ALCOVES ASSOCIATED TO SPECIAL FIBERS OF LOCAL MODELS

THOMAS J. HAINES AND NGÔ BAO CHÂU

Abstract. The special fiber of the local model of a PEL Shimura variety with Iwahori-type level structure admits a cellular decomposition. The set of strata is in a natural way a finite subset of the affine Weyl group determined by the Shimura data. In this paper, we generalize and give a new proof of a theorem of Kottwitz and Rapoport concerning the description of this set for the case of linear and symplectic groups. For higher rank groups not of type A_{n}, we produce counterexamples showing the situation is not as nice in general. We also recover a theorem of Deodhar characterizing the Bruhat order on S_{n}.

1. Introduction

Let G be a classical group over the p-adic field ℚ_{p}, and let μ be a dominant minuscule coweight of G. We assume that the pair (G, μ) comes from a PEL-type Shimura datum (G, X, K) where the p-component K_{p} of the compact open subgroup K is an Iwahori subgroup of G(ℚ_{p}) = G(ℚ_{p}). We assume that the corresponding Shimura variety Sh(G, X, K) has a model over a p-adic number ring O_{𝔭}. In this case Sh(G, X, K) is said to have Iwahori-type bad reduction at the prime 𝔭. A fundamental problem is to understand the geometry of the special fiber. In the prototypical example of the modular curve Y_{0}(p), the special fiber is a union of two smooth curves which intersect transversally. In the general case the global geometry cannot be easily described and the singularities can be quite complicated (see [11]).

The local geometry can be approached using the Rapoport-Zink local model M_{μ} [12]. This projective O_{𝔭}-scheme is a local model for the singularities in the special fiber of Sh(G, X, K), but it is defined in terms of the pair (G, μ) using linear algebra and is somewhat easier to deal with than the Shimura variety itself.

In this paper we address certain combinatorial questions which arise in the study of the special fiber M_{μ,\overline{𝔽}_{p}} of M_{μ}. To fix ideas, consider the case where G is split. It turns out that M_{μ,\overline{𝔽}_{p}} can be regarded as a finite-dimensional union of Iwahori-orbits in the affine flag variety for G(\overline{𝔽}_{p}((t))), see [5] and [7]. It follows that M_{μ,\overline{𝔽}_{p}} has a stratification indexed by a finite set Perm(μ) of the extended affine Weyl group W̃(G) for G. Equivalently, Perm(μ) can be regarded as a finite set of alcoves in the affine Coxeter complex determined by a splitting for G. Let W_{0} denote the finite Weyl group of G. For each translation λ in the W_{0}-orbit of μ there is an Iwahori-orbit in M_{μ,\overline{𝔽}_{p}} corresponding to λ; in other words, the element t_{λ} in W̃(G) is contained in Perm(μ). We let Adm(μ) denote the subset of Perm(μ) indexing those strata which lie in the closure of the stratum indexed by t_{λ}, for some λ ∈ W_{0}(μ).

We are primarily concerned with the equality Adm(μ) = Perm(μ), which has important geometric content. For example, if the O_{𝔭}-scheme M_{μ} is flat, this equality automatically holds. On the other hand, the equality has been established combinatorially when G = GL_{n} or GSp_{2n} by Kottwitz and Rapoport [10], and this was exploited by U. Görtz in his proof of the flatness of the local models associated to Res_{E/F} GL_{n}, where E/F is an unramified extension of p-adic fields.

In [10], Kottwitz and Rapoport propose purely combinatorial definitions of the sets Adm(μ) and Perm(μ), for every split group G and every dominant coweight μ. An element x ∈ W̃(G) belongs to Adm(μ) if x ≤ t_{λ} in the Bruhat order on W̃(G), for some λ ∈ W_{0}(μ). The element x belongs to Perm(μ) if the vertices of the corresponding alcove satisfy certain inequalities similar to those arising from the definition of local models; see section 3 for details. (These definitions agree with the geometric interpretations above when G = GL_{n} or GSp_{2n} and μ is minuscule, see loc. cit..) Kottwitz and Rapoport show that the inclusion Adm(μ) ⊂ Perm(μ) is always valid. We adopt their point of view and assume from now on that G is split. We address the question of whether the equality Adm(μ) = Perm(μ) holds in full generality, in particular for non-minuscule coweights μ.

The main result of this paper is that the situation is as nice as possible for groups of type A_{n−1}.

Theorem 1. For any root system of type A_{n−1}, the equality Adm(μ) = Perm(μ) holds for every dominant coweight μ.

Actually, we derive this theorem from a more general result available for any root system. We introduce the new notion of the set of μ-strongly permissible alcoves Perm^{st}(μ). Like the Kottwitz-Rapoport notion of Perm(μ), this is a set of alcoves determined by imposing conditions vertex-by-vertex. Although the condition we require on vertices might look more technical than that of Kottwitz-Rapoport, a similar notion for finite Weyl groups already occurred in a classical theorem of Bernstein-Gelfand-Gelfand [1]. We prove the following statement.

Proposition 2. For any root system R, the inclusion Adm(μ) ⊃ Perm^{st}(μ) holds for every dominant coweight μ.

When the root system R is of type A_{n−1}, we prove that μ-permissibility and μ-strong permissibility are equivalent and derive Theorem 1 from Proposition 2 and the inverse inclusion proved by Kottwitz-Rapoport.

In this paper we introduce some notions of cones, acute or obtuse, in the set of alcoves. These notions play an important role in the proof of Proposition 2 and may also be useful for other combinatorial questions. Another ingredient is a lemma due to Deodhar [3] which is also used in [4].

The equality Adm(μ) = Perm(μ) turns out to be false in general.

Theorem 3. If R is an irreducible root system of rank ≥ 4 and not of type A_{n−1}, then Adm(μ) ≠ Perm(μ) for every sufficiently regular dominant coweight μ.

Theorem 3 is based on counter-examples due to Deodhar [4]. Deodhar determines which finite Weyl groups W have the property that w ≤ w′ in the Bruhat order if and only if w(λ) − w′(λ) is a sum of positive coroots for every dominant coweight λ. He proves that W has this property if and only if the irreducible components of the associated root system R are of type A_{n−1} or of rank ≤ 3. Our proof of Theorem 3 uses the counter-examples he gives explicitly in the other cases. However Theorem 1 is proved independently and in fact in section 8 we give an alternate, perhaps more conceptual, proof of Deodhar’s theorem for root systems of type A_{n−1}.

In light of Theorem 3, and motivated by the study of local models attached to certain nonsplit groups, Rapoport proposed that the equality Adm(μ) = Perm(μ) might be valid for all root systems as long as μ is a sum of minuscule coweights. In section 10 we show this is indeed the case for the symplectic group.

Theorem 4. Let μ be a sum of minuscule coweights for the group GSp_{2n}. Then the equality Adm(μ) = Perm(μ) holds.

Theorem 4 relies on a description of the root system of GSp_{2n} as the “fixed-point” root system R^{[Θ]} with respect to the nontrivial automorphism Θ of the root system R for GL_{2n}, à la Steinberg [14]. This idea was also exploited in [10]. In fact in section 9 we prove the following statement is valid for every dominant coweight, at least in the case of the symplectic group.

Proposition 5. The equality Adm^{Θ}(μ) = Perm(μ)∩W̃(GSp_{2n}) holds for any dominant coweight μ of the root system R^{[Θ]} for GSp_{2n}.

Here the sets Adm^{Θ}(μ) and Perm^{Θ}(μ) are the subsets of W̃(GSp_{2n}) analogous to the subsets Adm(μ) and Perm(μ) of W̃(GL_{2n}). Moreover, the equality Perm^{Θ}(μ) = Perm(μ)∩W̃(GSp_{2n}) can be proved in the case where μ is a sum of minuscule coweights by a method of Kottwitz-Rapoport [10], yielding Theorem 4. It is also worthwhile to remark that the set Perm(μ) ∩ W̃(GSp_{2n}) has itself some geometric meaning for every μ. This set parametrizes the strata in the special fiber of the local model considered in [7], since the definition thereof uses the standard representation of the symplectic group.

