The Satake isomorphism for special maximal parahoric Hecke algebras

Let G denote a connected reductive group over a nonarchimedean local field F. Let K denote a special maximal parahoric subgroup of G(F). We establish a Satake isomorphism for the Hecke algebra H of K-bi-invariant compactly supported functions on G(F). The key ingredient is a Cartan decomposition describing the double coset space K\G(F)/K. We also describe how our results relate to the treatment of Cartier, where K is replaced by a special maximal compact open subgroup K' of G(F) and where a Satake isomorphism is established for the Hecke algebra of K'-bi-invariant compactly supported functions on G(F).


Introduction
The Satake isomorphism plays an important role in automorphic forms and in representation theory of p-adic groups. For global applications, one may often work with unramified groups. We begin by recalling the Satake isomorphism in this context. Let G denote an unramified group over a nonarchimedean local field F . Let v F denote a special vertex in the Bruhat-Tits building B(G ad (F )). Let K = K v F denote a special maximal compact open subgroup of G(F ) which fixes v F . Let The group Λ M is a finitely generated abelian group which can be explicitly described and which has the property that Λ M = Λ M /torsion. Moreover, K/K ∼ = Λ M,tor , the torsion subgroup of Λ M .
When G is unramified over F or when G is semi-simple and simply connected, it turns out that K = K and Λ M ∼ = Λ M (see section 11) so that our theorem does not give any new information in those cases. However our results are new in case K = K, and different methods from [Car] are needed to prove them. For ramified groups in particular, our results are expected to play some role in the study of Shimura varieties with parahoric level structure at p. For more about ramified groups and Shimura varieties with parahoric level the reader should consult [Rap], [PR], and [Kr].
In order to describe Λ M , we need to recall some notation and results of Kottwitz [Ko97]. Let F s denote a separable closure of F , and let F un denote the maximal unramified extension of F in F s . Let L = F un denote the completion of F un with respect to the valuation on F un which extends the normalized valuation on F . Let I = Gal(F s /F un ) ∼ = Gal(L s /L) denote the inertia subgroup of Gal(F s /F ), and let σ ∈ Aut(L/F ) denote the Frobenius automorphism. In [Ko97] Kottwitz defined a surjective homomorphism κ G : G(L) → X * (Z( G)) I , and in loc. cit. §7.7 he also proved that this induces a surjective homomorphism κ G : G(F ) → X * (Z( G)) σ I of the groups of σ-invariants. Set G(L) 1 := ker(κ G ) and G(F ) 1 := G(F ) ∩ G(L) 1 . (When G = M , this is consistent with our definition of M (F ) 1 above, see Lemmas 4.1.1, 4.2.1.) The Iwahori-Weyl group W for G carries a natural action under σ and contains a σinvariant abelian subgroup Ω G (the subgroup of length-zero elements). By choosing representatives in the normalizer of A we may embed W σ set-theoretically into G(F ), and then Ω σ G is mapped by κ G isomorphically onto X * (Z( G)) σ I (see section 2). The following is the sought-after explicit description of Λ M : Proposition 1.0.2. The Kottwitz homomorphism induces an isomorphism Λ M = M (F )/M (F ) 1 ∼ = X * (Z( M )) σ I . We can also identify Λ M with Ω σ M via the Kottwitz isomorphism κ M : Ω σ M → X * (Z( M )) σ I . As before, the main step in the proof of Theorem 1.0.1 is an appropriate Cartan decomposition. Equivalently, via the isomorphism κ M : Ω σ M → X * (Z( M )) σ I , we have a bijection W (G, A)\X * (Z( M )) σ I → K\G(F )/K. We give additional information about the finitely generated abelian group Λ M in section 11. For example, we prove that if G is an inner form of a split group, then Λ M = X * (Z( M )) = X * (T ) σ (see Corollary 11.3.2).
Finally, let G * denote the quasi-split inner form of F , and consider special maximal parahoric subgroups K * ⊂ G * (F ) and K ⊂ G(F ). In section 12, we define a canonical transfer homomorphism t : H K * (G * ) → H K (G), and we establish some of its basic properties.
This article relies heavily on the ideas of Kottwitz, especially as they are manifested in the article [HR]. The main theorems of [HR] provide the starting points for the proof of Theorem 1.0.3.

