AA NOTE ON INTEGRAL SATAKE ISOMORPHISMS
XINWEN ZHU
Abstract.
We formulate a Satake isomorphism for the integral spherical Hecke algebra of anunramified p -adic group G and generalize the formulation to give a description of the Hecke algebra H G ( V ) of weight V , where V is a lattice in an irreducible algebraic representation of G . Contents
1. The Satake isomorphism 21.1. The C -group 21.2. Affine monoids 31.3. Invariant theory 41.4. Classical Satake isomorphism over Z V
72. Compatibility with the geometric Satake 92.1. The geometric Satake equivalence 92.2. The representation ring 122.3. From Sat to Sat cl canonical integral Satake isomorphism, by identifying the integralspherical Hecke algebra H G of an unramified p -adic group G with a Z -algebra associated to an affinemonoid V ˆ G,ρ ad of the Langlands dual group ˆ G . The precise statement can be found in Proposition5. There are two motivations. First, the usual Satake isomorphism depends on a choice of q / and therefore only gives a description of H G ⊗ Z [ q ± / ] in terms of the dual group ˆ G (e.g. see[Gr96]). Using Deligne’s modification of the Langlands dual group (e.g. see [BG11]), one canformulate a canonical Satake isomorphism for H G ⊗ Z [ p − ]. On the other hand, there is the mod p Satake isomorphism, as discussed in [He11, HV15]. However, the Langland duality does not appearexplicitly in these works. Therefore, it is desirable to extend the classical Satake isomorphism to Z , which after mod p recovers the mod p Satake isomorphism. Another motivation comes from therecent work of R. Cass ([Ca19]) on the geometric Satake equivalence for perverse F p -sheaves onthe affine Grassmannian. In his work, what controls the tensor category is not the Langlands dualgroup, but some affine monoid M G . I hope V ˆ G,ρ ad and M G are closely related .Our formulation generalizes easily to give a description of the Hecke algebra H G ( V ) of weight V interms of the affine monoid V ˆ G,λ ad + ρ ad , where V is a lattice in an irreducible algebraic representationof G of highest weight λ . See Proposition 14 for the precise formulation. It follows from ourformulation that the ring of invariant functions on the Vinberg monoid specializes to all H G ( V )’s. Supported by the National Science Foundation under agreement Nos. DMS-1902239. However, as explained to me by Cass, the monoid appearing in his work is solvable and therefore cannot beisomorphic to the one appearing in this note. a r X i v : . [ m a t h . R T ] M a y . The Satake isomorphism
The C -group. Let G be a connected reductive group over a field F . Let ( ˆ G, ˆ B, ˆ T , ˆ e ) be apinned dual group of G over Z . There is a natural action of the Galois group Γ F of F on ( ˆ G, ˆ B, ˆ T , ˆ e ),induced by its action on the root datum of G . This action factors as Γ F (cid:16) Γ (cid:101) F /F ξ (cid:44) → Aut( ˆ G, ˆ B, ˆ T , ˆ e ),where Γ (cid:101) F /F is the Galois group of a finite Galois extension (cid:101)
F /F .Let 2 ρ : G m → ˆ T denote the cocharacter given by sum of positive coroots of ˆ G (with respect toˆ B ), and 2 ρ ad its projection to the adjoint torus ˆ T ad ⊂ ˆ G ad = ˆ G/Z ˆ G of ˆ G . It admits a unique squareroot ρ ad : G m → ˆ T ad . Define an action of G m on ˆ G byAd ρ ad : G m ρ ad → ˆ T ad Ad → Aut( ˆ G ) . Note that Ad ρ ad still preserves ( ˆ B, ˆ T ), but not ˆ e . The two actions Ad ρ ad and ξ on ˆ G commutewith each other. Following the terminology of Buzzard-Gee [BG11], we define the C -group of G asan affine reductive group scheme over Z as C G := ˆ G (cid:111) ( G m × Γ (cid:101) F /F ) . This group scheme appears naturally from the geometric Satake equivalence, as to be reviewedin Theorem 16 below. Note that our definition is different from [BG11, Definition 5.3.2], but isequivalent to it (see below). We may write C G ∼ = L G (cid:111) G m ∼ = ˆ G T (cid:111) Γ (cid:101) F /F , where L G := ˆ G (cid:111) ξ Γ (cid:101) F /F is the usual Langlands dual group, and where ˆ G T := ˆ G (cid:111) Ad ρ ad G m , whichfits into the short exact sequence(1.1) 1 → ˆ G → ˆ G T d ρ ad −−−→ G m → . Note that there is an isomorphism(1.2) ( ˆ G × G m ) / (2 ρ × id)( µ ) ∼ = ˆ G T , ( g, t ) (cid:55)→ ( g ρ ( t ) − , t ) , and the left hand side is the more familiar form of Deligne’s modification of the Langlands dualgroup (e.g. see [BG11, Proposition 5.3.3]). Remark 1.
We may regard ˆ G T as the dual group of a reductive group G T , which is a centralextension of G by G m over F , and then regard C G ∼ = L G T as the usual Langlands dual group of G T . This is the definition given in [BG11, Definition 5.3.2].We discuss a few examples. Example 2. (1) If G = T is a torus, then ˆ T T = ˆ T × G m and C T = ( ˆ T (cid:111) Γ ˜ F /F ) × G m .(2) If G is an inner form of a split reductive group, then C G = ˆ G T . For example, if G = PGL , C G = ˆ G T = GL and d ρ ad is the usual determinant map. In particular, ˆ G T is not isomorphicto ˆ G × G m in general.(3) However, if ρ ad lifts to a cocharacter ˜ ρ ∈ X • ( ˆ T ) (which does not necessarily satisfy 2 ˜ ρ = 2 ρ ),then ˜ ρ induces an isomorphism(1.3) ˆ G T (cid:39) ˆ G × G m , ( g, t ) (cid:55)→ ( g ˜ ρ ( t ) , t ) . For example, if G = GL n so ˆ G = GL n , we can choose ˜ ρ ( t ) = diag { t n − , · · · , t, } ; If G = PGL n +1 so ˆ G = SL n +1 , we can choose ˜ ρ ( t ) = diag { t n , t n − , · · · , t − n } .
4) In general, even if ρ ad lifts to an element ˜ ρ ∈ X • ( ˆ T ), so ˆ G T ∼ = ˆ G × G m , C G may notbe isomorphic to L G × G m , unless ˜ ρ can be chosen to be Γ (cid:101) F /F -invariant. For example, if G = U n is an even unitary group, associated to a quadratic extension (cid:101) F /F , then ˆ G = GL n ,and ξ : Γ (cid:101) F /F = { , c } → Aut(GL n ) is defined by ξ ( c )( A ) = J n ( A T ) − J − n , where J n isthe anti-diagonal matrix with ( i, n + 1 − i )-entry ( − i . In this case, ˆ G T is isomorphic toˆ G × G m , but C G is not isomorphic to L G × G m .1.2. Affine monoids.