Theorem 1 is expected to play a role in proving the flatness of the Pappas-Rapoport local models attached to the group Res_{E/F} GL_{n}, where E/F is a totally ramified extension of p-adic fields. For details we refer the reader to forthcoming work of U. Görtz [6]. Likewise, Theorem 4 (or Proposition 5) is expected to play a role in proving the flatness of analogous local models attached to Res_{E/F} GSp_{2n}. Thus, our study of Adm(μ) and Perm(μ) for non-minuscule coweights μ of split groups has ramifications for Shimura varieties attached to certain non-split groups.

For applications to the bad reduction of PEL-type Shimura varieties attached to orthogonal groups one would like to determine whether the analogs of Theorem 4 and Proposition 5 hold for the split orthogonal groups O(2n) (PEL Shimura varieties arise for groups of type A, C, and D). The methods of this paper do not seem to give much information in that situation. However, in section 11 we show that the analog of Proposition 5 for the odd orthogonal groups does not hold. This can be understood in terms of a non-inheritance property of the Bruhat order: the group SO(2n + 1) can be realized as the fixed point group SL(2n + 1)^{Θ} for a certain involution Θ, and this gives a corresponding embedding of affine Weyl groups W_{aff}(B_{n}) ↪ W_{aff}(A_{2n}). However, the Bruhat order on the former is not inherited from the Bruhat order on the latter (in contrast to the symplectic case, cf. Proposition 9.6).

2. Affine Weyl group attached to a root system

Let us fix notation and recall basic facts about affine Weyl group and alcoves. For proofs, we refer to Humphreys’ book [8], especially chapter 4.

Let (X^{∗}, X_{∗}, R, Ř) be a root system. We assume throughout this paper that the root system is reduced and irreducible. When there is no chance of confusion we will denote the root system (X^{∗}, X_{∗}, R, Ř) simply by R. Let Π be a base of R. Let R^{+} (resp. R^{−}) denote the set of positive (resp. negative) roots. The cardinality of Π will be called the rank of R. Denote by ⟨ , ⟩ : X^{∗} × X_{∗} → ℤ the perfect pairing making X^{∗} and X_{∗} dual free abelian groups.

Corresponding to α ∈ R we have a reflection s_{α}, acting on V = X_{∗}⊗ℝ by s_{α}(x) = x− ⟨α, x⟩α̌. The Weyl group W_{0} is the subgroup of GL(V) generated by these reflections. It is known that W_{0} is generated by S = {s_{α} | α ∈ Π} as a finite Coxeter group.

Let H_{α} denote the hyperplane (wall) fixed by the reflection s_{α}, for every α ∈ R. The connected components of the set V −⋃_{α∈R} H_{α} will be called chambers. The finite Weyl group W_{0} acts simply transitively on the set C of chambers. There is a distinguished chamber

C_{0} = {x ∈ V | ⟨α, x⟩ > 0, for every α ∈ Π}

which will be called the dominant chamber.

Corresponding to α ∈ R and k ∈ ℤ we have an affine reflection s_{α,k}, acting on V = X_{∗} ⊗ℝ by s_{α,k}(x) = x−(⟨α, x⟩−k)α̌. The affine Weyl group W_{aff} is the subgroup of Aut(V) generated by these reflections. It is known that W_{aff} is generated, as a Coxeter group, by S_{aff} = S ∪ {s_{α̃,1}} where α̃ is the unique highest root of R. Moreover, W_{aff} is the semi-direct product W_{aff} = W_{0} ⋉ Q̌ where Q̌ is the lattice generated by Ř acting on V by translation.

Let us denote R̃ = R×ℤ. Let H_{α,k} denote the hyperplane in V fixed by the reflection s_{α,k} for every (α, k) ∈ R̃. The connected components of the set V − ⋃_{(α,k)∈R̃} H_{α,k} will be called alcoves. The affine Weyl group W_{aff} acts simply transitively on the set A of the alcoves. There is a distinguished alcove

A_{0} = {x ∈ C_{0} | ⟨α̃, x⟩ < 1}

that will be called the base alcove.

The extended Weyl group W̃ = W_{0} ⋉ X_{∗} also acts on the set A. Indeed for any coweight λ ∈ X_{∗} the translation t_{λ} by the vector λ sends a wall on another wall since ⟨α, λ⟩ ∈ ℤ for every α ∈ R, therefore sends an alcove to another alcove. Let Ω be the isotropy group in W̃ of the base alcove. It is known that W̃ = W_{aff} ⋊ Ω. With the help of this decomposition, the Bruhat order and the length function on the Coxeter group W_{aff} can be extended to W̃, which is not a Coxeter group in general. For w, w′ ∈ W_{aff}, τ, τ′ ∈ Ω, we say wτ ≤ w′τ′ if and only if w ≤ w′ and τ = τ′. We put l(wτ) = l(w).

Let us recall some basic facts on minimal galleries. A gallery of length l is a sequence of alcoves A′ = A′_{0}, …, A′_{l} = A′′ such that A′_{i−1} and A′_{i} share a wall H_{i}, for i = 1, …, l.

The gallery is minimal if there does not exist a gallery going from A′ to A′′ with length strictly less than l. In general there exist more than one minimal gallery going from A′ from A′′, but the set of walls H_{i} depends only on A′ and A′′: it is the set of walls H = H_{α,k} separating A′ and A′′, i.e., those such that A′ and A′′ lie in different connected components of V − H.

Minimal galleries are very closely related to reduced expressions in the Coxeter system (W_{aff}, S_{aff}). Let x ∈ W_{aff} and x = s_{1}s_{2} … s_{l} be a reduced expression with s_{1}, …, s_{l} ∈ S_{aff}, i.e., such an expression with minimal length l = l(x). Let A_{0}, A_{1}, …, A_{l} be the alcoves defined by A_{i} = s_{1}s_{2} … s_{i}A_{0}. Obviously, A_{i−1} and A_{i} share a wall H_{i}. Hence, this sequence of alcoves forms a gallery going from the base alcove A_{0} to A_{l} = xA_{0}. In this way, the reduced expressions of x correspond bijectively to the minimal galleries going from A_{0} to xA_{0}. The cardinality of the set of hyperplanes H_{α,k} with (α, k) ∈ R̃ separating A_{0} and xA_{0} is equal to the length l(x).

Another basic fact that will be used in the sequel is the following. If an expression x = s_{1}s_{2} … s_{l} is not reduced, there are integers i < j such that we can delete s_{i} and s_{j} without changing x:

x = s_{1} … ŝ_{i} … ŝ_{j} … s_{l}.

Equivalently, a gallery A′_{0}, …, A′_{l} is not minimal if there exist integers i < j such that H_{i} = H_{j}, where for every i, H_{i} denotes the wall shared by the consecutive alcoves A′_{i−1} and A′_{i}.

3. The μ-admissible and μ-permissible sets

In the sequel, a dominant coweight μ will be fixed and we will denote by τ the unique element of Ω such that t_{μ} ∈ W_{aff}τ. Let Conv(μ) denote the convex hull of the set {λ | λ ∈ W_{0}(μ)}. In [10], R. Kottwitz and M. Rapoport introduce the notions of the μ-admissible and μ-permissible subsets of W̃.

Definition 3.1. (Kottwitz-Rapoport)

(1) Let Adm(μ) = {x ∈ W̃ | x ≤ t_{λ} for some λ ∈ W_{0}(μ)}. An element x ∈ Adm(μ) is called μ-admissible.

(2) Let Perm(μ) be the set of elements x ∈ W_{aff}τ such that x(a) − a ∈ Conv(μ) for every vertex a ∈ A_{0}. An element x ∈ Perm(μ) is called μ-permissible.

We remark that in general the facets of A_{0} having minimal dimension are not points, so our use of the word “vertex” in the definition above may be considered inaccurate. Throughout this paper we shall ignore this subtlety and continue to imagine that the facets of A_{0} of minimal dimension are in fact points. It is not difficult to translate our resulting arguments and statements into ones which do not abuse terminology.

The following theorem is proved in [10].

Theorem 3.2. (Kottwitz-Rapoport)

(1) For any root system R and dominant coweight μ, Adm(μ) ⊂ Perm(μ).

(2) Let R be the root system attached to GL_{n} or GSp_{2n}. Let μ be a minuscule coweight of R. Then Adm(μ) = Perm(μ).