Notation
2.1. Ring-theoretic notation. Let O = O F (resp. O L ) denote the ring of integers in the field F (resp. L). Let ̟ denote a uniformizer of F (resp. L), and let k F denote the residue field of F . We may identify the residue field k L with an algebraic closure of k F . Let Γ := Gal(F s /F ).
Throughout this paper, if J ⊂ G(F ) denotes a compact open subgroup, we make H J := C ∞ c (J\G(F )/J) a convolution algebra by using the Haar measure on G(F ) which gives J volume 1.
2.2. Buildings notation. Let B(G(L)) (resp. B(G(F ))) denote the Bruhat-Tits building of G(L) (resp. G(F )). The building B(G(L)) carries an action of σ. By [BT2], 5.1.25, we have an identification B(G(F )) = B(G(L)) σ . Moreover, there is a bijection a J → a σ J from the set of σ-stable facets in B(G(L)) to facets in B(G(F )) ( [BT2], 5.1.28). This bijection sends alcoves to alcoves ( [BT2], 5.1.14). It also follows from loc. cit. that every σ-stable facet a J in B(G(L)) is contained in the closure a of a σ-stable alcove a.
Let v F denote a special vertex in B(G ad (F )) ( [Tits], 1.9). Let A denote a maximal F -split torus in G whose corresponding apartment in B(G ad (F )) contains v F . Let A (resp. A ad ) denote the apartment in B(G(F )) (resp. B(G ad (F ))) corresponding to A. Let V G(F ) denote the real vector space X * (Z(G)) Γ ⊗ R. There is a simplicial isomorphism ( [Tits], 1.2) Therefore, there is a minimal dimensional facet a σ 0 in A associated to a σ-stable facet a 0 ⊂ B(G(L)), such that We consider parahoric (or Iwahori) subgroups in the sense of [BT2], 5.2. That is, to a facet a J ⊂ B(G(L)) we associate an O L -group scheme G • a J with connected geometric fibers, whose group of O L -points fixes identically the points of a J . We often write J(L) := G • a J (O L ). By [BT2], 5.2, if a J is σ-stable we get a parahoric subgroup J(F ) := J(L) σ in G(F ) and this is associated to the facet a σ J in B(G(F )). Moreover, every parahoric subgroup of G(F ) is of this form for a unique σ-stable facet a J . Now fix a σ-stable alcove a whose closure contains a 0 . Let I(L) (resp. K(L)) denote the Iwahori (resp. parahoric) subgroup of G(L) corresponding to the σ-stable alcove a (resp. facet a 0 ). Then I := I(F ) = I(L) σ is the Iwahori subgroup of G(F ) corresponding to a σ . Also, K := K(F ) = K(L) σ is a special maximal parahoric subgroup of G(F ) corresponding to a σ 0 (or equivalently, to v F ).
2.3. Weyl groups and Iwahori-Weyl groups. For a torus S in G, let N G (S) = Norm G (S) denote its normalizer and C G (S) = Cent G (S) its centralizer. Let W (G, S) := N G (S)/C G (S) denote its Weyl group. Fix the torus A as before. From now on, let S be a maximal L-split torus that is defined over F and contains A ( [BT2], 5.1.12). Let T = C G (S), a maximal torus of G (defined over F ) since G L is quasi-split by Steinberg's theorem.
We need to recall definitions and facts about Iwahori-Weyl groups; we refer the reader to [HR] for details. Let T (L) 1 = ker(κ T ), a normal subgroup of N G (S)(L). Let W := N G (S)(L)/T (L) 1 denote the Iwahori-Weyl group for G. It carries an obvious action of σ. Let A L denote the apartment of B(G(L)) corresponding to S, which we may assume contains the alcove a we fixed above. We let W aff denote the affine Weyl group, which is a Coxeter group generated by the reflections through the walls of a. The group W acts on the set of all alcoves in the apartment of B(G(L)) corresponding to S; let Ω G = Ω G,a denote the stabilizer of a. There is a σ-equivariant decomposition We extend the Bruhat order ≤ and the length function ℓ from W aff to W in the obvious way. We can identify W aff with the Iwahori-Weyl group associated to the pair G sc , S sc , where S sc is the pull-back of (S ∩ G der ) • via G sc → G der .
We can embed W set-theoretically into G(L) by choosing a set-theoretic section of the surjective homomorphism N G (S)(L) → W . Since T (L) 1 ⊂ ker(κ G ), we easily see that the restriction of κ G to W ֒→ G(L) gives a homomorphism κ G : W → X * (Z( G)) I which is surjective and σ-equivariant and whose kernel is W aff .

