ON THE SATAKE ISOMORPHISM

In a 1983 paper, the author has established a (decategorified) Satake equivalence for affine Hecke algebras. In this paper, we give new proofs for some results of that paper, one based on the theory of J-rings and one based on the known character formula for rational representations of a reductive group in positive, large characteristic. We also give an extension of that formula to disconnected groups.

Introduction 0.1.Let H q be the affine Hecke algebra over C (with equal parameters q, a prime power) associated to an affine Weyl group W (defined in terms of the dual G * of an adjoint group G).Let H 0,q be the Hecke algebra over C (with equal parameters q) associated to the corresponding finite Weyl group W 0 ⊂ W .Let H sph q the vector subspace of H q consisting of elements which are eigenvectors for the left and right multiplication by H 0,q with eigenvalue defined by the one dimensional representation of H 0,q corresponding to the unit representation of W 0 .Then H sph q is an algebra for the product f * f ′ = ( w∈W 0 q |w| ) −1 f f ′ where f f ′ is the product in H q and || is the standard length function on W 0 .Let Q be the group of translations in W .The classical Satake isomorphism states that the algebra H sph q is isomorphic to the algebra of W 0 -invariants in the group algebra C [Q].In [L83] we gave a refinement of this isomorphism in which the basis of C[Q] formed by the irreducible representations of a semisimple group with Weyl group W 0 and for which Q is the lattice of roots corresponds to a basis β of H sph q formed by certain elements of the basis [KL79] of H q , suitably normalized.This shows in particular that (a) the structure constants of the algebra H sph q with respect to β are integers independent of q.This is the starting point of "geometric Satake equivalence" (which we do not discuss in this paper).0.2.In this paper we show (see 1.5) that the structure constants in 0.1(a) can be interpreted as structure constants for a certain subring J * of the J-ring attached to W with respect to the standard basis of J * .(We actually prove a more general statement involving a weight function on W .)This gives a new (and simpler) proof of 0.1(a).We also give another approach to 0.1(a) based on the character formula for simple rational modules of a semisimple group in characteristic p ≫ 0. At the time when [L83] was written, this character formula was only conjectured and providing evidence for the conjecture was one of the motivations which led Supported by NSF grant DMS-1855773.e) l u .This proves (b).

Typeset by
In the setup of (b) we have L03, 13.4], for any w, w ′ in W we have ]T w ′′ .(In the case where L = || this is proved in [L85, §7]; the proof for general L is entirely similar.)As in [L03, 13.5], we deduce that for any w, w ′ in W we have (a) c w c w ′ = w ′′ ∈W h w,w ′ ,w ′′ c w ′′ (finite sum) where h w,w ′ ,w ′′ = N w,w ′ ,w ′′ ;L v L(M 0 ) mod v L(M 0 )−1 with N w,w ′ ,w ′′ ;L ∈ Z.
Let J be the free abelian group with basis {τ w ; w ∈ W }. We define a bilinear multiplication J × J − → J by τ w τ w ′ = w ′′ ∈W N w,w ′ ,w ′′ ;L τ w ′′ , (this is a finite sum.)It is known [L03, 18.3] that this multiplication is associative if the conditions in [L03, 18.1] are satisfied.
Let J * be the subgroup of J with basis {τ M x ; x ∈ Q + }.From 1.2(b) we see that J * is closed under the multiplication in J; thus for x, y in Q + we have τ M x τ M y = z∈Q + N M x ,M y ,M z ;L τ M z , (this is a finite sum.)Theorem 1.5.(a) For x, y in Q + we have (c) The isomorphism of abelian groups R ∼ − → J * given by c x → τ M x is compatible with the multiplication.In particular, the multiplication in J * is associative.
From the definition of c w , we have hw,w ′ ,w ′′ = h w,w ′ ,w ′′ for any w, w ′ , w ′′ in W , where ¯: A − → A is the ring involution which takes v n to v −n for any n.Using this and the fact that π L v −L(M 0 ) is fixed by ¯, we see that the left hand side of (d) is fixed by¯hence is necessarily in Z.
Taking the coefficient of v L(M 0 ) in the two sides of the equality This completes the proof of (a).Now (b),(c) are immediate consequences of (a).
1.6.The ring R has unit element c 0 and is known to be commutative; it follows that the ring J * has unit element τ M 0 and is commutative.In the case where L = ||, 1.5(b) recovers a result in [L83].For general L, 1.5(b) recovers a result in [K05].But the present proof is simpler than that in these references.