Recall that a coweight μ is minuscule if ⟨α, μ⟩ ∈ {−1, 0, 1}, for every root α. For root systems of type A_{n−1}, the second statement can be generalized to every dominant coweight μ.

Theorem 3.3. For any root system of type A_{n−1}, the equality Adm(μ) = Perm(μ) holds for every dominant coweight μ.

According to Kottwitz-Rapoport, it is sufficient to prove the inclusion Adm(μ) ⊃ Perm(μ).

4. Obtuse cones and μ-strongly permissible sets

Let B_{0} denote the negative obtuse cone in V generated by the coroots −α̌ with α ∈ R^{+}. The convex hull Conv(μ) occurring in the definition of the μ-permissible set may be usefully described as the intersection of |W_{0}| obtuse cones

Conv(μ) = ⋂_{w∈W_{0}} wμ + w(B_{0}).

We can rephrase the definition of μ-permissibility by saying that x ∈ W_{aff}τ is μpermissible if and only if for every vertex v of the base alcove and for any w ∈ W_{0}, we have

x(v) ∈ t_{wμ}(v) + w(B_{0}).

The notion of μ-strong permissibility consists in requiring a little bit more than the last inclusion.

Let v be an element of V, W_{aff}(v) the orbit of v under the action of the affine Weyl group W_{aff}. In the spirit of a paper of Bernstein-Gelfand-Gelfand [1], it seems reasonable to consider the subset B(v, w) of W_{aff}(v) defined as follows.

Definition 4.1. For every v ∈ V and w ∈ W_{0}, let B(v, w) be the set of elements v′ of the form s_{r} … s_{1}(v) with s_{i} = s_{α_{i},k_{i}} with (α_{i}, k_{i}) ∈ R̃ such that for all i = 1, …, r,

s_{i}s_{i−1} … s_{1}(v) ∈ s_{i−1} … s_{1}(v) + w(B_{0}).

We often call the set B(v, w) the obtuse cone with apex v in the w-opposite direction; notice however that B(v, w) is just a discrete set of points.

We obviously have the inclusion

B(v, w) ⊂ W_{aff}(v) ∩ (v + w(B_{0})).

In contrast to the notion of μ-permissibility, we use these smaller sets B(v, w) to define the notion of μ-strong permissibility.

Definition 4.2. For every dominant coweight μ ∈ X_{∗}, let Perm^{st}(μ) be the set of x ∈ W_{aff}τ such that for every vertex a of A_{0} and every w ∈ W_{0} the element x(a) lies in B(t_{wμ}(a), w). Such an element x is said to be μ-strongly permissible.

We obviously have the inclusion Perm^{st}(μ) ⊂ Perm(μ). We will prove the following stronger statement.

Proposition 4.3. For any root system R and dominant coweight μ ∈ X_{∗}, the inclusion Adm(μ) ⊃ Perm^{st}(μ) holds.

To prove this theorem we need the notion of acute cone to be discussed in section 5. The proof itself is postponed to section 6.

The following lemma shows that in the case of a root system of type A_{n−1} the notions of μ-permissibility and μ-strong permissibility coincide. Thus Theorem 3.3 follows from Proposition 4.3.

Lemma 4.4. Suppose R is of type A_{n−1}. Let v be a vertex of some alcove A. For every w ∈ W_{0}, the equality

B(v, w) = W_{aff}(v) ∩ (v + w(B_{0}))

holds. In particular Perm(μ) = Perm^{st}(μ) for every dominant coweight μ.

Proof. For a root system R of type A_{n−1}, the vertices of alcoves belong to the coweight lattice P̌ = {v ∈ V | ⟨α, v⟩ ∈ ℤ for every root α ∈ R}. Moreover, if v ∈ P̌, the orbit W_{aff}(v) is exactly the set of elements v′ ∈ P̌ such that v′ − v ∈ Q̌, as one sees using the relation tα̌ = s_{α,1}s_{α}, for any α ∈ R. Since Q̌ admits {w(α̌) | α ∈ Π} as ℤ-basis, any element

v′ ∈ W_{aff}(v) ∩ (v + w(B_{0}))

can be written uniquely in the form v′ = v − ∑_{α∈Π} n_{α}w(α̌) with n_{α} ∈ ℕ. Thus, it is sufficient to prove the inclusion v − w(α̌) ∈ B(v, w) for any simple root α ∈ Π. Let k be the integer ⟨w(α), v⟩. Obviously, v − w(α̌) = s_{w(α),k−1}(v) and we are done. □

Remark 4.5. Let R be any root system and let v be a special vertex for R, i.e. an element of P̌. Then for every element w ∈ W_{0} we have inclusions

{v − ∑_{α∈Π} ℕw(α̌)} ⊂ B(v, w) ⊂ W_{aff}(v) ∩ (v + w(B_{0})) ⊂ {v − ∑_{α∈Π} ℕw(α̌)},

the first and last inclusions being proved as in the proof of Lemma 4.4; in particular the displayed equality in Lemma 4.4 holds provided that v is a special vertex. The importance of the first set above stems from its interpretation as the coweights appearing in the character of the Verma module attached to the vertex v and the system of positive roots w(R^{+}). One can see from the example of nonspecial vertices for the root systems B_{2} and G_{2} that in general {v − ∑_{α∈Π} ℕw(α̌)} ⊉ B(v, w). One might hope that the second inclusion is always an equality (and thus that Perm^{st}(μ) always equals Perm(μ)). However, this also fails in general, as we shall see in §7 using the examples of Deodhar [4]. Of the three sets above, B(v, w) seems to be the most related to the Bruhat order but also the most difficult to be visualized.

The equality of the sets in Lemma 4.4 for every vertex imposes rather strong conditions on the underlying root system R. In fact, as we shall see in section 8 (Remark 8.3), if w_{0} is the longest element of W_{0} and

B(v, w_{0}) = W_{aff}(v) ∩ (v + w_{0}(B_{0}))

for every vertex v, then R is necessarily of type A_{n−1} or of rank ≤ 3.

Figure 1. The obtuse cone for G2 with marked vertex as apex in the anti-dominant direction. The base alcove is shaded.

5. Acute cones

Let C̄_{0} be the closure of the dominant chamber C_{0}. For any x ∈ V and w ∈ W_{0}, the acute cone C̄(x, w) = x+w(C̄_{0}) is the translation of w(C̄_{0}) by the vector x. For A ∈ A and w ∈ W, we want to define a subset C(A, w) of A which is as close as possible to a cone C̄(x, w).

The easiest way to do so consists in choosing a point a inside the base alcove A_{0}. Since W_{aff} acts simply transitively on A, each alcove A contains a unique point A(a) conjugate to a.

Definition 5.1. C_{a}(A, w) = {A′ ∈ A | A′(a) ∈ A(a) + w(C̄_{0})}.

The disadvantage of this definition is that it really depends on the choice of a ∈ A_{0}. In order to give a more intrinsic definition, we need the notion of direction of a gallery.

Let w ∈ W_{0}. A wall H = H_{α,k} can also be given by H_{−α,−k}. But only one root from {α, −α} lies in w(R^{+}); let us assume α ∈ w(R^{+}). We define the w-positive half-space H^{w+} by H^{w+} = {v ∈ V | ⟨α, x⟩ > k}. Let H^{w−} be the other half-space.

Definition 5.2. A gallery A′_{0}, …, A′_{l} is said to be in the w-direction if for any i = 1, …, l, letting H_{i} denote the common wall of A′_{i−1} and of A′_{i}, the alcove A′_{i−1} lies in the w-negative half-space H_{i}^{w−} and the alcove A′_{i} lies in the w-positive half-space H_{i}^{w+}.

Obviously, the gallery A′_{0}, …, A′_{l} is in the w-direction if and only if the inverse gallery A′_{l}, …, A′_{0} is in the ww_{0}-direction where w_{0} is the maximal length element of W_{0}. We will sometimes say the gallery A′_{l}, …, A′_{0} is in the w-opposite direction.

Lemma 5.3. Any gallery A′_{0}, …, A′_{l} in some w-direction is automatically minimal. If there exists a gallery going from A′_{0} to A′_{l} in the w-direction, then any other minimal gallery going from A′_{0} to A′_{l} is in the w-direction too.

