a Haar measure on L such that the volume of L ( O ) equals 1. For every function f ∶ G ( F ) → C , define f c by(3.9) f c ( x ) = ∫ L (O) µ L ( l ) f ( l − x ) dl Then f ↦ f c defines a splitting of the natural inclusion of left ( L ( O ) , µ L ) -invariant functions on G ( F ) into the space of all functions on G ( F ) . Note that this splitting obviously commutes withthe action of G ( F ) on the space of all functions by right translation. Therefore, the map f ↦ f c defines a splitting of the natural inclusions of G ( F ) -modules V ↪ W and Θ ↪ Π. Our goal is toshow that the diagram(3.10) W (cid:9) (cid:9) Φ / / Π (cid:10) (cid:10) V ?(cid:31) O O / / Θ ?(cid:31) O O commutes, where the averaging maps are denoted by the dotted arrows. The key computation isthe following: Proposition 46. Φ ( f c ) = a c .Proof. By definition, f c is the left L ( O ) -symmetrization of f , f c = ∣ K / J ∣ − µ ⋅ char ( K ) . This is,however, also the right symmetrization of f c . Since Φ commutes with the right action of G ( F ) (it is a morphism of representations), Φ ( f c ) is also the right L ( O ) -symmetrization of a . This, inturn, is the function supported on P I K which sends pk to ∣ K / J ∣ − µ I ( p ) µ ( k ) for p ∈ P I and k ∈ K .On the other hand, a c is the left L ( O ) -symmetrization of a , i.e., the function supported on P J sending pj to ∣ P / P I ∣ − µ L ( p ) µ J ( j ) . Since ∣ P / P I ∣ = ∣ K / J ∣ , this also coincides with the right-symmetrization, i.e., with Φ ( f c ) . (cid:3) We now resume the proof of Theorem 10. Note that every element of W can be written as afinite linear combination of elements of the form f .g , where g ∈ G ( F ) . Next, the morphisms Φand f ↦ f c (which represent morphisms W → V and Π → Θ) are G ( F ) -equivariant. Therefore, theabove proposition implies that for all f ∈ W , we have(3.11) Φ ( f c ) = Φ ( f ) c . Now given v ∈ V , we can write v = w c for some w ∈ W . Therefore, Φ ( v ) = Φ ( w c ) = Φ ( w ) c ∈ Θ.Therefore, Φ ∣ V defines a morphism of G ( F ) -modules V → Θ. This clearly creates the commutative quare (3.10). Since the dotted arrows are surjective and the top horizontal arrow is an isomorphism,Φ ∣ V ∶ V → Θ is surjective. Since it is the restriction of Φ, which is an isomorphism, it is also injective.Thus it is an isomorphism. (cid:3)
Proof of Theorem 7.
We will continue with the notation of the previous subsection.
Proposition 47.
We have a canonical isomorphism
End G ( F ) ( Θ ) ≃ H ( L ( F ) , L ( O ) , µ L ) .Proof. The fact that ¯ µ is parabolic with Levi L means that W ¯ µ = N G ( ¯ µ )/ T = N L ( T )/ T . In particu-lar, N G ( ¯ µ ) ⊂ L ( F ) . By the main theorem of [Roc02], parabolic induction with respect to P definesan equivalence of categories between Bernstein block of L corresponding to the pair ( T ( O ) , ¯ µ ) andthat of G . Under this equivalence, the L ( F ) -module ind L ( F ) L (O) µ L is mapped to Θ. Thus, we obtaina canonical isomorphism End G ( F ) ( Θ ) ≃ End L ( F ) ( ind L ( F ) L (O) µ L ) ≃ H ( L ( F ) , L ( O ) , µ L ) . (cid:3) Note that the algebra H ( G ( F ) , K, µ ) acts by convolution on the module V = ind G ( F ) K µ . It isa standard fact that H ( G ( F ) , K, µ ) ≃ End G ( F ) ( V ) . By Theorem 10, V is canonically isomorphicto Θ = ι GP ( ind L ( F ) L (O) µ L ) . By the preceding paragraph, the endomorphism ring of Θ is canonicallyisomorphic to the endomorphism ring of the L ( F ) -module ind L ( F ) L (O) µ L . Therefore, we obtain acanonical isomorphism H ( G ( F ) , K, µ ) ≃ H ( L ( F ) , L ( O ) , µ L ) .Finally, recall that µ L = µ L ( F ) ∣ L (O) , where µ L ( F ) ∶ L ( F ) → C × is a character of L ( F ) . Then multi-plication by µ L ( F ) defines a canonical isomorphism of algebras H ( L ( F ) , L ( O )) ≃ H ( L ( F ) , L ( O ) , µ L ) .Moreover, by the Satake isomorphism, we have a canonical isomorphism H ( L ( F ) , L ( O )) ≃ C [ ˇ T / W L ] = C [ ˇ T / W ¯ µ ] . Theorem 7 is established. (cid:3) References [Ber84] J. N. Bernstein. Le “centre” de Bernstein. In
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