Cartan decomposition: reduction to the key lemma
Changing slightly the notation of [HR], we set We write W σ K := ( W K ) σ . Our starting point is the following fact (see [HR], esp. Remark 9): the map K(L)nK(L) → n ∈ W induces a bijection K(L)\G(L)/K(L) ∼ = W K \ W / W K , and taking fixed-points under σ yields a bijection The Cartan decomposition follows immediately from the key lemma below, which allows us to describe the right hand side of (3.0.1) in the desired way. To state this we note that the σ-stable alcove a is contained in a unique σ-stable alcove a M in the apartment A M L ⊂ B(M (L)) corresponding to S. As before, we define Ω M ⊂ W M to be the stabilizer of a M under the action of W M on the alcoves in A M L . Lemma 3.0.1. [Ko97], 7.7). Putting this together with the lemma we get Theorem 1.0.3. The proof of Lemma 3.0.1 will occupy the next four sections.

Some ingredients about parahoric subgroups
4.1. Parahoric subgroups of F -Levi subgroups. As before, let A denote a maximal F -split torus in G, let S ⊇ A be a maximal L-split torus which is defined over F , and let T = C G (S), a maximal torus of G which is defined over F . Let A M denote any subtorus of A, and let M = C G (A M ). Thus M is a semi-standard F -Levi subgroup of G. The extended buildings B(M (L)) and B(G(L)) share an apartment (which corresponds to S), but the affine hyperplanes in the apartment A M L for M (L) form a subset of those in the apartment A L for G(L). Hence any facet a J in A L is contained in a unique facet in A M L , which we will denote by a M J . The following result was proved in [H08] in the special case where G splits over L. of [HR], Prop. 8. Using this we may assume m ∈ N M (S)(L). We will show that for such an element m which fixes a point of a M J we have m ∈ T (L) 1 N Msc (S M sc )(L), which will prove the first statement of the claim. It will also prove the second statement, since then m determines a type-preserving automorphism of the apartment A M L , hence fixes a M J if it fixes any of its points. Choose a special vertex a M 0 contained in the closure of a M , and let K 0 denote the corresponding special maximal parahoric subgroup of M (L). We may write m = tn, where t ∈ T (L) and n ∈ N M (S)(L) ∩ K 0 (cf. [HR], Prop. 13). Define ν ∈ X * (T ) I to be κ T (t) and Let Σ ∨ denote the coroots associated to the unique reduced root system Σ such that the set of affine roots Φ af (G(L), S) on A L are given by Clearly w is the image of an element from N Msc (S M sc )(L)∩K 0 , since the latter also surjects onto W (M, S). Thus we need only show that First, we will prove that ν ∈ Q ∨ (Σ). Indeed, by construction t ∈ G(L) 1 , and using X * (T ) I /X * (T sc ) I ∼ = X * (Z( G)) I (cf. [HR]) we see that ν ∈ X * (T sc ) I = Q ∨ (Σ).
Next, let r denote the order of w ∈ W (M, S). The element m r maps to (t ν w) r ∈ W M , which is the translation by the element µ := r−1 i=0 w i ν ∈ Q ∨ (Σ). But as this translation fixes a point of a M J , we must have This completes the proof of the claim, and thus the lemma.