Proof. Suppose the gallery A′_{0}, …, A′_{l} is not minimal. This means there exist two integers i < j such that H_{i} = H_{j}. Let H denote this hyperplane and suppose H_{k} ≠ H for any integer k in between i < k < j. By hypothesis the alcove A′_{i} lies in H^{w+}. By induction, we know that A′_{k} also lies in H^{w+} for any k = i, …, j − 1 since the only hyperplane separating A′_{k−1} and A′_{k} is H_{k} ≠ H. Thus A_{j−1} ∈ H_{j}^{w+}, which contradicts the hypothesis A_{j−1} ∈ H_{j}^{w−}.

In an arbitrary minimal gallery A′_{0}, …, A′_{l}, for every i, the alcoves A′_{0} and A′_{i−1} lie in the same connected component of V − H_{i}, and the alcoves A′_{i} and A′_{l} lie in the other connected component. If the gallery A′_{0}, …, A′_{l} is supposed to be in the w-direction, A′_{i−1} ∈ H_{i}^{w−} and A′_{i} ∈ H_{i}^{w+} and consequently we have A′_{0} ∈ H_{i}^{w−} and A′_{l} ∈ H_{i}^{w+}. This means for any hyperplane H separating A′_{0} and A′_{l}, A′_{0} ∈ H^{w−} and A′_{l} ∈ H^{w+}. The same argument now proves that any other minimal gallery going from A′_{0} to A′_{l} is also in the w-direction. □

Definition 5.4. Let C(A, w) be the set of alcoves A′ ∈ A such that there exists a gallery A′_{0}, A′_{1}, …, A′_{l} going from A to A′ in the w-direction.

We call C(A, w) the acute cone from A in the w-direction.

Proposition 5.5. We have the inclusion C_{a}(A, w) ⊂ C(A, w) for any point a ∈ A_{0}.

Proof. If A′(a) ∈ A(a) + w(C̄_{0}) then for every root α ∈ w(R^{+}), the inequality

⟨α, A(a)⟩ ≤ ⟨α, A′(a)⟩

holds. Hence, for any hyperplane H separating A and A′, A lies in the w-negative half-space H^{w−} and A′ lies in the w-positive half-space H^{w+}. As shown in the proof of the above lemma, any minimal gallery going from A to A′ is in the w-direction. □

Corollary 5.6. For any two alcoves A, A′, there exists w ∈ W_{0} and a gallery going from A to A′ in the w-direction. In other words

A = ⋃_{w∈W_{0}} C(A, w).

Proof. The union ⋃_{w∈W_{0}} C_{a}(A, w) obviously covers the whole set A for any a ∈ A_{0}. □

Corollary 5.7. Let μ be a dominant coweight. Then t_{wμ}(A_{0}) ∈ C(A_{0}, w).

Proof. If μ is dominant and lies in Q̌, t_{wμ}(A_{0}) belongs to C_{a}(A_{0}, w) for all a ∈ A_{0}. If now μ is dominant but not necessarily in Q̌, let us choose a point a in A_{0} fixed by Ω, for instance we can take the barycenter of A_{0}. Then t_{wμ}(A_{0}) belongs to C_{a}(A_{0}, w) a fortiori to C(A, w). □

There exists another characterization of the acute cone C(A, w) which will be useful later.

Lemma 5.8. An alcove A′ lies in C(A, w) if and only if it is contained in every wpositive halfspace H^{w+} containing A:

A′ ∈ ⋂_{A∈H^{w+}} H^{w+}.

Proof. This is only a matter of rephrasing the proof of Lemma 5.3. In fact, we have seen there that A′ ∈ C(A, w) if and only if for any wall H separating A and A′, we have A ∈ H^{w−} and A′ ∈ H^{w+}. In other words, A′ ∈ ⋂_{A∈H^{w+}} H^{w+}. □

Figure 2. An acute cone in A2 with marked alcove as apex

6. Proof of Proposition 4.3

Suppose μ ∈ X_{∗} is dominant and μ ∈ W_{aff}τ for τ ∈ Ω, the isotropy group of the base alcove. Given an element x ∈ W_{aff}τ which is μ-strongly permissible, we want to prove there exists w ∈ W_{0} such that x ≤ t_{wμ}. According to the Corollary 5.6, there exists w ∈ W_{0} such that x(A_{0}) ∈ C(A_{0}, w). According to Corollary 5.7, t_{wμ}(A_{0}) lies in the same acute cone C(A_{0}, w). We will prove x ≤ t_{wμ} for this w.

Let us write x = x_{1}τ and t_{wμ} = x_{2}τ with x_{1}, x_{2} ∈ W_{aff} and τ ∈ Ω. To prove Proposition 2 we thus have to prove the following statement.

Proposition 6.1. Let x_{1}, x_{2} be two elements of W_{aff} and w ∈ W_{0} such that • for every vertex v of A_{0}, x_{1}(v) lies in B(x_{2}(v), w);

• the two alcoves x_{1}(A_{0}) and x_{2}(A_{0}) lie in C(A_{0}, w).

Then the inequality x_{1} ≤ x_{2} holds.

First, we prove that the statement does not change when we move the base alcove in the w-opposite direction. By this operation we can then move the alcoves x(A_{0}) and t_{wμ}(A_{0}) far from the walls. As we will see, the proof is much easier in that case, compared to the proof of Kottwitz-Rapoport in the minuscule case, which deals with alcoves very close to the origin.

Lemma 6.2. Let x_{1}, x_{2} ∈ W_{aff} be such that the alcoves x_{1}(A_{0}) and x_{2}(A_{0}) lie in C(A_{0}, w). Let y ∈ W_{aff} be such that A′_{0} ∈ C(A_{0}, ww_{0}) where y(A′_{0}) = A_{0}. Then x_{1} ≤ x_{2} if and only if yx_{1} ≤ yx_{2}.

Note that the condition A′_{0} ∈ C(A_{0}, ww_{0}) means there exists a gallery going from A′_{0} to A_{0} in the w-direction.

Proof. According to Bernstein-Gelfand-Gelfand [2] (see also [3]) every Coxeter group satisfies the following property : let x_{1}, x_{2} ∈ W_{aff} and s ∈ S_{aff} be a simple reflection, then

x_{1} ≤ x_{2} ⇔ sx_{1} ≤ sx_{2}

whenever l(sx_{1}) > l(x_{1}) and l(sx_{2}) > l(x_{2}). It follows easily by induction on l(y) that x_{1} ≤ x_{2} is equivalent to yx_{1} ≤ yx_{2} whenever l(yx_{i}) = l(y) + l(x_{i}) for i = 1, 2.

Therefore what we really have to prove is that the lengths add

l(yx_{1}) = l(y) + l(x_{1})  l(yx_{2}) = l(y) + l(x_{2}).

Of course, we only need to prove the first of these equalities.

Let us choose a minimal gallery A′_{0}, …, A′_{l} going from A′_{0} to A_{0}, with l = l(y), and A_{0}, …, A_{r} a minimal gallery going from A_{0} to x_{1}(A_{0}), with r = l(x_{1}). According to Lemma 5.3, any minimal galleries going from A′_{0} to A_{0} or going from A_{0} to x_{1}(A_{0}) are automatically in the w-direction. Obviously, the concatenation of two galleries in the w-direction is still in the w-direction. By Lemma 5.3 again, the concatenated gallery A′_{0}, …, A′_{l} = A_{0}, …, A_{r} is minimal.

Let us prove that the concatenated gallery A′_{0}, …, A_{0}, …, A_{r} = x_{1}yA′_{0} is minimal only if the equality l(yx_{1}) = l + r holds.