1.7.
In this subsection we assume that L = ||.In this case the ring J in [L03, 18.3] is associative.In [L97] we have categorified J to a monoidal tensor category with simple objects indexed by W .In particular J * is categorified to a monoidal tensor category J * .It is known that (as a consequence of 1.5(b)) R can be also categorified to a monoidal category S known as the "Satake category".The ring isomorphism R ∼ − → J * in 1.5(c) gives rise to an equivalence of monoidal categories S ∼ − → J * .

Use of modular representations
2.1.In this section we assume that L = || : W − → N. Let k be an algebraically closed field of characteristic p ≥ 0. Let G be an adjoint semisimple group over k with a fixed pinning (involving a maximal torus T ).We assume that the Weyl group of G is W 0 , the lattice of roots of G with respect to T is Q and that W = W 0 Q is the affine Weyl group associated in the usual way to the dual group 2.2.We now assume that p is a prime number, p ≫ 0. Following Verma [Ve] we identify W with the subgroup W p of the group of affine transformations of Q R generated by the reflections in the hyperplanes in H which preserve the set H.
Let x ∈ Q + be such that x / ∈ ∪ α,m H α,m and α0 (x) ≤ p(p − h + 2) where α0 is the highest coroot and h is the Coxeter number.It is known [AJS,KL94,KT95] that, as virtual T -modules, we have (a) L x = y∈Z x (−1) |w y w x | dim(V w y ,w x )V y , where Z x is the set of all y ∈ Q + in the same W p -orbit as x; w x , w y are certain well defined explicit elements of W p ; V w y ,w x is a C-vector space of dimension P w y ,w x ;|| (1) defined in terms of the stalks of the intersection cohomology complex of an affine Schubert variety associated to G * .
As shown in [L17, comments to [53]], from (a) with x of the form ) This provides a new proof of one of the main results in [L83].

2.3.
In this subsection we assume that p = 0. Let A be the subring of Q(v) consisting of elements which have no pole for v = 1.Let H A be the A-submodule of H spanned by {T w ; w ∈ W } or equivalently by {c w ; w ∈ W }. This is a subring of H.We define a group homomorphism ξ from H A to the group ring Q[W ] by w f w T w → w f w (1)w; here f w ∈ A. This is a ring homomorphism.Recall that for x ∈ Q + , P w,M x ;|| (1) depends only on the (W 0 , W 0 ) double coset of w ∈ W . Hence (We have used 2.2(b).)We have also a∈W 0 e ′ a. Indeed, and the first equality in (a) is established.The second equality in (a) follows the first by the substitution e ′ = aea −1 .

Now let
On the other hand we have Comparing with (b) we deduce for any e ′′ ∈ Q. Hence x,y,z;|| for any x, y, z in Q + .Thus, we recover one of the main results in [L83].

Folding
3.1.In this section we assume that W, S, s 0 , W 0 , Q, Q + in 1.1 are such that W is irreducible of simply laced type.We asume given an automorphism σ of (W, S) of order δ ∈ {2, 3} preserving s 0 .Let ′ W = {w ∈ W ; σ(w) = w}.For each σ-orbit O in S let s O be the longest element in the subgroup of W generated by the elements in O. Let ′ S be the subset of ′ W consisting of the elements s O for various O as above.Note that ( ′ W, ′ S) is an affine Weyl group.Let L : ′ W − → N be the restriction to ′ W of the usual length function of W ; this is a weight function on ′ W .
We preserve the setup of 2.1.We assume that G is simple of simply laced type.We fix an automorphism of G preserving the pinning of G which induces the automorphism σ of W considered above.This automorphism of G is denoted again by σ.
again by σ (they act as identity on a highest weight vector).We have where k * δ = {θ ∈ k * ; θ δ = 1} and V x,θ , L x,θ are the θ-eigenspaces of σ. 3.2.We now assume that p ≫ 0 and that x in 2.2(a) satisfies in addition σ(x) = x.The proof of 2.2(a) is sufficiently functorial to imply that we have also (equality in the representation ring of T /{σ(t)t −1 ; t ∈ T } tensored with C; here θ → θ is an imbedding of k * δ into C * ).Note that σ(w x ) = w x and that when y ∈ Z x , σ(y) = y, we have σ(w y ) = w y , so that σ acts naturally on V w y ,w x .We now substitute (b) tr(σ, V w y ,w x ) = P w y ,w x ;L (1) where P w y ,w x ;L is defined in terms of ′ W and L : ′ W − → N as in 3.1.(See 4.5, 4.6.)We obtain (c) This is an extension of the character formula 2.2(a) to disconnected groups.Note that the coefficients P w y ,w x ;L (1) are computable by an algorithm in [L03, §6] (which is somewhat more involved than that for the unweighted case in [KL79]).