4.2.
Parahoric subgroups of minimal F -Levi subgroups. Now we return to the usual notation, where M := C G (A) is a minimal F -Levi subgroup of G. In this case M ad is anisotropic over F and the semi-simple building B(M ad (F )) = B(M ad (L)) σ is a singleton. The apartment (A M L ) σ is the empty apartment (no affine hyperplanes). Therefore, M (F ) has only one parahoric subgroup. Proof. By Lemma 4.1.1, the inclusion "⊆" is clear. Let m ∈ M (F ) 1 . Since m acts trivially on the apartment A σ L in the building B(G(F )) = B(G(L)) σ , it fixes a point of the σinvariant facet a J (e.g. its barycenter). But then since m ∈ G(F ) 1 , by the Claim in the proof of Lemma 4.1.1 (taking M = G), m fixes every point in a J . Clearly then m By [HR], Lemma 6, the right hand side is By [HR], Remark 9, any element of W σ K is represented by an element of N G (S)(F ). Let x ∈ N G (S)(F ). Then xSx −1 = S contains xAx −1 and A, which being maximal F -split tori in S, must coincide. Thus, there is a tautological homomorphism By Lemma 4.2.2, this factors to give an injective homomorphism . The next statement furnishes the proof of Lemma 3.0.1, (I).
Lemma 5.0.1. The homomorphism W σ K → W (G, A) is an isomorphism. This allows us to regard W (G, A) as a subgroup of W σ .
Proof. It is enough to prove the domain and codomain have the same order. Let k L denote the residue field of O L , which can be identified with an algebraic closure of k F . Consider the special fiber G (this is well-known, but one can also use the argument which yields Remark 6.1.3 below).
On the other hand, by [Tits], 3.5.1, the root system of G

the root system consisting of the vector parts of the affine roots for
These remarks imply that W σ K and W (G, A) are abstractly isomorphic groups and in particular they have the same order.

A decomposition of the Iwahori Weyl group
The goal here is to prove Lemma 3.0.1, (II).
6.1. A lemma on finite Weyl groups. Let w ∈ W (G, A) and choose a representative g ∈ N G (A)(F ) for w; write [g] = w. The tori gSg −1 and S are both maximal L-split tori in M , hence there exists m ∈ M (L) such that mgSg −1 m −1 = S. We claim that the map To show the map is welldefined, we suppose w = w 0 and we show that (mg) −1 m 0 g 0 ∈ N M (S). It will suffice to show (mg) −1 m 0 g 0 belongs to M (L). Since g normalizes M = C G (A) and g −1 g 0 ∈ M , this is obvious. To show the map is injective we suppose Arguing as before, we deduce that g −1 g 0 ∈ M . This shows that w = w 0 and so we get the injectivity. Next we prove the surjectivity. Suppose x ∈ N G (S) projects to an element in W (G, S) which represents a σ-fixed coset C in W (G, S)/W (M, S), that is, x −1 σ(x) ∈ M . Then the subtorus xAx −1 ⊂ S is defined over F . The inner automorphism Int(x) : S → S, restricted to A gives an isomorphism Int(x) : A → xAx −1 which is defined over F . It follows that xAx −1 is F -split. Since A and xAx −1 are maximal F -split tori in S, they coincide. Thus x ∈ N G (A), and the image of x is the coset C.
Remark 6.1.3. If G is quasi-split over F , then M = T and we recover the well-known result that W (G, A) = W (G, S) σ . 6.2. Proof of the decomposition. We keep the notation of the previous subsection. There is a commutative diagram of exact sequences with σ-equivariant morphisms and injective vertical maps [HR], Prop. 13). The canonical map W M \ W → W (M, S)\W (G, S) is bijective and σ-equivariant, so we get Using the map W (G, A) ֒→ W σ constructed in Lemma 5.0.1 we get a commutative diagram The commutativity of this diagram follows using Remark 6.1.1. Since the diagonal arrow is a bijection by the above discussion, and the vertical arrow is obviously an injection, it follows that all arrows in the diagram are bijections. The decomposition , the maximal parahoric subgroup of G(F ) corresponding to v F . By [HR], Prop. 3 and Remark 9, we have the equality Using the Claim from the proof of Lemma 4.1.1 in the case M = G, we derive the equality Our goal is to prove the analogous description of K.
Lemma 8.0.1. The special maximal compact subgroups of G(F ) are precisely the subgroups of the form where v F ranges over the special vertices in the building B(G ad (F )).