Let S′_{aff} = {y^{−1}sy | s ∈ S_{aff}} be the set of reflections by walls of A′_{0}; obviously (W_{aff}, S′_{aff}) is also a Coxeter system. Now, by viewing A′_{0} as base alcove, the gallery A′_{0}, …, A_{0}, …, A_{r} = x_{1}yA′_{0} gives rise to a reduced expression

x_{1}y = s′_{1} … s′_{l+r}

with s′_{1}, …, s′_{l+r} in S′_{aff}. Thus the length of x_{1}y in the Coxeter system (W_{aff}, S′_{aff}) is equal to l + r. Consequently, the length of

yx_{1} = (ys′_{1}y^{−1}) … (ys′_{l+r}y^{−1})

in the Coxeter system (W_{aff}, S_{aff}) is equal to l + r. □

Another important ingredient in the proof is a lemma due to Deodhar [3], which is available for any Coxeter system, in particular for the Coxeter system (W_{aff}, S_{aff}). Let J be a subset of S_{aff} ; let W_{J} be the subgroup of W_{aff} generated by J. It is known that there exists a subset W^{J} of W_{aff} such that any element x ∈ W_{aff} admits a unique decomposition x = x^{J}x_{J} with x^{J} ∈ W^{J} and x_{J} ∈ W_{J} such that l(x) = l(x^{J}) + l(x_{J}).

Lemma 6.3 (Deodhar [3]). Let J be a set whose elements are subsets of S_{aff} such that ⋂_{J∈J} J = ∅. Let x, y ∈ W_{aff} and x = x^{J}x_{J} and y = y^{J}y_{J} be the unique decompositions of x and y for J ∈ J. Then x ≤ y if and only if x^{J} ≤ y^{J} for every J ∈ J.

Elements of S_{aff} correspond to walls of the base alcove A_{0}. For any vertex a of A_{0}, let J_{a} denote the subset of elements of S_{aff} corresponding to walls containing a. Obviously, the intersection of such subsets J_{a} for all vertices a of the base alcove is empty. Thus, the lemma of Deodhar applies.

Let us fix a vertex a of the base alcove A_{0} and let J denote J_{a}. It is useful to visualize the decomposition x = x^{J}x_{J} in terms of alcoves. Let x be an element of W_{aff} and A = x(A_{0}) the corresponding alcove. The coset xW_{J} is in bijection with the set of alcoves sharing with A the vertex x(a). The element x^{J} corresponds to the minimal element among those alcoves.

Lemma 6.4. Let a be a vertex of the base alcove A_{0} and let J denote J_{a}. Let x_{1}, x_{2} be elements of W_{aff} such that x_{1}(a) ∈ B(x_{2}(a), w). There exists an alcove A such that for every y ∈ W_{aff} satisfying y^{−1}A_{0} ∈ C(A, ww_{0}) we have (yx_{1})^{J} ≤ (yx_{2})^{J}.

In more intuitive terms, the inequality (yx_{1})^{J} ≤ (yx_{2})^{J} holds for the alcove y^{−1}A_{0} far enough in w-opposite direction.

Proof. Let us denote v_{1} = x_{1}(a) and v_{2} = x_{2}(a). We can easily reduce to the case v_{1} = sv_{2}, where s is some reflection s_{α,k} and v_{1} ∈ v_{2} + w(B_{0}). In other words v_{1} and v_{2} are symmetric with respect to the hyperplane H = H_{α,k} ; v_{1} lies in the w-negative half-space H^{w−} ; v_{2} lies in the w-positive half-space H^{w+}. We can also suppose v_{1} ≠ v_{2}, thus v_{1} and v_{2} lie in the interiors of the corresponding half-spaces determined by H.

Let A be an alcove in the w-negative half-space H^{w−}. The whole acute cone C(A, ww_{0}) belongs then to H^{w−}.

Let A′_{0} = y^{−1}(A_{0}) be an alcove lying in C(A, ww_{0}), a fortiori in the w-negative halfspace H^{w−}. Just as in the proof of the last lemma, it will be convenient to consider A′_{0} as base alcove. Let S′_{aff} denote the set of reflections by walls of A′_{0}. Let a′ = y^{−1}(a) and let J′ denote the subset of reflections by the walls of A′_{0} containing the vertex a′. Since the conjugation by y^{−1} makes the Coxeter (W_{aff}, S_{aff}) isomorphic to the Coxeter system (W_{aff}, S′_{aff}), the inequality we want to prove in the former system

(yx_{1})^{J} ≤ (yx_{2})^{J}

is equivalent to the inequality we are going to prove

(x_{1}y)^{J′} ≤ (x_{2}y)^{J′}

in the latter system. In the remainder of the proof, all inequalities are understood to be relative to the Coxeter system (W_{aff}, S′_{aff}).

Obviously, (x_{1}y)^{J′}(a′) = x_{1}y(a′) = v_{1} and (x_{2}y)^{J′}(a′) = x_{2}y(a′) = v_{2}. The alcove (x_{1}y)^{J′}(A′_{0}), resp. (x_{2}y)^{J′}(A′_{0}), is minimal among the alcoves sharing the vertex v_{1}, resp. v_{2}. Like v_{1}, the alcove (x_{1}y)^{J′}(A′_{0}) lies in the w-negative half-space H^{w−}. Like v_{2}, the alcove (x_{2}y)^{J′}(A′_{0}) lies in the w-positive half-space H^{w+}.

Let A′_{0}, A′_{1}, …, A′_{l} be a minimal gallery going from A′_{0} to A′_{l} = (x_{2}y)^{J′}A′_{0}. Since A′_{0} ∈ H^{w−} and A′_{l} ∈ H^{w+}, there exists an integer j such that A′_{j−1} and A′_{j} share the wall H. The minimal gallery A′_{0}, A′_{1}, …, A′_{l} corresponds to a reduced expression

(x_{2}y)^{J′} = s′_{1}s′_{2} … s′_{l}

with s′_{1}, …, s′_{l} ∈ S′_{aff}. By removing the reflection s′_{j} from the reduced expression, we get

s′_{1} … ŝ′_{j} … s′_{l} = s(x_{2}y)^{J′}.

Therefore s(x_{2}y)^{J′} ≤ (x_{2}y)^{J′}.

But the alcove s(x_{2}y)^{J′}A′_{0} contains the vertex s(x_{2}y)^{J′}(a′) = v_{1}. Thus by minimality, we know (x_{1}y)^{J′} ≤ s(x_{2}y)^{J′}, therefore (x_{1}y)^{J′} ≤ (x_{2}y)^{J′}. □

Proof of Proposition 6.1. The proposition follows from Lemmas 6.2, 6.3 and 6.4 as follows. By hypothesis we know that x_{1}(A_{0}), x_{2}(A_{0}) ∈ C(A_{0}, w) and that, for every vertex a of the base alcove, x_{1}(a) ∈ B(x_{2}(a), w). By 6.4, for every vertex a ∈ A_{0} there exists an alcove A_{a} such that, if y ∈ W_{aff} satisfies y^{−1}(A_{0}) ∈ C(A_{a}, ww_{0}), then (yx_{1})^{J_{a}} ≤ (yx_{2})^{J_{a}}. Now choose y such that

y^{−1}(A_{0}) ∈ C(A_{0}, ww_{0}) ∩ ⋂_{a} C(A_{a}, ww_{0}).

(Note that the intersection of finitely many acute cones in the same direction is nonempty.) Then (yx_{1})^{J_{a}} ≤ (yx_{2})^{J_{a}} for every a and thus by Lemma 6.3, yx_{1} ≤ yx_{2}. But then Lemma 6.2 implies x_{1} ≤ x_{2}.

□

7. Deodhar’s counter-examples

In [4] Deodhar considers the following question: which finite Weyl groups W_{0} have the property that w ≤ w′ in the Bruhat order if and only if w(λ) − w′(λ) is a sum of positive coroots for every dominant coweight λ? Surprisingly enough, at least for us, he proves that W_{0} has this property if and only if the irreducible components of the associated root system R are of type A_{n−1} or of rank ≤ 3. For part of his proof, for an irreducible root system of rank ≥ 4 and not of type A_{n−1}, he proves the following statement by giving explicit examples.

Proposition 7.1 (Deodhar [4]). Let R be an irreducible root system of rank ≥ 4 and not of type A_{n−1}. Then there exist elements w, w′ ∈ W_{0}, w ≠ w′ such that

• for every dominant coweight λ, w(λ) − w′(λ) is a sum of positive coroots;

• l(w) = l(w′).

We derive the following statement about μ-admissible and μ-permissible sets, thus proving Theorem 3.

Proposition 7.2. For R an irreducible root system of rank ≥ 4 and not of type A_{n−1}, we have Adm(μ) ≠ Perm(μ) for every sufficiently regular dominant coweight μ.