We define an affine monoid scheme V ˆ G,ρ ad equipped with a faithfully flatmonoid morphism(1.4) d ρ ad : V ˆ G,ρ ad → A which extends the homomorphism d ρ ad : ˆ G T → G m from (1.1). It will be obtained as the pullbackof a universal monoid (called the Vinberg monoid) associated to ˆ G , whose definition we first brieflyrecall following the approach in [XZ19, § § loc. cit. are given overa field. But as they work over any field, they actually work over the base Z .We will use notations from [XZ19] with ( ˆ G, ˆ B, ˆ T ) in place of ( G, B, T ) in loc. cit. . We identify X • ( ˆ T ad ) ⊂ X • ( ˆ T ) with the root lattice inside the weight lattice. Let X • ( ˆ T ad ) pos be the submonoidgenerated by simple roots { ˆ α , . . . , ˆ α r } of ˆ G with respect to ( ˆ B, ˆ T ), let X • ( ˆ T ) ? ⊂ X • ( ˆ T ) for ? = + , − be the submonoids of dominant and anti-dominant weights, and let X • ( ˆ T ) +pos ⊂ X • ( ˆ T ) be thesubmonoid generated by X • ( ˆ T ad ) pos and X • ( ˆ T ) + . Letˆ T +ad = Spec Z [ X • ( ˆ T ad ) pos ] , which is an affine monoid containing ˆ T ad as the group of invertible elements. Note that everydominant coweight λ : G m → ˆ T ad of ˆ G ad can be extended to a monoid homomorphism(1.5) λ + : A → ˆ T +ad . We equip X • ( ˆ T ) with the usual partial order (cid:22) , i.e. ˆ λ (cid:22) ˆ λ if ˆ λ − ˆ λ ∈ X • ( ˆ T ad ) pos . Forˆ ν ∈ X • ( ˆ T ), let ˆ ν ∗ := − w (ˆ ν ). Recall that the left and the right multiplication of ˆ G on itself inducea natural ( ˆ G × ˆ G )-module structure on Z [ ˆ G ], which admits a canonical multi-filtration Z [ ˆ G ] = (cid:88) ˆ ν ∈ X • ( ˆ T ) +pos fil ˆ ν Z [ ˆ G ]by ( ˆ G × ˆ G )-submodules indexed by X • ( ˆ T ) +pos . Here fil ˆ ν Z [ ˆ G ] denotes the saturated Z -module thatis maximal among all ( ˆ G × ˆ G )-submodules V ⊂ Z [ ˆ G ] with the following property: for any pair(ˆ µ, ˆ µ (cid:48) ) ∈ X • ( ˆ T × ˆ T ), if the weight space V (ˆ µ, ˆ µ (cid:48) ) (cid:54) = 0, then ˆ µ (cid:22) ˆ ν ∗ and ˆ µ (cid:48) (cid:22) ˆ ν . One knows thatfil ν Z [ ˆ G ] is finite free over Z and thatgr Z [ ˆ G ] = (cid:77) ˆ ν ∈ X • ( ˆ T ) + S ˆ ν ∗ ⊗ S ˆ ν , where S ˆ ν denotes the Schur module of highest weight ˆ ν (i.e. the induced ˆ G -module from thecharacter − ˆ ν of ˆ B ). See [XZ19, § §
3] for detailed discussions, including the definition of associatedgraded of a multi-filtration. Now the Vinberg monoid V ˆ G of ˆ G is defined as V ˆ G = Spec R X • ( ˆ T ) +pos Z [ ˆ G ] , where R X • ( ˆ T ) +pos Z [ ˆ G ] := ⊕ ˆ ν ∈ X • ( ˆ T ) +pos fil ˆ ν Z [ ˆ G ] denotes the Rees algebra associated to the above de-fined multi-filtration. It is an affine monoid Z -scheme of finite type which admits a faithfully flatmonoid morphism d : V ˆ G → ˆ T +ad such that d − ( ˆ T ad ) contains the group of invertible elements of V ˆ G and is isomorphic to the quotientof ˆ G × ˆ T by Z ˆ G with respect to the diagonal action z · ( g, t ) = ( gz, tz ) (so in particular ˆ G acts on V ˆ G by left and right translations); • d − (0) ∼ = As ˆ G as ( ˆ G × ˆ G )-schemes, whereAs ˆ G := Spec gr Z [ ˆ G ]is called the asymptotic cone of ˆ G .In fact, the Vinberg monoid of ˆ G can be characterized as the unique affine monoid scheme M equipped with a faithfully flat monoid morphism d : M → ˆ T +ad satisfying the above two properties.Now, for a dominant coweight λ : G m → ˆ T ad , let(1.6) d λ : V ˆ G,λ := A × λ + , ˆ T ad V G → A be the pullback of of the homomorphism d : V ˆ G → ˆ T +ad along the map λ + from (1.5). Thehomomorphism ˆ G × G m id × ρ −−−→ ˆ G × ˆ T induces a homomorphism ˆ G T → ˆ G × Z ( ˆ G ) ˆ T by (1.2), andtherefore we obtain desired map (1.4) by setting λ = ρ ad in (1.6).Note that V ˆ G and ˆ T +ad are acted by Γ (cid:101) F /F (induced by its action ξ ) and the homomorphism d : V ˆ G → ˆ T +ad is Γ (cid:101) F /F -equivariant. If λ : G m → ˆ T ad is a dominant coweight fixed by Γ (cid:101) F /F , themonoid V ˆ G,λ is also acted by Γ (cid:101)
F /F and the map d λ extends to a homomorphism˜ d λ : V ˆ G,λ (cid:111) Γ (cid:101) F /F → A × Γ (cid:101) F /F . Note that ρ ad is Γ (cid:101) F /F -invariant. It follows that there is a natural isomorphism(1.7) ˜ d − ρ ad ( G m × Γ (cid:101) F /F ) ∼ = C G. Invariant theory.
