On Parahoric Subgroups

We give the proofs of some simple facts on parahoric subgroups and on Iwahori Weyl groups used in [H], [PR] and in [R]. 2000 Mathematics Subject Classification: Primary 11E95, 20G25; Secondary 22E20. Let G be a connected reductive group over a strictly henselian discretely valued field L. Kottwitz defines in [Ko] a functorial surjective homomorphism (1) κG : G(L) −→ X (Ẑ(G)). Here I = Gal(L̄/L) denotes the absolute Galois group of L. Let B be the Bruhat-Tits building of the adjoint group of G. Then G(L) operates on B. Definition 1. A parahoric subgroup of G(L) is a subgroup of the form KF = Fix (F ) ∩Ker κG, for a facet F of B. An Iwahori subgroup of G(L) is the parahoric subgroup associated to an alcove of B. Remark 2. If F ′ = gF , then KF ′ = gKF g . In particular, since G(L) acts transitively on the set of all alcoves, all Iwahori subgroups are conjugate. We shall see presently that this definition coincides with the one of Bruhat and Tits [BTII], 5.2.6. They associate to a facet F in B a smooth group scheme GF over Spec OL with generic fiber G and the open subgroup G F of it with the same generic fiber and with connected special fiber, and define the parahoric subgroup attached to F as P ◦ F = G ◦ F (OL); from their definition it follows that P ◦ F ⊆ Fix(F ). We denote by G(L)1 the kernel of κG. For a facet F , let as above P ◦ F be the associated parahoric subgroup in the BT-sense. Our first goal is to prove the following proposition. Proposition 3. For any facet F in B, we have P ◦ F = KF . Date: April 23, 2008. *Partially supported by NSF Focused Research Grant 0554254. 1 2 T. HAINES AND M. RAPOPORT Proof. a) If G = T is a torus, then T (L)1 is the unique Iwahori subgroup, cf. Notes at the end of [R], n1. Hence the result follows in this case. b) If G is semisimple and simply connected, then G(L)1 = G(L). The assertion therefore follows from [BTII], 4.6.32, which also proves that GF = G ◦ F , where GF is the group scheme defined in loc. cit. 4.6.26. c) Let G be such that Gder is simply connected. Let D = G/Gder. We claim that there is a commutative diagram with exact rows 1 −→ Gder,F (OL) −→ G ◦ F (OL) −→ D ◦(OL) −→ 1 ∥∥ y ∥∥ 1 −→ Kder −→ KF −→ D(L)1 . The bottom row involves parahoric subgroups associated in our sense to the facet F , and the top row involves those defined in [BTII] and comes by restricting the exact sequence 1 −→ Gder(L) −→ G(L) −→ D(L) −→ 1 to OL-points of the appropriate Bruhat-Tits group schemes (and more precisely in the case of D◦(OL), the group D is the lft Neron model of DL; cf. the notes at the end of [R]). The vertical equalities result from a) and b) above. The vertical arrow is an inclusion (but we need to justify its existence, see below) making the entire diagram commutative. Let us first construct the top row. The key point is to show that the map G(L) −→ D(L) restricts canonically to a surjective map G F (OL) −→ D ◦(OL). We shall derive this from the corresponding statement involving the lft Neron models of D and of a maximal torus T ⊂ G. Let S denote a fixed maximal L-split torus in G, and define T = CentG(S), a maximal torus in G since by Steinberg’s theorem G is quasisplit. Also define Tder := Gder ∩ T = CentGder(Sder), where Sder = (S ∩Gder) , a maximal L-split torus in Gder. Consider the lft Neron models T , Tder, and D associated to T , Tder, and D, cf. [BLR]. The map T (OL) −→ D(OL) is surjective, and this implies that T ◦(OL) −→ D ◦(OL) is also surjective, cf. [BLR], 9.6, Lemma 2. By [BTII], 4.6.3 and 4.6.7 we have decompositions G F (OL) = T ◦(OL)UF (OL) and Gder,F (OL) = Tder(OL)UF (OL), where UF denotes the group generated by certain root-group OL-schemes UF,a, which depend on F ; these all fix F . These remarks show that G(L) −→ D(L) restricts to a map G F (OL) −→ D ◦(OL), and also that the latter map is surjective. The kernel of the latter map is contained in the subgroup of Gder(L) which fixes F , hence is precisely Kder = Gder,F (OL). This completes our discussion of the first row of the diagram above. Since Gder is simply connected, G(L)1 is the inverse image of D(L)1 under the natural projection G −→ D; hence G F (OL) belongs to G(L)1 hence is contained in KF ; this yields the inclusion which fits in the above diagram and makes it commutative. A diagram chase then shows that the inclusion G F (OL) −→ KF is a bijection. ON PARAHORIC SUBGROUPS 3 d) To treat the general case choose a z-extension G −→ G, with kernel Z where the derived group of G is simply connected. Since X∗(Z)I = X (Ẑ) is torsion-free, the induced sequence (2) 0 −→ X(Ẑ) −→ X(Ẑ(G)) −→ X(Ẑ(G)) −→ 0 is exact. We therefore obtain an exact sequence 1 −→ Z(L)1 −→ G ′(L)1 −→ G(L)1 −→ 1. It follows that we also have an exact sequence 1 −→ Z(L)1 −→ K ′ F −→ KF −→ 1. As in c) one shows that G(L) → G(L) maps G F (OL) = P ′◦ F onto G ◦ F (OL) = P ◦ F ; in particular, since P ′◦ F = K ′ F by c), we deduce P ◦ F ⊂ G(L)1 and thus P ◦ F ⊆ KF . The equality P ◦ F = KF then follows from P ′◦ F = K ′ F . Remark 4. Proposition 3 makes sense and still holds true if F is replaced with any bounded non-empty subset Ω ⊂ B which is contained in an apartment. Indeed, one can follow the same proof, making only the following adjustment in the proof of b) where G = Gsc: although [BTII] 4.6.32 is restricted to Ω contained in a facet, the equality G Ω(OL) = GΩ(OL) (= fixer of Ω) which we need requires only the connectedness of the group scheme T occurring in loc.cit., which holds here by loc. cit. 4.6.1. Let F be a facet contained in the apartment associated to the maximal split torus S. Let T be the centralizer of S, a torus since by Steinberg’s theorem G is quasisplit. Let N be the normalizer of S. Let KF be the parahoric subgroup associated to F . Let κT : T (L) −→ X (T̂ ) = X∗(T )I be the Kottwitz homomorphism associated to T , and T (L)1 its kernel. Lemma 5. T (L) ∩KF = T (L)1. Proof. By functoriality of the Kottwitz homomorphisms, we see T (L)1 ⊂ G(L)1. The elements of Ker κT act trivially on the apartment associated to S, hence the inclusion ”⊃” is obvious. The converse follows from the fact that T (L)1 equals T ◦(OL) where T ◦ is the identity component of the lft Neron model of T (cf. Notes at the end of [R], n1.) and the fact that T ◦(OL) is the centralizer of S in KF , comp. [BTII], 4.6.4 or [L], 6.3. Let K0 be the Iwahori subgroup associated to an alcove contained in the apartment associated to S. Lemma 6. N(L) ∩K0 = T (L)1. 4 T. HAINES AND M. RAPOPORT Proof. An element of the LHS acts trivially on the apartment associated to S, hence is contained in T (L) ∩K0 = T (L)1 by the previous lemma. Definition 7. Let S ⊂ T ⊂ N be as before (maximal split torus, contained in a maximal torus, contained in its normalizer). The Iwahori-Weyl group associated to S is W̃ = N(L)/T (L)1. Let W0 = N(L)/T (L) (relative vector Weyl group). We obtain an obvious exact sequence (3) 0 −→ X∗(T )I −→ W̃ −→ W0 −→ 0. Proposition 8. Let K0 be the Iwahori subgroup associated to an alcove contained in the apartment