Proof.
A compact subgroup of G(F ) is automatically contained in G(F ) 1 . This follows from the alternative description of G(L) 1 as the intersection of the kernels of the homomorphisms |χ| : G(L) → R >0 , where χ ranges over L-rational characters on G. Thus, using [BT1], Cor. (4.4.1), every maximal compact subgroup K of G(F ) (equiv., of G(F ) 1 ) is the stabilizer in G(F ) 1 of a well-defined facet in the building B(G der (F )). By definition, such a K is special if and only if the facet it stabilizes is a special vertex v F . In that case, we have To show the converse, we must check that G(F ) 1 ∩ Fix(v F ) is compact (the argument above will then show it is (special) maximal compact).
and since the former is compact, so is the latter. This completes the proof.

Statement of the Satake isomorphism
In this section, let P = M N denote any F -rational parabolic subgroup of G with unipotent radical N , which has M as a Levi factor. 9.1. Iwasawa decomposition. In light of Lemma 8.0.1, the following version of the Iwasawa decomposition can be derived easily from similar statements in the literature (cf. [BT1], Rem. (4.4.5) or Prop. (7.3.1)): Proposition 9.1.1. There is an equality of sets We need the variant of this where K(F ) is replaced by K(F ). It will be enough to prove that K(F ) = ( K ∩ M (F )) · K(F ).
We have thus proved the first part of the following corollary.
Corollary 9.1.2 (Iwasawa decomposition). There is an equality of sets Proof. We need only show the second equality, which can be rewritten as . This follows from [BT2], 5.2.4 (taking the set denoted by Ω there to be {v F }).

9.2.
Construction of the Satake transform. We will follow the approach taken in [HKP], which treated the case of F -split groups.
Recall that H K := C c (K(F )\G(F )/K(F )), the spherical Hecke algebra of K(F )-biinvariant compactly-supported functions on G(F ). The convolution is defined using the Haar measure on G(F ) which gives K(F ) volume 1. Set Here dm 1 is the Haar measure on M (F ) giving M (F ) 1 volume 1, and δ P is the modular function on P Proof. This follows directly from Proposition 9.1.2.
Given f ∈ H K , let f ∨ ∈ R denote the unique element satisfying the identity

It is obvious that
Evaluating both sides of (9.2.1) on m ∈ M (F ) and using the usual G = M N K integration formula (see [Car]), we get the familiar expression where dn gives N (F ) ∩ K(F ) measure 1.