3.3.
Note that σ acts naturally on G * .Let ′ G be the simply connected group over k isogenous to the dual group of the identity component of the σ-fixed point set on G * .By a theorem of Jantzen [Ja], the expression θ∈k * θ θV y,θ in (c) can be expressed in terms of the character of a Weyl module of ′ G. Using this one can deduce as in §2 the analogues of 2.3(b), 2.4(c) with (W, S, ||) replaced by ( ′ W, ′ S, L). (This recovers in our case a result in [K05]).
3.4.Assume that (W, S) is of (affine) type A 2 with σ of order 2. In this case ( ′ W, ′ S) is of (affine) type A 1 and the values of L|′ S are 1 and 3.In this case the ring J * associated to ( ′ W, ′ S, L) in 1.5 is isomorphic together with its basis to the representation ring of SL 2 (C) with its standard basis, see [L03, 18.5].This shows that the group ′ G in 3.3 cannot be replaced by the corresponding adjoint group (even though G was adjoint).
3.5.In the setup of 3.1, 3.2 with k = C, we identify W 0 with the group W 0 of affine transformations of Q R generated by the reflections in the (finitely many) hyperplanes in H and which preserve H. Let g be the Lie algebra of G. Let x ∈ Q be such that x / ∈ ∪ αH α,0 .Let z ∈ Q.Then the Verma g-module V x , its irreducible quotient L x and their z-weight spaces V z x , L z x are defined.It is known that the following equality (conjectured in [KL79]) holds: (a) dim L z x = y∈Z x (−1) |ω y ω x | P ω y ,ω x ;|| (1) dim V z y , where Z x is the set of all y ∈ Q in the same W 0 -orbit as x; ω x , ω y are certain well defined explicit elements of W 0 .Now assume that x, z are fixed by σ.Then σ : G − → G induces automorphisms of L z x and of V z x denoted again by σ.We have (b) tr(σ, L z x ) = y∈Z x ,σ(y)=y (−1) L(ω y ω x ) P ω y ,ω x ;L (1)tr(σ, V z y ).This follows from the proof of (a) in the same way as 3.2(c) follows from the proof of 2.2(a) (using 4.5).

4.
A geometric interpretation of P y,w;L 4.1.Let W 0 be a (finite) Weyl group with a set S 0 of simple reflections and let σ : W 0 − → W 0 be an automorphism preserving S 0 .For each σ-orbit O in S 0 we denote by σ O the longest element in the subgroup of W 0 generated by the reflections in O. Let ′ W 0 = {w ∈ W 0 ; σ(w) = w} and let ′ S 0 be the subset of ′ W 0 consisting of the elements s O for various O as above.Then ′ W 0 is a Weyl group with set of simple reflections ′ S 0 .Let L : ′ W 0 − → N be the restriction to ′ W 0 of the standard length function of W 0 ; it is known that L is a weight function on ′ W 0 so that the Hecke algebra over A with its bases {T w ; w ∈ ′ W 0 }, {c w ; w ∈ ′ W 0 } can be defined as in 1.1 (in terms of ′ W 0 , ′ S 0 , L instead of W, S, L).This Hecke algebra specialized at v = √ q with q a prime power is a C-algebra denoted by H 0,q;L .
For w ∈ ′ W 0 we write ] is 0 unless y ≤ w and w 0 is the longest element of W 0 (or ′ W 0 ).Note the following inductive formulas for R y,w;L .(Here s ∈ S.) We have P y,w;L = 0 unless y ≤ w and P w,w;L = 1.For y, w in ′ W 0 we have