Proof. The idea of the proof is the remark, due to Kottwitz and Rapoport, that nearby the extreme elements t_{wμ}, the picture looks like the Bruhat order in the finite Weyl group W_{0}. (This is proved in a precise form in Lemma 7.5 below.)

Let μ be a regular dominant coweight. For simplicity, suppose μ ∈ Q̌. Let w, w′ ∈ W_{0} be chosen as in Proposition 7.1. Let us consider the element

x = t_{w^{−1}μ}w^{−1}w′.

We prove x is μ-permissible but not μ-admissible, if μ is sufficiently regular.

Since w ≠ w′, we know x ≠ t_{μ′} for any μ′ ∈ W_{0}(μ). Therefore to prove x is not μ-admissible, it is enough to show l(x) = l(t_{μ}). Since μ is regular dominant, by a theorem of Iwahori-Masumoto [9] the maximal length element in the coset t_{w^{−1}μ}W_{0} is t_{w^{−1}μ}w^{−1}. Moreover we have

l(x) = l(t_{w^{−1}μ}w^{−1}) − l(w′)
= l(t_{w^{−1}μ}w^{−1}) − l(w)
= l(t_{w^{−1}μ}),

so l(x) is equal to l(t_{μ}).

Let us prove that x is μ-permissible. Let a be a vertex of A_{0}. By construction of the elements w and w′, the difference

t_{w^{−1}μ}(a) − t_{w^{−1}μ}w^{−1}w′(a) = a − w^{−1}w′(a) = w^{−1}(w(a) − w′(a))

lies in −w^{−1}(B_{0}), since a is dominant.

For μ sufficiently regular dominant, we have

t_{w^{−1}μ}w^{−1}w′(a) − a ∈ w^{−1}(C̄_{0}).

(This relation can be used as the definition of “sufficiently regular”.) According to the following well-known statement, x is μ-permissible. □

Lemma 7.3. The equality

wC̄_{0} ∩ (wμ + wB_{0}) = wC̄_{0} ∩ Conv(μ)

holds for any μ dominant and w ∈ W_{0}.

In some sense, this lemma was the starting point of this work.

Remark 7.4. The following lemma, which is not used in the sequel, makes precise the remark of Kottwitz-Rapoport which inspired Proposition 7.2 above.

Lemma 7.5. Let R be any irreducible root system. Suppose λ ∈ X_{∗}, and let t_{λ}w_{λ} be the unique element of minimal length in the coset t_{λ}W_{0}. Suppose w_{1}, w_{2} ∈ W_{0}. Then t_{λ}w_{λ}w_{1} ≤ t_{λ}w_{λ}w_{2} if and only w_{1} ≤ w_{2}.

Proof. By a formula of Iwahori-Matsumoto (loc. cit.), l(t_{λ}w_{λ}w_{i}) = l(t_{λ}w_{λ}) + l(w_{i}) for i = 1, 2. The lemma is a consequence of these equalities, as explained already in the proof of Lemma 6.2. □

The geometric meaning of the lemma can be stated as follows.

Corollary 7.6. Suppose x, y ∈ W_{aff}τ are such that the alcoves x(A_{0}) and y(A_{0}) share a vertex v = t_{λ}(0), where t_{λ} ∈ W_{aff}τ. Then x < y if and only if there exists a sequence of reflections s_{H_{1}}, …, s_{H_{n}} such that

x < s_{H_{1}}x < s_{H_{2}}s_{H_{1}}x < ⋯ < s_{H_{n}} … s_{H_{1}}x = y,

where every hyperplane H_{i}, i = 1, …, n, contains the vertex v.

8. Proof of Deodhar’s theorem for A_{n−1}

The goal of this section is to present a short proof of the following theorem of Deodhar [4] characterizing the Bruhat order on W_{0} = S_{n−1}. As in the proof of Theorem 1 we rely on Proposition 6.1 as the key ingredient.

Proposition 8.1 (Deodhar [4]). Let R be a root system of type A_{n−1}. Let w, w′ ∈ W_{0}. Then w′ ≤ w in the Bruhat order on W_{0} if and only if for every dominant coweight λ, w′(λ) − w(λ) is a sum of positive coroots.

Proof. We need to show that if w′(λ)−w(λ) is a sum of positive coroots for all dominant coweights λ, then w′ ≤ w, the reverse implication being a general fact for all root systems which is easily proved. Recall that for root systems of type A_{n−1} every vertex v of the base alcove is either a dominant fundamental coweight or zero. Let w_{0} denote the longest element of W_{0}.

The assumption on w and w′ implies that, for every vertex v ∈ A_{0},

w′(v) ∈ W_{aff}(w(v)) ∩ (w(v) + w_{0}(B_{0}))

and thus

w′(v) ∈ B(w(v), w_{0}),

by Lemma 4.4. By Lemma 8.2 below, w(A_{0}) and w′(A_{0}) both belong to C(A_{0}, w_{0}). Therefore by Proposition 6.1, w′ ≤ w, as desired.

□

We have used a special case (μ = 0) of the following general lemma. Intuitively, it says that all the alcoves “on the boundary” of the antidominant Weyl chamber are contained in the antidominant acute cone of alcoves.

Lemma 8.2. Let R be an irreducible root system with Weyl group W_{0}. Let w_{0} denote the longest element of W_{0}. Then for any dominant coweight μ and any w ∈ W_{0} we have

t_{w_{0}μ}w(A_{0}) ∈ C(A_{0}, w_{0}).

Proof. Since the concatenation of two galleries in the w_{0}-direction is still in the w_{0}- direction, it suffices to prove that t_{w_{0}μ}(A_{0}) ∈ C(A_{0}, w_{0}) and t_{w_{0}μ}w(A_{0}) ∈ C(t_{w_{0}μ}(A_{0}), w_{0}). The first statement results from Corollary 5.7, and since the translation of a gallery in the w_{0}-direction is still in the w_{0}-direction, the second is equivalent to w(A_{0}) ∈ C(A_{0}, w_{0}).

Choose a reduced expression w = s_{1}s_{2} ⋯ s_{r}, where for each i = 1, … r, s_{i} = s_{α_{i}} is the reflection corresponding to a simple root α_{i}. We claim that the gallery A_{0}, s_{1}(A_{0}), …, s_{1} ⋯ s_{r}(A_{0}) is in the w_{0}-direction. Fix i and let H_{i} = H_{s_{1}···s_{i−1}(α_{i})} denote the hyperplane separating s_{1} ⋯ s_{i−1}(A_{0}) and s_{1} ⋯ s_{i}(A_{0}). We need to show that

s_{1} ⋯ s_{i−1}(A_{0}) ⊂ H_{i}^{w_{0}−}

and

s_{1} ⋯ s_{i}(A_{0}) ⊂ H_{i}^{w_{0}+}.

But since s_{1} ⋯ s_{i−1}(α_{i}) > 0, we have

H_{i}^{w_{0}−} = {x ∈ V | ⟨s_{1} ⋯ s_{i−1}(α_{i}), x⟩ > 0}

and

H_{i}^{w_{0}+} = {x ∈ V | ⟨s_{1} ⋯ s_{i−1}(α_{i}), x⟩ < 0},

yielding the desired inclusions.

□

Remark 8.3. If R is any root system such that

B(v, w_{0}) = W_{aff}(v) ∩ (v + w_{0}(B_{0}))

for every vertex v, then as in the proof of Proposition 8.1. we see that W_{0} enjoys the property that w′ ≤ w in the Bruhat order if and only if w′(λ) − w(λ) is a sum of positive coroots for every dominant coweight λ. (Indeed, letting λ range over fundamental coweights and recalling that every vertex of A_{0} is a positive scalar multiple of a fundamental coweight, we see that the condition on the differences w′(λ) − w(λ) ensures that w′(v) ∈ W_{aff}(w(v)) ∩ (w(v) + w_{0}(B_{0})) for every vertex v ∈ A_{0}, and the rest of the proof goes over word-for-word.) Consequently, Deodhar’s theorem (see Proposition 7.1) implies that R is of type A_{n−1} or of rank ≤ 3 (cf. Remark 4.5).