associated to the maximal split torus S. Then G(L) = K0.N(L).K0 and the map K0nK0 7→ n ∈ W̃ induces a bijection K0\G(L)/K0 ≃ W̃ . More generally, let K resp. K ′ be parahoric subgroups associated to facets F resp. F ′ contained in the apartment associated to S. Let W̃ = (N(L) ∩K)/T (L)1 , W̃ ′ = (N(L) ∩K )/T (L)1 . Then K\G(L)/K ′ ≃ W̃\W̃/W̃ ′ . Proof. To F there are associated the subgroups UF ⊂ PF ⊂ G(L), cf. [BTI], 7.1 (where PF is denoted P̂F ) or [L], 8.8. By [BTI], 7.1.8 or [L], 8.10 we have PF ⊂ UF .N(L). Since UF is contained in the parahoric subgroup P ◦ F ⊂ PF , the equalityG(L) = PF .N(L).PF ′ ([BTI], 7.4.15 resp. [L], 8.17) implies the first assertion G(L) = KF .N(L).KF ′ . To see the second assertion, we follow closely the proof of [BTI], 7.3.4. Assume that n, n ∈ N(L) with n ∈ P ◦ F nP ◦ F ′ , i.e. n.n ∈ P ◦ F · P ◦ nF ′ . We choose points f ∈ F and f ′ ∈ n · F ′ with P ◦ F = P ◦ f , P ◦ n.F ′ = P ◦ f ′ . We choose an order on the root system of S, with associated vector chamber D such that f ∈ f ′ +D. ON PARAHORIC SUBGROUPS 5 Then U f · U − f ′ ⊂ U − f ′ , in the notation of [L], 8.8. Hence, by [BTII], 4.6.7 (comp. [L], 8.10), P ◦ f .P ◦ f ′ = [ (N(L) ∩ P ◦ f ) · U + f .U − f ] · [ U f ′U + f ′ · (N(L) ∩ P ◦ f ′) ] = (N(L) ∩ P ◦ f ) · U + f .P ◦ f ′ = (N(L) ∩ P ◦ f ) · U + f .U + f ′ .U − f ′(N(L) ∩ P ◦ f ′). It follows that there exist m1 ∈ N(L) ∩ P ◦ f and m2 ∈ N(L) ∩ P ◦ f ′ such that m1 · n nm2 ∈ U (L).U(L). From the usual Bruhat decomposition it follows that m1 · n nm2 = 1, i.e., m1n (nm2n) = n . Since m1 ∈ N(L) ∩KF and n m2n ∈ N(L) ∩KF ′ , the last equality means that n ≡ n in W̃\W̃/W̃ ′ . Remark 9 (Descent). Let σ denote an automorphism of L having fixed field L such that L is the strict henselization of L. Let us assume G is defined over L; we may assume S, and hence T and N , are likewise defined over L ([BTII], 5.1.12). Assume F and F ′ are σ-invariant facets in B. Write K(L) = K and K (L) = K . We have a canonical bijection (4) K(L)\G(L)/K (L) →̃ [K\G(L)/K ]. To prove that the map is surjective, we use the vanishing of H(〈σ〉,K) and H(〈σ〉,K ). To prove it is injective, we use the vanishing of H(〈σ〉,K ∩ gK g) for all g ∈ G(L). The vanishing statements hold because K, K , and K ∩ gK g are the OL-points of group schemes over OL♮ with connected fibers, by virtue of Proposition 3 and Remark 4. Next, note that, using a similar cohomology vanishing argument, W̃ σ = N(L)/T (L) ∩ T (L)1 =: W̃ (L ) (W̃) = N(L) ∩K/T (L) ∩ T (L)1 =: W̃ (L). Now suppose that F and F ′ are σ-invariant facets contained in the closure of a σ-invariant alcove in the apartment of B associated to S. Then the canonical map (5) W̃(L)\W̃ (L)/W̃ ′ (L) → [W̃\W̃/W̃ ′ ] is bijective. Indeed, note first that W̃ and W̃ ′ are parabolic subgroups of the quasiCoxeter group W̃ (see Lemma 14 below), and that any element x ∈ W̃ has a unique expression in the form wx0w ′ where w ∈ W̃ , w ∈ W̃ ′ , such that x0 is the unique minimal-length element in W̃x0W̃ K ′, and wx0 is the unique minimal-length element in 6 T. HAINES AND M. RAPOPORT wx0W̃ K ′. Secondly, note that σ preserves these parabolic subgroups as well as the quasiCoxeter structure and therefore the Bruhat-order on W̃ . These remarks imply the bijectivity just claimed. Putting (4) and (5) together, we obtain a bijection K(L)\G(L)/K (L) →̃ W̃(L)\W̃ (L)/W̃ ′ (L). Remark 10. We now compare the Iwahori Weyl group with a variant of it in [T]. In [T], p. 32, the following group is introduced. Let T (L)b be the maximal bounded subgroup