We fix a finite order automorphism σ of ( ˆ G, ˆ B, ˆ T , ˆ e ) and a positive integer q . We consider the σ -twisted conjugation action of ˆ G on V ˆ G given by c σ ( g )( x ) = gxσ ( g ) − , g ∈ ˆ G, x ∈ V ˆ G . Let λ : G m → ˆ T ad be a dominant coweight. Then the conjugation action of ˆ G on V ˆ G (cid:111) (cid:104) σ (cid:105) restrictsto an action on V ˆ G | d = λ ( q ) σ . Then we have the isomorphism Z [ V ˆ G | d = λ ( q ) σ ] ˆ G = Z [ V ˆ G | d = λ ( q ) ] c σ ( ˆ G ) .Let V ˆ T be the closure of ˆ T × Z ˆ G ˆ T ⊂ ˆ G × Z ˆ G ˆ T inside V ˆ G . This is a commutative submonoid of V ˆ G . If we write the ring of regular functions of ˆ T × Z ˆ G ˆ T by Z [ ˆ T × Z ˆ G ˆ T ] = (cid:77) (ˆ λ, ˆ ν ) Z ( e ˆ λ ⊗ e ˆ ν ) , where the sum is taken over pairs of weights (ˆ λ, ˆ ν ) of ˆ T such that ˆ ν + ˆ λ ∈ X • ( ˆ T ad ) , and where e ˆ λ ⊗ e ˆ ν denotes the corresponding regular function on ˆ T × Z ˆ G ˆ T , then the ring of regular functionson V ˆ T is the subring Z [ V ˆ T ] = (cid:77) (ˆ λ, ˆ ν ) , ˆ ν +ˆ λ − ∈ X • ( ˆ T ad ) pos Z ( e ˆ λ ⊗ e ˆ ν ) , where ˆ λ − ∈ X • ( ˆ T ) − denotes the anti-dominant weight in the Weyl group orbit of ˆ λ . In [XZ19] the condition reads as ˆ ν − ˆ λ ∈ X • ( ˆ T ad ). The sign convention here is more suitable for our purpose. or ˆ λ ∈ X • ( ˆ T ), let e ˆ λ denote the corresponding regular function on ˆ T , and for ˆ λ ∈ X • ( ˆ T ad ) pos ,let ¯ e ˆ λ be the corresponding regular function on ˆ T +ad . The homomorphism d : V ˆ T → ˆ T +ad is given bythe ring map Z [ X • ( ˆ T ad ) pos ] → Z [ V ˆ T ] sending ¯ e ˆ λ to 1 ⊗ e ˆ λ , which admits a section s : ˆ T +ad → V ˆ T Z [ V ˆ T ] → Z [ X • ( ˆ T ad ) pos ] , e ˆ λ ⊗ e ˆ ν (cid:55)→ ¯ e (ˆ ν +ˆ λ ) . Note that s | ˆ T ad : ˆ T ad → ˆ T × Z ˆ G ˆ T is induced by the diagonal embedding ˆ T → ˆ T × ˆ T . We still denoteby s the composed section ˆ T +ad → V ˆ G . Its restriction to ˆ T ad induces a section ˆ T ad → ˆ G × Z ˆ G ˆ T , andtherefore an isomorphism ˆ G × Z ˆ G ˆ T ∼ = ˆ G (cid:111) ˆ T ad , whose pullback along ρ ad : G m → ˆ T ad gives theisomorphism (1.2).There is a natural injective map i : ˆ T → ˆ T × Z ˆ G ˆ T → V ˆ T , where the first map is the inclusioninto the first factor. Then we obtain a map ( i , s ) : ˆ T × ˆ T +ad → V ˆ T over ˆ T +ad , which induces the ringmap Z [ V ˆ T ] → Z [ X • ( ˆ T )] ⊗ Z [ X • ( ˆ T ad ) pos ] , e ˆ λ ⊗ e ˆ ν (cid:55)→ e ˆ λ ⊗ ¯ e (ˆ ν +ˆ λ ) . It in turn induces an injective map(1.8) Z [ V ˆ T | d = λ ( q ) ] ⊂ Z [ X • ( ˆ T )] , e ˆ λ ⊗ e ˆ ν (cid:55)→ q (cid:104) λ, ˆ ν +ˆ λ (cid:105) e ˆ λ . Let W = N ˆ G ( ˆ T ) / ˆ T be the Weyl group of ( ˆ G, ˆ T ) and let W = W σ be the subgroup of elementsfixed by σ , which naturally acts on X • ( ˆ T ) σ . Let ˆ N be the preimage of W in N ˆ G ( ˆ T ). The σ -twisted conjugation c σ of ˆ G on V ˆ G induces the σ -twisted action of ˆ N on V ˆ T . Recall that there isthe Chevalley restriction isomorphism (see [XZ19, Proposition 4.2.3], which was denoted as Res σ + , )Res : Z [ V ˆ G ] c σ ( ˆ G ) ∼ = Z [ V ˆ T ] c σ ( ˆ N ) , compatible with the Z [ X • ( ˆ T ad ) pos ]-structure on both sides. The same argument as in [XZ19, Lemma4.2.1] implies that the isomorphism Res induces isomorphisms(1.9) Res : Z [ V ˆ G | d = λ ( q ) ] c σ ( ˆ G ) ∼ = Z [ V ˆ G ] c σ ( ˆ G ) ⊗ Z [ X • ( ˆ T ad ) pos ] , ¯ e ˆ αi (cid:55)→ q ( λ, ˆ αi ) Z ∼ = Z [ V ˆ T ] c σ ( ˆ N ) ⊗ Z [ X • ( ˆ T ad ) pos ] , ¯ e ˆ αi (cid:55)→ q ( λ, ˆ αi ) Z ∼ = Z [ V ˆ T | d = λ ( q ) ] c σ ( ˆ N ) . Remark 3.
The c σ -action of ˆ N on V ˆ T | d = λ ( q ) induces an action of ˆ N on Z [ V ˆ T | d = λ ( q ) ]. On theother hand, the c σ -action of ˆ N on ˆ T induces an action of ˆ N on Z [ X • ( ˆ T )]. The inclusion (1.8) is not equivariant with respect to these two actions. Indeed, the base change of (1.8) to Q becomes anisomorphism, under which the action of ˆ N on Q [ V ˆ T | d = λ ( q ) ] induces an action of W on Q [ X • ( ˆ T ) σ ]given by(1.10) w • λ e ˆ λ = q (cid:104) λ,w ˆ λ − ˆ λ (cid:105) e w ˆ λ , w ∈ W , ˆ λ ∈ X • ( ˆ T ) σ . The following lemma follows from (1.8).
Lemma 4.
The image of the map (1.11) Z [ V ˆ T | d = λ ( q ) ] c σ ( ˆ N ) ⊂ Z [ V ˆ T | d = λ ( q ) ] (1.8) −−−→ Z [ X • ( ˆ T )] is the subring of Z [ X • ( ˆ T ) σ ] with a Z -basis given by elements of the form (cid:88) ˆ λ (cid:48) ∈ W ˆ λ q (cid:104) λ, ˆ λ (cid:48) − ˆ λ (cid:105) e ˆ λ (cid:48) , ˆ λ ∈ X • ( ˆ T ) σ, − := X • ( ˆ T ) σ ∩ X • ( ˆ T ) − . .4. Classical Satake isomorphism over Z . Now, we assume that F is a non-archimedean localfield, with O its ring of integers and k (cid:39) F q the residue field. Let G be a connected reductive groupover O . In this case, (cid:101) F /F is an unramified extension and Γ (cid:101)
F /F ∼ = (cid:104) σ (cid:105) is generated by the geometric q -Frobenius σ of k . We also fix a uniformizer (cid:36) ∈ O . Then every ˆ λ ∈ X • ( T ) σ gives a well definedpoint ˆ λ ( (cid:36) ) ∈ T ( F ) ⊂ G ( F ).Let K = G ( O ), and let H G = C c ( K \ G ( F ) /K, Z ) be the space of compactly supported bi- K -invariant Z -valued functions on G ( F ). This is a natural algebra under convolution (with the chosenHaar measure on G ( F ) such that the volume of K is 1). Let T be the abstract Cartan of G , whichis defined as the quotient of a Borel subgroup B ⊂ G over O by its unipotent radical U ⊂ B . (But T is canonically defined, independent of the choice of B , e.g. see [Zhu17b, 0.3.2]). For a choice ofBorel subgroup B ⊂ G over O , we define the classical Satake transform(1.12) CT cl : H G → Z [ X • ( ˆ T ) σ ] , f (cid:55)→ CT cl ( f ) = (cid:88) ˆ λ ∈ X • ( ˆ T ) σ (cid:0) (cid:88) u ∈ U ( F ) /U ( O ) f (ˆ λ ( (cid:36) ) u ) (cid:1) e ˆ λ . This map is independent of the choice of B and the uniformizer (cid:36) . Proposition 5.