The Satake transform is an isomorphism
10.1. Weyl group invariance. The first step is to prove that f ∨ belongs to the subring R W (G,A) of W (G, A)-invariants in R. Once this is proved, the functoriality of the Kottwitz homomorphism A) , as well. The argument is virtually the same as Cartier's [Car]. 10.2. Upper triangularity. The second step is to show that with respect to natural Cbases of H K and R W (G,A) , the map f → f ∨ is "invertible upper triangular", hence is an isomorphism of algebras.
The set W σ K \ W σ / W σ K ∼ = W (G, A)\Ω σ M provides a natural C-basis for H K and for R W (G,A) . Recall that W has a natural structure of a quasi-Coxeter group W = W aff ⋊ Ω (cf. [HR], Lemma 14). We extend the Bruhat order ≤ and the length function ℓ from W aff to W in the usual way (cf. loc. cit.). Given x ∈ W , denote byx ∈ W the unique minimal element in W K x W K . (Note that W K is finite and that the usual theory of such minimal elements for Coxeter groups goes over to handle quasi-Coxeter groups.) By [HR], Remark 9, we may regard W σ K \ W σ / W σ K as a subset (the σ-invariant elements) we define the partial order by requiring y y ′ ⇔ỹ ≤ỹ ′ , resp.
The set W (G, A)\Ω σ M is countable and every element y has only finitely many predecessors with respect to the partial order . Therefore there is a total ordering y 1 , y 2 , . . . on this set which is compatible with , meaning that y i y j only if i ≤ j. Similar remarks apply to the partially ordered set W σ K \ W σ / W σ K , and we get an analogous total ordering x 1 , x 2 , . . . for it.
We claim that the matrix for f → f ∨ in terms of the bases {y i } ∞ 1 and {x i } ∞ 1 is upper triangular and invertible. The upper triangularity is the content of the next lemma.
Proof. Let I denote the Iwahori subgroup of G(L) associated to the σ-stable alcove a, as defined earlier. We shall need two BN-pair relations. The first is the relation This follows easily using [HR], Prop. 8. The second is the relation and hence y = x ′′ for some x ′′ ∈ W with x ′′ ≤ x ′ . Thusỹ ≤ x ′ . A standard argument then shows thatỹ ≤x, which is what we wanted to prove.
Finally, the invertibility follows from the obvious fact that This completes the proof that f → f ∨ is an isomorphism.
11. The structure of Λ M It is clear that Λ M = X * (Z( M )) σ I is a finitely-generated abelian group. In this section we make it more concrete in various situations. 11.1. General results. As before, in this subsection T denotes the centralizer in G of the torus S. Recall that we can assume S is defined over F , and so T is also defined over F . Recall also that T M sc denotes the pull-back of T via M sc → M .
Proof. Use the long exact sequence for H i ( σ , −) associated to the short exact sequence (For a discussion of this short exact sequence, see [HR], proof of Prop. 13.) Note that X * (T M sc ) σ I ⊂ W σ M,aff = 1 (cf. Lemma 7.1.1). Also, X * (T M sc ) Γ is finite because M sc is anisotropic over F . The lemma follows easily using this remarks.
Proof. Part (a). Since G is quasi-split over F , we have M = T , and the desired formula follows directly from the definition of Λ M .
Part (b) follows immediately from Lemma 11.1.1. Part (c) follows as a special case of either (a) or (b). Part (c) was known previously (cf. [Bo], 9.5).
Remark 11.1.3. If G is semi-simple and anisotropic, then Λ M is finite. There are examples, At the opposite extreme, let E/F denote a finite totally ramified extension. Consider the "diagonal" embedding G m ֒→ R E/F G m and set G = (R E/F G m )/G m . Then Λ G is torsion, and non-zero if E = F . The next proposition tells us how to measure the difference between the subgroups K and K of G(F ) attached to a special vertex v F . This will complete the proof of Theorem 1.0.1. For an abelian group H let H tor denote its torsion subgroup.
Proposition 11.1.4. There is a set-theoretic inclusion Ω σ M,tor ⊂ K which induces an isomorphism of groups Λ M,tor → K/K.