9. Admissible sets and automorphisms of root systems

In this section we study the behavior of admissible sets relative to an automorphism Θ of the underlying root system. This will yield information about the admissible and permissible sets in the extended affine Weyl group for GSp_{2n}, to be discussed at the end of this section and in section 10. We begin by recalling some facts about automorphisms of root systems; the main reference is Steinberg’s article [14], especially §1.30 − 1.33.

Let R = (X^{∗}, X_{∗}, R, Ř, Π) be an irreducible and reduced (based) root system, as in section 2. We may equip V = X_{∗} ⊗ ℝ with a W_{0}-invariant inner product ( , ). By assumption X^{∗} = Hom(X_{∗}, ℤ), and we let ⟨ , ⟩ : X^{∗} × X_{∗} → ℤ denote the canonical pairing. A root β ∈ R is a linear functional on V. Using the isomorphism of the vector space V with its dual defined with the help of ( , ), we have the identification β = 2β̌/(β̌, β̌); in this way β may be regarded as an element of V.

Let Θ be an automorphism of R, in the sense of [14]. This means that Θ is a linear automorphism of (V, ( , )) leaving stable the sets X^{∗}, R, and Π, when these are regarded as functionals on V. Hence, Θ leaves stable the subsets X_{∗} and Ř of V as well. Clearly Θ induces automorphisms of the groups W̃, W_{aff}, W_{0}, and Q̌; in particular we may consider the fixed-point subgroups W_{aff}^{Θ}, W_{0}^{Θ}, and Q̌^{Θ}.

The group W_{0}^{Θ} is the Weyl group of a root system R^{[Θ]}, which is defined as follows. Define Z = {x ∈ X_{∗} | ⟨α, x⟩ = 0, for all α ∈ R}. It is known that Z ∩ Q̌ = {0} and Z + Q̌ has finite index in X_{∗} (see Lemma 1.2 of [13]). Let X_{∗}^{[Θ]} = {x ∈ X_{∗} | x − Θ(x) ∈ Z}, and let V^{[Θ]} = X_{∗}^{[Θ]} ⊗ℝ. For any β ∈ R, let Θ̄(β) denote the average of the Θ-orbit of β. Then according to [14] §1.33, R^{[Θ]} is the subset of V^{[Θ]} consisting of all elements Θ̄(β), β ∈ R, except those which are smaller multiples of others. Moreover, for any Θ-orbit π ⊂ Π, let V_{π} denote the real vector space generated by π and let α̃_{π} be any highest root of the (possibly reducible) root system R∩V_{π}. Then the set Π^{[Θ]} of simple roots in R^{[Θ]} consists of the elements α_{π} := Θ̄(α̃_{π}) as π ranges over all Θ-orbits in Π. (The highest roots of R ∩ V_{π} form a single Θ-orbit, so the choice of α̃_{π} for each orbit π is immaterial.) By loc. cit. §1.32 − 1.33, (V^{[Θ]}, R^{[Θ]}, Π^{[Θ]}) is a root system in the sense of loc. cit., §1.1 − 1.6.

For any Θ-orbit π ⊂ Π, let s_{π} denote the longest element of the Weyl group W_{π} generated by {s_{α} | α ∈ π}. The restriction of s_{π} to V^{[Θ]} is the reflection s_{α_{π}} corresponding to α_{π}. In the sequel we will abuse notation and write s_{α_{π}} instead of s_{π}.

Let X^{∗[Θ]} = Hom(X_{∗}^{[Θ]}, ℤ). Regarding R^{[Θ]} as a set of linear functionals on V^{[Θ]}, it follows easily that R^{[Θ]} ⊂ X^{∗[Θ]}. Regarding R^{[Θ]} as a subset of V^{[Θ]}, we define another subset Ř^{[Θ]} = {2β/(β, β) | β ∈ R^{[Θ]}}. The following statement is certainly well-known, but lacking a convenient reference, we provide some details.

Proposition 9.1. R^{[Θ]} = (X^{∗[Θ]}, X_{∗}^{[Θ]}, R^{[Θ]}, Ř^{[Θ]}, Π^{[Θ]}) is a reduced and irreducible (based) root system in the vector space V^{[Θ]}. Its affine Weyl group is W_{aff}^{Θ} = Q̌^{Θ} ⋊ W_{0}^{Θ}.

Proof. In light of loc. cit. §1.32, to prove R^{[Θ]} is a based root system it remains only to show that, (i) every element of R^{[Θ]} is an integral linear combination of elements of Π^{[Θ]}, and (ii) Ř^{[Θ]} ⊂ X_{∗}^{[Θ]}. For statement (i), consider the following property of a Θ-orbit π ⊂ Π:

(∗) The elements of π are pairwise orthogonal.

If (∗) holds for every π, then each α ∈ π is a highest root and thus Θ̄(α) = α_{π}. Therefore (i) follows from the analogous property of the root system R = (X^{∗}, X_{∗}, R, Ř, Π). By examining automorphisms of simple Dynkin diagrams, we see that (∗) holds for every π unless R is of type A_{2n} (which we assume realized in V = ℝ^{2n+1}), Θ(x_{1}, …, x_{2n+1}) = (−x_{2n+1}, …, −x_{1}), and π = {e_{n} − e_{n+1}, e_{n+1} − e_{n+2}}; therefore, α_{π} = e_{n} − e_{n+2}. In this case one can verify by direct calculation that (i) holds (and in fact R^{[Θ]} is of type C_{n}). We note that (i) is equivalent to the statement that 2(α, β)/(β, β) ∈ ℤ for every α, β ∈ R^{[Θ]}.

Next, one can show, following loc. cit. §1.32, that W_{aff}^{Θ} is the group of affine transformations of V^{[Θ]} generated by the reflections through the hyperplanes β +k = 0, for β ∈ R^{[Θ]} and k ∈ ℤ. It follows that W_{aff}^{Θ} is the affine Weyl group for the root system R^{[Θ]} in V^{[Θ]}, and therefore the fixed-point group Q̌^{Θ} is precisely the corresponding coroot lattice. But then Ř^{[Θ]} ⊂ Q̌^{Θ} ⊂ X_{∗}^{[Θ]}, and (ii) is proved.

Finally, if α̃ is the highest root of R, then Θ(α̃) is also highest and thus Θ(α̃) = α̃ is the unique highest root of R^{[Θ]}. Therefore R^{[Θ]} is irreducible. □

Viewing roots as linear functionals on V, we have the inclusion

R^{[Θ]} ⊂ R|_{V^{[Θ]}},

since α|_{V^{[Θ]}} can be identified with Θ̄(α), for any α ∈ R. Although the opposite inclusion does not always hold, every element of R|_{V^{[Θ]}} is of the form cβ for some β ∈ R^{[Θ]} and c ∈ [0, 1]. In fact c ∈ \{\frac{1}{2}, 1\}, as is shown by the following lemma which is implicit in Steinberg’s book [14]; we provide a proof since we could not locate a reference.

Lemma 9.2. Given a root α ∈ R, Θ̄(α) is either a root α′ or a half-root \frac{1}{2}α′, for α′ ∈ R^{[Θ]}. In other words

Θ̄(R) ⊂ W_{0}^{Θ}Π^{[Θ]} ∪ \frac{1}{2}W_{0}^{Θ}Π^{[Θ]}.

Conversely, if α′ is any root in R^{[Θ]}, there is at least one root α ∈ R such that Θ̄(α) = α′. Moreover α′ is a positive root in R^{[Θ]} if and only if α is a positive root in R.

Proof. First note that if α ∈ V_{π} ∩R, then Θ̄(α) is either α_{π} or \frac{1}{2}α_{π} (the latter occurring only in the case R is of type A_{2n} and π = {e_{n} − e_{n+1}, e_{n+1} − e_{n+2}}; cf. the proof of Prop. 9.1). Thus it is enough to prove the following statement.

(†) Given β ∈ R^{+}, there exist π ⊂ Π and w ∈ W_{0}^{Θ} such that wβ ∈ V_{π} ∩ R.

For any β ∈ R^{+}, we have Θ̄(β) = ∑_{π} c_{π}α_{π}, where c_{π} ≥ 0 for every Θ-orbit π ⊂ Π. We will prove (†) by induction on the quantity ΣΘ̄(β) = ∑_{π} c_{π}.