There exists a unique isomorphism
Sat cl : Z [ V ˆ G,ρ ad (cid:111) (cid:104) σ (cid:105)| ˜ d ρ ad =( q,σ ) ] ˆ G ∼ = −→ H G , which we call the Satake isomorphism, making the following diagram commutative Z [ V ˆ G,ρ ad (cid:111) (cid:104) σ (cid:105)| ˜ d ρ ad =( q,σ ) ] ˆ G ∼ =Sat cl (cid:47) (cid:47) Res ∼ = (cid:15) (cid:15) H G CT cl (cid:15) (cid:15) Z [ V ˆ T | d = ρ ad ( q ) ] c σ ( ˆ N ) (cid:31) (cid:127) (1.11) (cid:47) (cid:47) Z [ X • ( ˆ T ) σ ] . In particular, H G is finitely generated.Proof. The uniqueness is clear. To prove the existence, by Lemma 4, it is enough to show that theSatake transform (1.12) induces an isomorphismCT cl : H G ∼ = −→ Z [ X • ( ˆ T ) σ ] ∩ Q [ X • ( ˆ T σ ] W • ρ ad , where W • ρ ad denotes the action given in (1.10) (with λ = ρ ad ). Indeed, this follows from the usualSatake isomorphism by noticing that (1.12) differs from the usual Satake transform (e.g. see [Gr96,(3.4)] in the split case) by a square root of the modular character. (cid:3) Remark 6.
The above isomorphism might look artificial as we identify both sides with a subringof Z [ X • ( ˆ T ) σ ]. In the next section, we will deduce this isomorphism from the geometric Satake .This alternative approach has the advantage that it is more natural and is independent of the usualSatake isomorphism, and is useful for some arithmetic applications.Let us explain the relation of Sat cl with the classical Satake isomorphism (e.g see [Gr96, Propo-sition 3.6] in the split case) and the mod p Satake isomorphism as in [He11, HV15].First by (1.7), ( V ˆ G,ρ ad | ˜ d ρ ad =( q,σ ) ) Z [ q − ] ∼ = ( C G | d ρ ad =( q,σ ) ) Z [ q − ] , so the above isomorphism inducesa canonical isomorphism H G ⊗ Z [ q − ] ∼ = Z [ q − ][ C G | d ρ ad =( q,σ ) ] ˆ G . Using the arithmetic Frobenius will lead to a different formulation by taking the dual. In fact, this is how the formulation given here was first discovered. fter choosing a square root q / , there is a ˆ G -equivariant isomorphism (comparing with (1.3)) C G | d ρ ad =( q,σ ) ∼ = ˆ Gσ, ( g, ( q, σ )) ∈ ˆ G (cid:111) ( G m × (cid:104) σ (cid:105) ) (cid:55)→ g ρ ( q − ) σ ∈ L G. Note that the following diagram is commutative Z [ q ± ][ C G | d ρ ad =( q,σ ) ] ˆ G ∼ = (cid:47) (cid:47) (1.11) ◦ Res (cid:15) (cid:15) Z [ q ± ][ ˆ Gσ ] ˆ G Res (cid:15) (cid:15) Z [ q ± ][ X • ( ˆ T ) σ ] e ˆ λ (cid:55)→ ( q − ) (2 ρ, ˆ λ ) e ˆ λ (cid:47) (cid:47) Z [ q ± ][ X • ( ˆ T ) σ ] , where the right vertical map is the restriction map from functions on ˆ Gσ to function on ˆ T σ . Thecomposition of (1.12) with the bottom map in the above diagram is the usual Satake transform.We thus obtain the usual classical Satake isomorphism H G ⊗ Z [ q ± ] ∼ = Z [ q ± ][ ˆ Gσ ] ˆ G . On the other hand, after mod p , (1.12) is the formula used in [He11, HV15] to define the mod p Satake isomorphism. In addition, F p [ V ˆ G | d = ρ ad ( q ) ] = F p [As ˆ G ]. Corollary 7.
There is a canonical isomorphism H G ⊗ F p ∼ = F p [As ˆ G ] c σ ( ˆ G ) . This gives a natural description of the mod p Hecke algebra by Langlands duality. Note that bydefinition F p [As ˆ G ] c σ ( ˆ G ) = (cid:77) ν ∈ X • ( ˆ T ) + ( S ν ∗ ⊗ S ν ) c σ ( ˆ G ) ∼ = F p [ X • ( T ) σ, − ] . Therefore, we recover the mod p Satake isomorphism [He11, HV15] (for trivial V in loc. cit. ). Example 8.