Proof.
Clearly Ω σ M,tor lies in M (F ) 1 hence in G(F ) 1 . Also, every element of M (F ) 1 acts trivially on the apartment A σ L , and in particular, fixes a σ 0 . This shows that Ω σ M,tor ⊂ We claim the induced homomorphism Ω σ M,tor → K/K is an isomorphism. It is injective because (cf. Lemma 4.2.1). Let us prove surjectivity. Any coset in K/K can be represented by an element x ∈ Ω σ M . We need to show this element is torsion. Let r be such that x r ∈ K. But then x r ∈ Ω σ M ∩ K = {1} (see above), and we are done. Corollary 11.1.5. If M L is L-split group and M der = M sc , then Λ M is torsion-free, and for every special vertex v F , we have Proof. We have (11.1.1) X * (Z( M )) I = X * (Z( M )) and the latter is torsion free since M der = M sc is equivalent to Z( M ) being connected.
Remark 11.1.6. The hypotheses on M hold if G der = G sc and G L is an L-split group.
Corollary 11.1.7. If G = G sc , then K = K and Λ M is torsion-free.
11.2. Passing to inner forms. It is of interest to describe Λ M explicitly in terms of an appropriate maximal torus T in G. For quasi-split groups this has been done in Corollary 11.1.2, (a), which proves that Λ M = X * ( T ) σ I = X * ( T I ) σ . Here we study the effect of passing to an inner form of a quasi-split group.
Thus, we fix a connected reductive group G * which is quasi-split over F . Recall that an inner form of G * is a pair (G, Ψ) consisting of a connected reductive F -group G and a Γ-stable G * ad (F s )-orbit Ψ of F s -isomorphisms ψ : G → G * . The set of isomorphism classes of inner forms of G * corresponds bijectively to the set H 1 (F, G * ad ), by the rule which sends (G, Ψ) to the 1-cocycle τ → ψ • τ (ψ) −1 for any ψ ∈ Ψ (cf. [Ko97], 5.2). Now assume (G, Ψ) is an inner form of G * . Denote the action of τ ∈ Γ on G(F s ) (resp. G * (F s )) by τ (resp. τ * ).
Let A be a maximal F -split torus in G, and let S denote a maximal F un -split torus in G which is defined over F and contains A. Such a torus S exists by [BT2], 5.1.12, noting that that any F -torus which is split over L is already split over F un . Let T = C G (S) and M = C G (A). Then T is a maximal torus of G, since the group G F un is quasi-split. Let A * , S * , T * have the corresponding meaning for the group G * , and assume that T * is contained in an F -rational Borel subgroup B * = T * U * of G * . Of course T * = C G * (A * ) since G * is quasi-split over F . Let P = M N be an F -rational parabolic subgroup of G having Levi factor M and unipotent radical N . Let P * be the unique standard F -rational parabolic subgroup of G * which is G * (F s )-conjugate to ψ(P ) for all ψ ∈ Ψ (cf. [Bo], section 3). Let M * denote the unique Levi factor of P * which contains T * . Let Ψ M denote the set of ψ ∈ Ψ such that ψ(P ) = P * and ψ(M ) = M * . Then Ψ M is a non-empty Γ-stable M * ad (F s )-orbit of F s -isomorphisms M → M * ; hence M is an inner form of the F -quasi-split group M * .
It is clear that G F un and G * F un are isomorphic, since they are inner forms of each other and are both quasi-split (cf. [Tits], 1.10.3). In fact it is easy to see that any inner twisting G F un → G * F un over F un is G * (F s )-conjugate to an isomorphism of F un -groups. For this a key fact is that the image T * ad of T * in G * ad,F un is an induced F un -torus. The same remarks obviously apply to M F un and M * F un . Hence we may choose ψ 0 ∈ Ψ M such that ψ 0 : M → M * is an F un -isomorphism and ψ 0 (S) = S * (and thus also ψ 0 (T ) = T * ). Since ψ 0 restricted to A is defined over F , we see that ψ 0 (A) is an F -split subtorus of T * and hence ψ 0 (A) ⊆ A * .