of T (L). The affine Weyl group in the sense of [T], p.32, associated to S is the quotient W̃ ′ = N(L)/T (L)b. We obtain a morphism of exact sequences 0 −→ X∗(T )I −→ W̃ −→ W0 −→ 0 y y ∥∥ 0 −→ Λ −→ W̃ ′ −→ W0 −→ 0 . Here Λ = X∗(T )I/torsion. Of course, Λ = Hom(X(T ) ,Z) = T (L)/T (L)b , (see [Ko], 7.2.) It follows that the natural homomorphism from W̃ to W̃ ′ is surjective, with finite kernel isomorphic to T (L)b/T (L)1. The affine Weyl group in the sense of [T] also appears in a kind of Bruhat decomposition, as follows. Let vG : G(L) −→ X (Ẑ(G))/torsion be derived from κG in the obvious way. Let C be an alcove in the apartment corresponding to S. We consider the subgroup (6) K̃0 = Fix(C) ∩Ker vG. Then K0, the Iwahori subgroup corresponding to C, is a normal subgroup of finite index in K̃0. In fact Ker vG is the group denoted G 1 in [BTII] 4.2.16, and so by loc. cit. 4.6.28, K̃0 is the group denoted there P̂ 1 C , in other words the fixer of C in G . Using loc. cit. 4.6.3, 4.6.7, we have K0 = T (L)1 UC(OL) and K̃0 = T (L)b UC(OL), where UC is the group generated by certain root-group OL-schemes UC,a which fix C. It follows that K̃0 = T (L)bK0, and T (L)b/T (L)1 ≃ K̃0/K0. In [T], 3.3.1, the affine Weyl group W̃ ′ appears in the Bruhat decomposition with respect to K̃0 (7) K̃0\G(L)/K̃0 ≃ W̃ . The group T (L)b/T (L1) acts freely on both sides of the Bruhat decomposition K0\G(L)/K0 = W̃ of Proposition 8, and we obtain (7) by taking the quotients. ON PARAHORIC SUBGROUPS 7 Remark 11. In [BTII] the building of G(L) (sometimes called the enlarged building B) is also considered; it carries an action of G(L). There is an isomorphism B = B × VG, where VG := X∗(Z(G))I ⊗ R. Given a bounded subset Ω in the apartment of B associated to S, there is a smooth group scheme ĜΩ whose generic fiber is G and whose OL-points Ĝ(OL) is the subgroup of G(L) fixing Ω × VG, in other words the subgroup P̂ 1 Ω in G 1 fixing Ω. We have (ĜΩ) ◦ = G Ω. Thus, the above discussion and Proposition 3 show that (ĜΩ) ◦(OL) = ĜΩ(OL) ∩ kerκG. Proposition 12. Let K be associated to F as above. Let G F be the OL-form with connected fibres of G associated to F , and let Ḡ F be its special fiber. Then W̃ K is isomorphic to the Weyl group of Ḡ F . Proof. Let S̄ ⊂ T̄ ◦ ⊂ Ḡ F be the tori associated to S resp. T , cf. [BTII], 4.6.4 or [L], 6.3. The natural projection Ḡ F −→ Ḡ ◦ F,red = Ḡ ◦ F/Ru(Ḡ ◦ F ) induces an isomorphism S̄ ∼ −→ S̄ red, where the index ,,red” indicates the image group in Ḡ F,red. Let W̄ denote the Weyl group of S̄ red and consider the natural homomorphism N(L) ∩KF −→ W̄ . The surjectivity of this homomorphism follows from [BTII], 4.6.13 or [L], 6.10. An element n of the kernel centralizes S̄ red and hence also S̄ . But then n centralizes S because this can be checked via the action of n on X∗(S). But the centralizer of S in KF is T (L)∩KF , cf. [BTII], 4.6.4 or [L], 6.3. Hence n ∈ T (L) ∩KF , which proves the claim. We will give the Iwahori-Weyl group W̃ the structure of a quasi-Coxeter group, that is, a semi-direct product of an abelian group with a Coxeter group. Consider the real vector spaces V = X∗(T )I ⊗R = X∗(S)⊗R and V ′ := X∗(Tad)I ⊗R, where Tad denotes the image of T in the adjoint group Gad. The relative roots Φ(G,S) for S determine hyperplanes in V (or V ), and the relative Weyl group W0 can be identified with the group generated by the reflections through these hyperplanes. The homomorphism T (L) → X∗(T )I → V derived from κT can be extended canonically to a group homomorphism