Let G = PGL , so that C G = ˆ G T = GL and V ˆ G,ρ ad = V ˆ G = M is the monoid of 2 × d = det : M → A is the usual determinant map. Then Z [ M | det= q ] SL ∼ = Z [tr] is thepolynomial ring generated by the trace function. On the other hand, H G = Z [ T p ] is a polynomialring generated by the T p -operator. Under the canonical Satake isomorphism T p matches tr. Remark 9. (1) Proposition 5 is compatible with the Weil restriction of G along unramified exten-sions. We leave the verification as an exercise.(2) As suggested by Bernstein, it is the C -group rather than the L -group that should be usedin the formulation of the Langlands functoriality. Similarly, we expect the Vinberg monoid of ˆ G might be useful to formulate the more subtle arithmetic aspect of the Langlands functoriality.1.5. Satake isomorphism for the Hecke algebra of weight V . We retain notations from § Z -algebra, and let ( V, π ) be a Λ-module equipped with a Λ-linear action of K = G ( O ).We first briefly recall the general formalism of Hecke algebra H G ( V ) of “weight” V and theSatake transform. We refer to [HV15] for a more general treatment in a more abstract setting.First, we define(1.13) H G ( V ) := { f : G ( F ) → End Λ ( V ) | f ( k (cid:48) gk )( v ) = k (cid:48) ( f ( g )( kv )) , ∀ k, k (cid:48) ∈ K, Supp( f ) is compact } , with the ring structure given by convolution( f (cid:63) f )( g )( v ) = (cid:88) h ∈ G/K f ( h )( f ( h − g )( v )) , g ∈ G ( F ) , v ∈ V. xample 10. Let G = T be a torus over O . Let Λ = O L , where L is a non-archimedean localfield over F . Let V be the rank one free module over Λ on which T ( O ) acts through a continuouscharacter χ : T ( O ) → O × L . In this case, we write H T ( V ) as H T ( χ ). There is an isomorphism(1.14) H T ( χ ) ∼ = O L [ X • ( ˆ T ) σ ] , f (cid:55)→ (cid:88) ˆ λ ∈ X • ( T ) σ f (ˆ λ ( (cid:36) )) e ˆ λ . If χ is non-trivial, this isomorphism depends on the choice of a uniformizer (cid:36) ∈ F .Similar to (1.12), there is the Satake transform(1.15) CT cl V : H G ( V ) → H T ( V U ( O ) ) , f (cid:55)→ (cid:16) CT cl V ( f ) : t ∈ T ( F ) (cid:55)→ (cid:88) u ∈ U ( F ) /U ( O ) f ( tu ) | V U ( O ) (cid:17) . To justify the definition, note that the sum (cid:80) u ∈ U ( F ) /U ( O ) f ( tu ) | V U ( O ) is finite and that for v ∈ V U ( O ) , (cid:80) u ∈ U ( F ) /U ( O ) f ( tu )( v ) ∈ V U ( O ) . In addition, one verifies that (1.15) is a homomorphism,either by a direct computation (e.g. see Step 2 in the proof of [He11, Theorem 1.2]), or by con-sidering the action of H G ( V ) on the “principal series representation of weight V ” (e.g. see [HV15,Section 2]). By virtue of (1.14), (1.15) specializes to (1.12) when V = is the trivial representation. Remark 11.
In most literature, V is assumed to be a smooth K -module, i.e. the stabilizer of everyelement v ∈ V in K is open. But this assumption is in fact not necessary in the above discussions.In the above generality, there is very little one can say about (1.15). In the sequel, we specialize V to the following situation. Let L be a non-archimedean local field over F , with O L its ring ofintegers. Let V be a finite free O L -module arising from an algebraic representation of G over O L ,such that V U ( O ) is free of rank one. This is the case if and only if dim L V U L L = 1. In this case, T ( O ) = B ( O ) /U ( O ) acts on V U ( O ) via a dominant weight λ of T . It follows from (1.14) that H T ( V U ( O ) ) = H T ( λ ) is commutative. Lemma 12.
Under the above assumption, the map CT cl V is injective.Proof. The proof given below follows the same strategy in [ST06, He11, HV15]. First,
Lemma 13.
Fix a uniformizer (cid:36) ∈ F . For every ˆ µ ∈ X • ( ˆ T ) σ, − , There is a unique element f ˆ µ ∈ H G ( V ) satisfying • f ˆ µ is supported on K ˆ µ ( (cid:36) ) K ; • f ˆ µ (ˆ µ ( (cid:36) )) | V U ( O ) = id : V U ( O ) → V U ( O ) .In addition, The collection { f ˆ µ } ˆ µ ∈ X • ( ˆ T ) σ, − form an O L -basis of H G ( V ) .Proof. Clearly, restricting a map f : G ( F ) → End O L ( V ) to ˆ µ ( (cid:36) ) induces a bijection betweenelements in H G ( V ) satisfying the first condition and O L -linear maps ϕ : V → V satisfying(1.16) π (ˆ µ ( (cid:36) ) k ˆ µ ( (cid:36) ) − ) ϕ ( w ) = ϕ ( π ( k ) w ) , ∀ k ∈ K ∩ ˆ µ ( (cid:36) ) − K ˆ µ ( (cid:36) ) , w ∈ V. As K ∩ ˆ µ ( (cid:36) ) − K ˆ µ ( (cid:36) ) is Zariski dense in G ( F ), the rational map ϕ L : V L → V L must be equal to cπ (ˆ µ ( (cid:36) )) for some c ∈ L . Then ϕ preserves the integral lattice V if and only if c ∈ (cid:36) (cid:104) λ, − ˆ µ (cid:105) O L .It follows from the above considerations that f ˆ µ (ˆ µ ( (cid:36) )) = (cid:36) (cid:104) λ, ˆ µ (cid:105) π (ˆ µ ( (cid:36) )) is the desired elementas in the lemma. In addition { f ˆ µ } ˆ µ ∈ X • ( ˆ T ) σ, − form an O L -basis of H G ( K ). (cid:3) ote that the natural map H G ( V ) ⊗ L → H G ( V L ) is an isomorphism, and there is the followingcommutative diagram(1.17) H G ( ) ⊗ L CT cl (cid:47) (cid:47) (cid:15) (cid:15) H T ( ) ⊗ L (cid:15) (cid:15) H G ( V ) ⊗ L CT cl V (cid:47) (cid:47) H T ( V U ( O ) ) ⊗ L where the left vertical map sends f : G ( F ) → L to ˜ f : G ( F ) → End( V L ) , g (cid:55)→ f ( g ) π ( g ), and theright vertical map sends f : T ( F ) → L to ˜ f : T ( F ) → End( V U L L ) = L, t (cid:55)→ f ( t ) λ ( t ).The left vertical map sends 1 K ˆ µ ( (cid:36) ) K to (cid:36) (cid:104) λ, ˆ µ (cid:105) f ˆ µ , and therefore is an isomorphism. Similarly, theright vertical map is an isomorphism. As CT cl ⊗ L is injective, we conclude Lemma 12 . (cid:3) In this sequel, we further assume that F = Q p , and choose the uniformizer (cid:36) = p . Then wewrite the Satake transform (1.15) as a homomorphism CT cl V : H G ( V ) → O L [ X • ( ˆ T ) σ ] using (1.14).For a dominant weight λ of G , let λ ad : G m → ˆ T → ˆ T ad be the corresponding cocharacter of ˆ T ad . Proposition 14.