4.2.
Let k be an algebraic closure of the finite prime field F p .Let G be a simply connected semisimple group over k with Weyl group (W 0 , S 0 ) and with a fixed pinning involving a maximal torus T and a Borel subgroup B containing T .We fix an F p -rational structure on G (with Frobenius map F : G − → G) compatible with the pinning such that T is split over F p hence B is defined over F p .We consider an automorphism of G preserving the pinning and compatible with the F p -structure; it induces an automorphism of W 0 which we assume to be σ.This automorphism of G is denoted again by σ; we have σF = F σ. Hence if t ≥ 1 then F t := F t σ = σF t is the Frobenius map for a rational structure over the subfield F p t with p t elements of k.Let B be the variety of Borel subgroups of G.Note that For any algebraic variety X of pure dimension let IC(X) be the intersection cohomology complex of X with coefficients in Ql (with l a prime = p).Let H i (X) (resp.H i c (X)) be the i-th cohomology (resp.i-th cohomology with compact support) of X with coefficients in IC(X).For x ∈ X let H i x (X) be the stalk at x of the i-th cohomology sheaf of IC(X).
Theorem 4.3.Let y ∈ ′ W 0 , w ∈ ′ W 0 be such that y ≤ w.We have (Note that σ acts naturally on H i y B ( Bw ).)The proof will use the following result (analogous to [KL79, A4(a)]).
Let F be the vector space of functions B F t − → C. Then F is an H 0,p t ;L -module in which for w ∈ ′ W 0 and f ∈ F we have T w f = f ′ where for B ′ ∈ B F t we have f ′ (B ′ ) = p −tL(w)/2 B ′′ ∈B F t ;pos(B ′ ,B ′′ )=w f (B ′′ ).Applying the equality 4.1(a) to f ∈ F and evaluating at B we see that for w ∈ ′ W 0 we have We now take f to be the function equal to 1 at C 0 = yw 0 B and equal to 0 on B F t − {C 0 }.We obtain The lemma is proved.
4.5.We now prove the theorem.When y = w the result is obvious.We can assume that y < w and that (a) the result is true when y, w is replaced by z, w with z ∈ ′ W 0 such that y < z ≤ w.Applying the Grothendieck-Lefschetz fixed point formula for F t on the F t -stable open subvariety Bw ∩ A y of Bw we obtain For such z we apply Lemma 4.4 and we obtain By Poincaré duality on Bw ∩ A y we have Using [KL80,4.5(a),1.5] we have By [KL80,4.2]we have H i y B ( Bw ) = 0 if i is odd while if i is even the eigenvalues of F t on H i y B ( Bw ) are equal to p it/2 .It follows that p tL (w)   i even p −it/2 tr(σ −1 , H i y B ( Bw )) = z∈ ′ W 0 ;y≤z≤w R y,z;L (p t )p tL(y) i even p it/2 tr(σ, H i z B ( Bw )).
Since this holds for t = 1, 2, . . .we can replace p t by v 2 where v is an indeterminate and we get an equality in Ql [v, v −1 ]: 4.6.The proof in 4.5 is written in such a way that it remains valid in the affine case so that it gives an analogous geometric interpretation for P y,w;L with y, w in ′ W where ′ W, L are as in 3.1.In this case B is an Iwahori subgroup and B * is an anti-Iwahori subgroup (opposed to B) as in [KL80,§5].The definition of A y still makes sense; it is the set of Iwahori subgroups opposed to a certaain fixed anti-Iwahori subgroup.Now R y,w;L as defined by 4.1(a) does not make sense in the affinne case; instead one can use the inductive definition in 4.1 (b).With this definition, the analogue of 4.1(c) remains valid; Lemma 4.4 remains valid but it is now proved by an (easy) induction on |z|.

F
t acts naturally on B and defines a Frobenius map on B. We say that B 1 , B 2 in B are opposed if B 1 ∩ B 2 is a maximal torus.We define B * ∈ B by the conditions that B ∩ B * = T .For B 1 , B 2 in B let pos(B 1 , B 2 ) ∈ W 0 be the relative position of B 1 , B 2 .For w ∈ W 0 we set B w = {B ′ ∈ B; pos(B, B ′ ) = w}.For y ∈ W 0 we define y B ∈ B by the conditions T ⊂ y B, y B ∈ B y ; we define y B * ∈ B by the conditions T ⊂ y B * , pos(B * , y B * ) = y.Let Bw be the closure of B w in B. For y ∈ W 0 we set A y = {B ′ ∈ B; B ′ , y B * opposed}.
tr(σ −1 , H i y B ( Bw )) = z∈ ′ W 0 ;y≤z≤w R y,z;L v 2L(y) i even v i tr(σ, H i z B ( Bw )).Using the induction hypothesis (a) we obtainv 2L(w) i even v −i tr(σ −1 , H i y B ( Bw )) − v 2L(y) i even v i tr(σ, H i y B ( Bw )) = z∈ ′ W 0 ;y<z≤w R y,z;L v 2L(y) P z,w;L .Using 4.1(c), the right hand side of this equality isv 2L(w)Py,w;L − v 2L(y) P y,w;L .Thus we have v L(w)−L(y) ( i evenv −i tr(σ −1 , H i y B ( Bw )) − Py,w;L ) = v L(y)−L(w) ( i even v i tr(σ, H i y B ( Bw )) − P y,w;L ).(b)By the known properties of H i , in both sides of (b) we can assume that i < dim B w − dim B y = L(w) − L(y).Moreover, we have v L(w)−L(y) Py,w;L ∈ vZ[v] and v L(y)−L(w) P y,w;L ∈ v −1 Z[v −1 ].Thus the left hand side of (b) is in v Ql [v] while the right hand side of (b) is in v −1 Ql [v −1 ].We see that both sides of (b) are zero.The theorem is proved.