Letσ denote any lift in Γ of the Frobenius element σ ∈ Gal(F un /F ). We may write for an element m * σ ∈ N M * (S * )(F s ) whose image in M * ad (F s ) is well-defined. As operators on X * (T * ) = X * ( T * ), we may write (11.2.1) ψ 0 • σ(ψ 0 ) −1 = w * σ for a well-defined element w * σ ∈ W (M * , S * )(F un ). Denote by w σ the preimage under the isomorphism ψ 0 : W (M, S)(F un ) → W (M * , S * )(F un ) of w * σ . Then (11.2.1) translates into the equality of operators on X * (T ) = X * ( T ). In defining w σ ∈ W (M, S), we fixed the objects A and S (needed to specify the ambient group W (M, S)) and along the way we also chose several additional objects: P , A * , S * , B * , and an element ψ 0 ∈ Ψ M such that ψ 0 (S) = S * and ψ 0 : M → M * is F un -rational. It is straightforward to check that the element w σ ∈ W (M, S) is independent of all of these additional choices.
11.3. Inner forms of split groups. In this subsection we assume G * is F -split. Then A * = S * = T * , and G F un and M F un are split groups. In particular, the relative Weyl group W (M * , S * ) coincides with the absolute Weyl group W (M * , T * ). Using ψ 0 as above, we may regard w σ as an element of W (M, S) = W (M, T ) I = W ( M , T ) I . For the next lemma, we need to recall the notion of cuspidal elements of Weyl groups. Let (W, S) be any Coxeter group with a finite set S of simple reflections. We say w ∈ W is cuspidal if every conjugate of w is elliptic, that is, every conjugate w ′ has the property that any reduced expression for w ′ contains every element of S. Note that the identity element of W is cuspidal if and only if S = ∅, in which case W itself is trivial. Proof. Part (a). We may assume M = T and hence W (M, T ) is not trivial. Suppose the assertion is false. Then there is a notion of simple positive root for M, T and a corresponding Coxeter group structure on W (M, T ), for which w σ is not an elliptic element. Let s i denote a simple reflection in W (M, T ) which does not appear in a reduced expression for w σ . Then the corresponding fundamental coweight λ i ∈ X * (T /Z(M )) for M ad is fixed by w σ . It is also fixed by ψ −1 0 • (σ * ) −1 • ψ 0 . Thus by (11.2.2) λ i is fixed by σ, and λ i (G m ) is an F -split torus in M ad . This contradicts the fact that M ad is anisotropic over F . Part (b). Since every anisotropic F -group is type A (cf. Kneser [Kn] and Bruhat-Tits [BT3], 4.3), the group M is type A. For type A groups, every cuspidal element in the Weyl group is Coxeter, as may be seen using cycle decompositions of permutations. Thus, the cuspidal element w σ is a Coxeter element of W (M, T ).
Part (c). It is enough to prove the following statement: if G is a type A connected reductive complex group with maximal torus T , and if w ∈ W (G, T ) is a Coxeter element, then Z(G) = T w . First, if G = PGL n , a simple computation shows that T w = 1 = Z(G). Since G ad is a product of projective linear groups and w corresponds to a product of Coxeter elements, this also handles the case of adjoint groups. In the general case, note that an element t ∈ T w maps to (T ad ) w = 1 in G ad , hence t ∈ ker(G → G ad ) = Z(G). Corollary 11.3.2. If G is an inner form of an F -split group, then Λ M = X * (Z( M )) = X * ( T σ ) = X * (T ) σ .
The second equality follows similarly using Lemma 11.3.1,(c), and the third equality is apparent.

The transfer homomorphism
Now we return to the conventions and notation of subsection 11.2. Let A S L (resp. A S * L ) denote the apartment of B(G(L)) (resp. B(G * (L))) corresponding to S (resp. S * ). The twisting ψ 0 gives an isomorphism X * (S) R → X * (S * ) R of the real vector spaces underlying these apartments. Let K (resp. K * ) denote a special maximal parahoric subgroup of G(F ) (resp. G * (F )) corresponding to a special vertex in (A S L ) σ (resp. (A S * L ) σ * ). Then our goal is to define a canonical algebra homomorphism t : H K * (G * ) → H K (G).