There exists a unique isomorphism
Sat cl : O L [ V ˆ G | d =( λ ad + ρ ad )( p ) ] c σ ( ˆ G ) ∼ = −→ H G ( V ) , making the following diagram commutative O L [ V ˆ G | d =( λ ad + ρ ad )( p ) ] c σ ( ˆ G )Res ∼ = (cid:15) (cid:15) ∼ =Sat cl (cid:47) (cid:47) H G ( V ) CT cl V (cid:15) (cid:15) O L [ V ˆ T | d =( λ ad + ρ ad )( p ) ] c σ ( ˆ N ) (cid:31) (cid:127) (1.11) (cid:47) (cid:47) O L [ X • ( ˆ T ) σ ] . In particular, H G ( V ) is finitely generated.Proof. As in the proof of Proposition 5, it is enough to proveCT cl V : H G ( V ) ∼ = O L [ X • ( ˆ T ) σ ] ∩ L [ X • ( ˆ T ) σ ] W • λ ad+ ρ ad . But this follows from the case V = and the commutative diagram (1.17). (cid:3) Remark 15. (1) It follows from (1.9) that the algebra O L [ V ˆ G ] c σ ( ˆ G ) specializes all H G ( V )’s.(2) Taking the p -adic completion and inverting p allows us to recover some results of [ST06] onthe Banach-Hecke algebra (cid:98) H G ( V )[1 /p ].(3) Again, the way to identify H G ( V ) with O [ V ˆ G | d =( λ ad + ρ ad )( p ) ] c σ ( ˆ G ) given above might lookartificial. It would be interesting to have a geometric version of Proposition 14.2. Compatibility with the geometric Satake
In this section, we deduce Proposition 5 from the geometric Satake equivalence.2.1.
The geometric Satake equivalence.
We refer to [Zhu17a, Zhu17b] and references citedthere for detailed discussions of the geometric Satake equivalence. We retain notations from § F /F be the splitting field of G . It is a finite unramified extension of F , with ˜ O its ring ofintegers and ˜ k the residue field. Let r = [ (cid:101) F : F ] = [˜ k : k ]. Let L + G denote the positive loop groupof G over k and let Gr denote its affine Grassmannian over k .For ˆ µ ∈ X • ( ˆ T ) + , let k ˆ µ ⊂ ˜ k be its field of definition and d ˆ µ := [ k ˆ µ : k ]. Let Gr ≤ ˆ µ denote theSchubert variety corresponding to ˆ µ , which is a (perfect) projective scheme defined over k ˆ µ . LetGr ˆ µ denote the open Schubert cell. e fix (cid:96) (cid:54) = p . Let IC ˆ µ be the intersection complex with Q (cid:96) -coefficient on Gr ≤ ˆ µ , so thatIC ˆ µ | Gr ˆ µ = Q (cid:96) [ (cid:104) ρ, ˆ µ (cid:105) ] . If k (cid:48) /k is an algebraic extension in ¯ k , let Sat G,k (cid:48) ,(cid:96) denote the category of L + G ⊗ k (cid:48) -equivariantperverse sheaves on Gr ⊗ k (cid:48) with Q (cid:96) -coefficients, which is a tensor abelian categories. Inside Sat G, ˜ k,(cid:96) ,there is a full semisimple monoidal abelian subcategory Sat TG, ˜ k,(cid:96) as defined in [Zhu17b] : it is thefull semisimple tensor abelian category generated by all { IC ˆ µ ( i ) , ˆ µ ∈ X • ( T ) + , i ∈ Z } . Now wedefine Sat TG,(cid:96) as the category of Gal(˜ k/k )-equivariant objects in Sat
TG, ˜ k,(cid:96) . I.e., objects are pairs( F , γ ), where F ∈
Sat G, ˜ k,(cid:96) and γ : σ ∗ F (cid:39) F is an isomorphism such that the induced isomorphism F = ( σ r ) ∗ F γσ ( γ ) ··· σ r − ( γ ) −−−−−−−−−→ F is the identity map, and morphisms from ( F , γ ) to ( F (cid:48) , γ (cid:48) ) aremorphisms from F to F (cid:48) in Sat TG, ˜ k,(cid:96) that are compatible with γ and γ (cid:48) . This is still a semisimpletensor category.For a (not necessarily connected) split reductive group H over a field E of characteristic zero,let Rep( H E ) denote the category of finite dimensional algebraic representations of H over E . Let σ − Mod Q (cid:96) denote the category of representations of σ on finite dimensional Q (cid:96) -vector spaces. Hereis the version of the geometric Satake equivalence we need in the sequel. Theorem 16.
There is a natural equivalence of tensor categories
Sat : Rep( C G Q (cid:96) ) ∼ = Sat TG,(cid:96) suchthat the composition with the cohomology functor H ∗ (Gr ¯ k , − ) : Sat TG,(cid:96) → σ − Mod Q (cid:96) is the restrictionfunctor Rep( C G Q (cid:96) ) → σ − Mod Q (cid:96) induced by the inclusion σ (cid:55)→ (1 , q, σ ) ∈ ˆ G (cid:111) ( G m × (cid:104) σ (cid:105) ) .Proof. If G is split (so C G = ˆ G T ), this was stated in [Zhu17b, Lemma 5.5.14]. So we obtain anatural equivalence Rep( ˆ G T Q (cid:96) ) ∼ = Sat TG, ˜ k,(cid:96) satisfying the desired properties as in the theorem with σ replaced by σ r . In addition, ˆ G T is equipped with a pinning (induced from the pinning of ˆ G asdescribed in [Zhu17b, Corollary 5.3.23]).Now for ( F , γ ) ∈ Sat
TG,(cid:96) , we have a canonical isomorphism H ∗ (Gr ¯ k , F ) ∼ = H ∗ (Gr ¯ k , σ ∗ F ) ∼ =H ∗ (Gr ¯ k , F ). Using the formalism as in [RZ14, Lemma A.3] and an argument similar to [RZ14,Proposition A.6] (see also [Zhu17b, Lemma 5.5.5] and the paragraphs before it), we see that theabove equivalence induces the pinned action of σ on ˆ G T Q (cid:96) , and the desired equivalence. (cid:3) Remark 17. (1) Without adding the G m factor, the Galois action of σ on ˆ G obtained by theformalism [RZ14, Lemma A.3] does not preserve the pinning. The semidirect product of Γ k withˆ G ( Q (cid:96) ) using this action was denoted by L G geom in loc. cit. (and later denoted by L G geo in [Ri14, §
5] and in [Zhu17b, § §
5] (see also [Zhu17b, Theorem5.5.12]), via restriction along the injective map L G geo → C G ( Q (cid:96) ).(2) One of the consequences of the above theorem is as follows. For a σ -invariant dominantweight ˆ µ , there is a unique (up to isomorphism) irreducible representation V ˆ µ of C G Q (cid:96) such that V ˆ µ | ˆ G is irreducible of highest weight ˆ µ and that the action of G m × (cid:104) σ (cid:105) on the lowest weight line of V ˆ µ (w.r.t. ( ˆ G, ˆ B )) is trivial. Namely, under the geometric Satake, V ˆ µ corresponds to IC ˆ µ equippedwith the natural Gal(˜ k/k )-equivariant structure. Of course, this fact is well-known.We will need the following properties of the geometric Satake equivalence.First, let T be the abstract Cartan of G . We can define the Satake category Sat TT,(cid:96) as a sub-category of perverse sheaves on Gr T . For a choice a Borel subgroup B ⊂ G over O , we have thecorrespondence Gr T r ←− Gr B i −→ Gr G . Recall that for ˆ λ ∈ X • ( ˆ T ), we have the point ˆ λ ( (cid:36) ) ∈ Gr T (˜ k ). Strictly speaking, only equal characteristic version was considered in [Zhu17b]. However its counterpart in mixedcharacteristic is obvious, using [Zhu17a]. The similar remark applies to the discussions in the sequel. et Gr B, ˆ λ = r − (ˆ λ ( (cid:36) )) be the (geometrically) connected component of Gr B given by ˆ λ . We write r ˆ λ (resp. i ˆ λ ) be the restriction of r (resp. i ) to Gr B, ˆ λ . Then we define the Mirkovi´c-Vilonen’sweight functor CT : Sat TG,(cid:96) → Sat
TT,(cid:96) , CT( F ) = (cid:77) ˆ λ r ˆ λ, ! i ∗ ˆ λ F [(2 ρ, ˆ λ )] . We note that CT( F ) is a sheaf on Gr T, ˜ k naturally equipped with a Gal(˜ k/k )-equivariant structureso that the above definition makes sense. Then the the geometric Satake fits into the followingcommutative diagram(2.1) Rep( C G Q (cid:96) ) Sat (cid:47) (cid:47)
Res (cid:15) (cid:15)
Sat
TG,(cid:96) CT (cid:15) (cid:15) Rep( C T Q (cid:96) ) Sat (cid:47) (cid:47)
Sat
TT,(cid:96) . This follows from the usual compatibility between the geometric Satake and the restriction to themaximal torus, with the Galois action taking into account.For
F ∈
Sat
TG,(cid:96) , its Frobenius trace function f F ∈ C c ( K \ G ( F ) /K, Q (cid:96) )makes sense as usual. The next fact we need is a description of f Sat( V ) for V ∈ C G Q (cid:96) . For thispurpose, we need to recall the so-called Brylinski-Kostant filtration. Let { ˆ e, ρ, ˆ f } be the principal sl -triple of ˆ G containing ˆ e ∈ Lie ˆ B and 2 ρ ∈ X • ( ˆ T ) ⊂ Lie ˆ T . For a representation V of ˆ G andˆ λ ∈ X • ( ˆ T ) − , we define the Brylinski-Kostant filtration on V (ˆ λ ) as F i V (ˆ λ ) := ker( ˆ f i +1 : V → V ) ∩ V (ˆ λ ) . Let gr F • V (ˆ λ ) denote its associated graded. Note that if ˆ λ is σ -invariant, then F i V (ˆ λ ) is σ -stableand therefore σ acts on gr F • V (ˆ λ ). Proposition 18.
Let V be a representation of C G Q (cid:96) , and ˆ λ ∈ X • ( ˆ T ) σ, − . Then f Sat( V ) (ˆ λ ( (cid:36) )) = ( − (cid:104) ρ, ˆ λ (cid:105) tr((1 , q, σ ) | gr Fi V (ˆ λ )) q − i . Proof.
This follows [Zhu15, § loc. cit. relies on the existence of “big open cell” of the affine Grassmannian, whichis not known in mixed characteristic. However, one can easily replace the purity argument in loc.cit. by a parity argument, e.g. the argument in the middle of p. 452 of [Zhu17a]. (cid:3) Remark 19.
Recall that for the representation V = V ˆ µ of C G Q (cid:96) as in Remark 17, and forˆ λ ∈ X • ( ˆ T ) σ , Jantzen’s twisted character formula ([Ja73, Satz 9]) expresses tr((1 , , σ ) | V ˆ µ (ˆ λ ))as the dimension of a representation of a reductive group ˆ G σ whose weight lattice is X • ( ˆ T ) σ . Itwould be very interesting to have its q -analogue, expressing (cid:80) i tr( σ | gr Fi V ˆ µ (ˆ λ )) q − i in terms ofrepresentations of ˆ G σ , which should, among other things, imply that tr( σ | gr Fi V ˆ µ (ˆ λ )) ∈ Z ≥ .We do not have such a formula at the moment. The following lemma is sufficient for our purpose. Lemma 20.
Let V ˆ µ be as above. Then for every ˆ λ ∈ X • ( ˆ T ) σ , tr( σ | gr Fi V ˆ µ (ˆ λ )) ∈ Z .Proof. We may assume that ˆ λ ∈ X • ( T ) σ, − . The root datum of G defines a reductive group G over C equipped with a C -automorphism σ . Let Gr G be its affine Grassmannian, acted by σ . We havethe geometric Satake for Gr G , and the analogous statement of Proposition 18 in this setting istr( σ | gr Fi V ˆ µ (ˆ λ )) = tr( σ | H − i + (cid:104) ρ, ˆ λ (cid:105) ˆ λ ( (cid:36) ) IC ˆ µ ) , here H − i + (cid:104) ρ, ˆ λ (cid:105) ˆ λ ( (cid:36) ) IC ˆ µ denotes the stalk cohomology of IC ˆ µ at ˆ λ ( (cid:36) ). Over C , the sheaf IC ˆ µ has anatural Z -structure preserved by the action of σ ([MV07, Proposition 8.1]). The lemma follows. (cid:3) The representation ring.
We generalize some well-known relations between the represen-tation ring of a reductive group and its ring of invariant functions. Let E be a characteristic zerofield. Let Rep + ( ˆ G TE ) ⊂ Rep( ˆ G TE ) denote the subcategory consisting of those objects on which allweights of G m ⊂ ˆ G (cid:111) Ad ρ ad G m = ˆ G T are ≥
0. Let Rep( V ˆ G,ρ ad ,E ) denote the category of finitedimensional algebraic representations of V ˆ G,ρ ad ,E . Lemma 21.
The inclusion ˆ G T ⊂ V ˆ G,ρ ad induces an equivalence of categories Rep + ( ˆ G TE ) ∼ = Rep( V ˆ G,ρ ad ,E ) .Proof. First, under the inclusion ˆ G × Z ˆ G ˆ T → V ˆ G , an irreducible representation V of ˆ G E × ˆ T E canbe extended to a representation of V ˆ G,E if and only if the following holds: if V | ˆ G E has the highestweight ˆ µ , and V | ˆ T E has the weight ˆ ν , then ˆ ν + w (ˆ µ ) is a sum of nonnegative roots of ˆ G . Therefore,an irreducible representation of ( ˆ G × G m ) / (2 ρ × id)( µ ) can be extended to a representation of V ˆ G,ρ ad if and only if the following two conditions hold:(i) G m acts on V by some weight n ; and(ii) If ˆ µ is the highest weight of V as a ˆ G -representation, then (cid:104) ρ, ˆ µ (cid:105) ≤ n and ( − (cid:104) ρ, ˆ µ (cid:105) =( − n .But under the isomorphism (1.2), this exactly corresponds to an object in Rep + ( ˆ G TE ). (cid:3) Now let σ be a finite order automorphism of ( ˆ G, ˆ B, ˆ T , ˆ e ) as before. We let Rep int ( G m ×(cid:104) σ (cid:105) ) denotethe full tensor subcategory of finite dimensional algebraic E -linear representations of G m × (cid:104) σ (cid:105) consisting of W satisfying the following two properties • all G m -weights in W are ≥ • the trace of σ on each G m -weight subspace of W is an integer.This is equivalent to requiring that the trace of ( q, σ ) on W is an integer.Let Rep int ( C G E ) be the full tensor subcategory of Rep( C G E ) consisting of those representations V such that for every σ -invariant weight ˆ λ of ˆ T and every i ≥
0, the space gr Fi V (ˆ λ ) as a represen-tation of G m × (cid:104) σ (cid:105) ⊂ ˆ G (cid:111) ( G m × (cid:104) σ (cid:105) ) belongs to Rep int ( G m × (cid:104) σ (cid:105) ). Note that the restriction ofsuch a representation to ˆ G T belongs to Rep + ( ˆ G TE ), and therefore we have the functorRep int ( C G E ) → Rep( V ˆ G,ρ ad ,E (cid:111) (cid:104) σ (cid:105) ) . When σ is trivial, we have Rep int ( C G E ) ∼ = Rep + ( ˆ G TE ) ∼ = Rep( V ˆ G,ρ ad ,E ). Lemma 22.
There exists a unique homomorphism tr : K (Rep int ( C G E )) → Z [ V ˆ G,ρ ad | d ρ ad = q ] c σ ( ˆ G ) making the following diagram commutative (2.2) K (Rep int ( C G E )) tr (cid:47) (cid:47) K (Res) (cid:15) (cid:15) Z [ V ˆ G,ρ ad | d ρ ad = q ] c σ ( ˆ G )(1.11) ◦ Res (cid:15) (cid:15) K (Rep int ( C T E )) [ V ] (cid:55)→ (cid:80) ˆ λ ∈ X • ( ˆ T ) σ tr(( q,σ ) | V (ˆ λ )) e ˆ λ (cid:47) (cid:47) Z [ X • ( ˆ T ) σ ] . Proof.
We define the maptr : K (Rep int ( C G E )) → K (Rep( V ˆ G,ρ ad ,E (cid:111) (cid:104) σ (cid:105) ) → E [ V ˆ G,ρ ad | d ρ ad = q ] c σ ( G ) , where the second arrow is induced by taking the trace of representations at elements in V ˆ G | d = ρ ad ( q ) σ .The diagram is clearly commutative when the right vertical map is tensored with E . To see that tr actors through Z [ V ˆ G,ρ ad | d = q ] c σ ( ˆ G ) , it is enough to note that Z [ V ˆ G,ρ ad | d = q ] c σ ( ˆ G ) = E [ V ˆ G,ρ ad | d = q ] c σ ( ˆ G ) ∩ Z [ X • ( ˆ T ) σ ] inside E [ X • ( ˆ T ) σ ] which follows easily from Lemma 4. The uniqueness is clear as theright vertical map is injective. (cid:3) Lemma 23.
The homomorphism in Lemma 22 is surjective.Proof.
For a σ -invariant dominant weight ˆ µ , let V ˆ µ be the unique (up to isomorphism) irreduciblerepresentation of C G E as in Remark 17 (2). We claim that V ˆ µ ∈ Rep int ( C G E ). Then the lemmafollows from the explicit description of Z [ V ˆ G,ρ ad | d ρ ad = q ] c σ ( ˆ G ) inside Z [ X • ( ˆ T ) σ ], as in Lemma 4.We need to show that tr((1 , q, σ ) | gr iF V ˆ µ (ˆ λ )) ∈ Z for σ -invariant weight ˆ λ . Note that under themap ˆ G × G m → ˆ G T from the isomorphism (1.2), the second factor G m acts on V ˆ µ by a fixed weight(namely (cid:104) ρ, ˆ µ (cid:105) ). Therefore, (1 , q ) ∈ ˆ G (cid:111) G m = ˆ G T acts on V ˆ µ (ˆ λ ) by q (cid:104) ρ ad , ˆ λ − w ˆ µ (cid:105) . Therefore, thelemma follows by noticing that the trace of (1 , , σ ) on V ˆ µ (ˆ λ ) is an integer, by Lemma 20. (cid:3) From
Sat to Sat cl . Let Sat T, int G,(cid:96) be the full subcategory of Sat
TG,(cid:96) consisting of those F suchthat f F ∈ H G , i.e. f F (ˆ λ ( (cid:36) )) ∈ Z for every ˆ λ ∈ X • ( ˆ T ) σ . By Proposition 18, we have the followingstatement. Lemma 24.
The geometric Satake induces an equivalence
Rep int ( C G E ) ∼ = Sat T, int G,(cid:96) . Lemma 25.
Taking the trace Frobenius function induces a surjective ring homomorphism tr : K (Sat T, int G,(cid:96) ) → H G . Proof.
For dominant ˆ µ ∈ X • ( ˆ T ) σ , by Proposition 18 and Lemma 23,(2.3) f IC ˆ µ = ( − (cid:104) ρ, ˆ µ (cid:105) K ˆ µ ( (cid:36) ) K + (cid:88) ˆ µ (cid:48) < ˆ µ a ˆ µ ˆ µ (cid:48) K ˆ µ (cid:48) ( (cid:36) ) K , a ˆ µ ˆ µ (cid:48) ∈ Z , where 1 K ˆ µ (cid:48) ( (cid:36) ) K denotes the characteristic function on K ˆ µ (cid:48) ( (cid:36) ) K and “ < ” denotes the Bruhat order.Then the lemma follows since these 1 K ˆ µ ( (cid:36) ) K ’s form a Z -basis of H G . (cid:3) Note that the Grothendieck-Lefschetz trace formula implies thattr ◦ K (CT) = CT cl ◦ tr : K (Sat T, int G,(cid:96) ) → Z [ X • ( ˆ T ) σ ] . Together with (2.1) and (2.2), we obtain the following commutative diagram K (Rep int ( C G Q (cid:96) )) K (Sat) ∼ = (cid:15) (cid:15) tr (cid:47) (cid:47) Z [ V ˆ G | d = ρ ad ( q ) ] c σ ( G ) Res (cid:47) (cid:47) Z [ V ˆ T | d = ρ ad ( q ) ] c σ ( ˆ N )(1.11) (cid:15) (cid:15) K (Sat T, int G,(cid:96) ) tr (cid:47) (cid:47) H G CT cl (cid:47) (cid:47) Z [ X • ( ˆ T ) σ ] . Both trace maps tr in the above diagram are surjective, by Lemma 23 and 25. Therefore, we cancomplete the diagram by adding an isomorphism Sat cl in the middle column, as desired. References [BG11] K. Buzzard, T. Gee, The conjectural connections between automorphic representations and Galois repre-sentations,
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