Fix a positive root β; we may assume that β is not supported in any π ⊂ Π. Since the pairing ( , ) is positive definite, we have (Θ̄(β), Θ̄(β)) > 0 and thus there is a Θ-orbit π such that (Θ̄(β), α_{π}) > 0. By [14] §1.15, s_{α_{π}} permutes the positive roots not supported on π, and thus s_{α_{π}}β is positive. On the other hand

Θ̄(s_{α_{π}}β) = s_{α_{π}} Θ̄(β) = Θ̄(β) − \frac{2(Θ̄(β), α_{π})}{(α_{π}, α_{π})}α_{π},

and thus ΣΘ̄(s_{α_{π}}β) < ΣΘ̄(β). By induction s_{α_{π}} Θ̄(β) ∈ W_{0}^{Θ}Π^{[Θ]} ∪ \frac{1}{2}W_{0}^{Θ}Π^{[Θ]}, whence the first statement follows. The converse statement, as well as the one concerning positivity, is obvious. □

Given a fixed affine root (α, k) for R, let k′ = k if Θ̄(α) = α′ and let k′ = 2k if Θ̄(α) = \frac{1}{2}α′. Given a hyperplane H_{α,k} for R in V we have

H_{α,k} ∩ V^{[Θ]} = H_{α′,k′},

where α′ and k′ are defined as above. Moreover, every hyperplane for R^{[Θ]} in V^{[Θ]} is of this form. This has the following consequence for the set of alcoves in V^{[Θ]} that arise from the root system R^{[Θ]}.

Proposition 9.3. Given an alcove A in V, the set A ∩ V^{[Θ]} is either empty or is an alcove A′ in V^{[Θ]}; moreover every alcove A′ in V^{[Θ]} arises as A′ = A ∩ V^{[Θ]} for a uniquely determined alcove A in V.

If A_{0} (resp. A′_{0}) is the base alcove in V (resp. in V^{[Θ]}), we have A′_{0} = A_{0} ∩ V^{[Θ]}.

Proof. If x, y ∈ V^{[Θ]} are in the same alcove A′, they belong to the same connected component of V − ⋃ H_{α,k}; therefore x, y ∈ A for some alcove A in V. If x, y ∈ V^{[Θ]} belong to two different alcoves in V^{[Θ]}, there exists (α′, k′) ∈ R^{[Θ]}×ℤ such that ⟨α′, x⟩ < k′ and ⟨α′, y⟩ > k′. Let (α, k) be a corresponding affine root in R (i.e., Θ̄(α) = α′ and k = k′). We have ⟨α, x⟩ < k and ⟨α, y⟩ > k, thus x and y belong to different alcoves in V.

An element x ∈ A′_{0} satisfies the inequalities ⟨Θ̄(α), x⟩ > 0 for all positive roots α of R, and the inequality ⟨Θ̄(α̃), x⟩ < 1 for the highest root α̃ of R. Since x ∈ V^{[Θ]} these inequalities are equivalent with ⟨α, x⟩ > 0 for all positive roots α of R, and ⟨α̃, x⟩ < 1. Therefore x ∈ A_{0}. □

Lemma 9.4. For any hyperplane H_{α,k} for R in V, the equality

H_{α,k}^{w+} ∩ V^{[Θ]} = H_{α′,k′}^{w+}

holds for any w ∈ W_{0}^{Θ}.

Proof. This follows easily from the definitions, noting that α ∈ w(R^{+}) if and only if α′ ∈ w(R^{[Θ]+}) under the assumption w ∈ W_{0}^{Θ}. □

It is now easy to prove an inheritance property for acute cones of alcoves. Let C(A_{0}, w) ∩ V^{[Θ]} denote the set of alcoves in V^{[Θ]} of the form A ∩ V^{[Θ]} such that A ∈ C(A_{0}, w).

Proposition 9.5. Let w ∈ W_{0}^{Θ}. Then

C(A_{0}^{[Θ]}, w) = C(A_{0}, w) ∩ V^{[Θ]}.

Proof. This follows easily from Lemma 9.4 and the description of acute cones of alcoves as intersections of half-spaces as in Lemma 5.8. □

As in section 2, define the extended affine Weyl group W̃^{[Θ]} = X_{∗}^{[Θ]} ⋊ W_{0}^{Θ}, which acts on the set A^{[Θ]} of alcoves of V^{[Θ]}. Let A′_{0} denote the base alcove and Ω^{[Θ]} the stabilizer of A′_{0} in W̃^{[Θ]}. Because W_{aff}^{Θ} acts simply transitively on A^{[Θ]}, we have the decomposition W̃^{[Θ]} = W_{aff}^{Θ} ⋊ Ω^{[Θ]}.

According to Proposition 9.3, we have A_{0} ∩V^{[Θ]} = A′_{0}, thus Ω∩W̃^{[Θ]} = Ω^{[Θ]}. We also have W_{aff} ∩ W̃^{[Θ]} = W_{aff}^{Θ}. This allows us to generalize the Kottwitz-Rapoport result on the inheritance property of the Bruhat order from affine Weyl groups to extended affine Weyl groups.

Proposition 9.6. The Bruhat order ≤ on W̃^{[Θ]} is inherited from the Bruhat order ⪯ on W̃. In other words, if x, y ∈ W̃^{[Θ]}, then x ≤ y if and only if x ⪯ y.

Proof. According to Proposition 2.3 of Kottwitz-Rapoport [10], the statement is already valid if W̃^{[Θ]} is replaced by W_{aff}^{Θ} and W̃ is replaced by W_{aff}. Since the Bruhat order on W_{aff}^{Θ} (resp. W_{aff}) is extended in the obvious way to W̃^{[Θ]} (resp. W̃) as explained in section 2, the proposition follows. □

Now let μ ∈ X_{∗}^{[Θ]} be a fixed coweight, which we suppose is dominant (note that it is dominant with respect to R if and only if it is dominant with respect to R^{[Θ]}). We denote by Adm^{Θ}(μ), Perm^{Θ}(μ), and Perm^{st,Θ}(μ) the subsets of W̃^{[Θ]} analogous to the subsets Adm(μ), Perm(μ), and Perm^{st}(μ) of W̃. The main goal of this section is the following proposition. If R is taken to be the root system for GL_{2n} and Θ its nontrivial automorphism (cf. section 10), then R^{[Θ]} is the root system for GSp_{2n}. Therefore this result implies Proposition 5.

Proposition 9.7. Suppose that the root system R is of type A_{m}. Then the equality

Adm^{Θ}(μ) = Perm(μ) ∩ W̃^{[Θ]}

holds for every dominant coweight μ of R^{[Θ]}.

Proof. Under the assumption that R is of type A_{m}, we have

Adm(μ) = Perm(μ) = Perm^{st}(μ).

If x ∈ Adm^{Θ}(μ), we have x ∈ Adm(μ) according to Proposition 9.6, thus x ∈ Perm(μ)∩ W̃^{[Θ]}.

Suppose x ∈ Perm(μ) ∩ W̃^{[Θ]}. Let A = x(A_{0}) and A′ = x(A′_{0}); the alcoves A in V and A′ in V^{[Θ]} satisfy A′ = A ∩ V^{[Θ]}. By Corollary 5.6, there is an element w ∈ W_{0}^{Θ} such that A′ ∈ C(A′_{0}, w). By Proposition 9.5, A ∈ C(A_{0}, w). Since x ∈ Perm^{st}(μ), we have x(v) ∈ B(t_{wμ}(v), w) for every vertex v ∈ A_{0}. We can now apply Proposition 6.1. to x and t_{wμ} and we obtain x ≤ t_{wμ} relative to the Bruhat order on W̃. By Proposition 9.6. the same inequality holds for the Bruhat order on W̃^{[Θ]}. □

Note that the set Perm(μ) ∩ W̃^{[Θ]} has some geometric meaning, when m = 2n − 1, which directly relates to the local models of [7] attached to GSp_{2n} and a dominant coweight μ. More precisely, let us write Θ for the automorphism of GL_{2n} given by X ↦ J̃^{−1}(X^{t})^{−1}J̃, where J̃ is the matrix

(\begin{matrix}0 & J \\ −J & 0 \end{matrix}) ,

and where J is the anti-diagonal matrix with entries equal to 1. This determines a symplectic group Sp_{2n} = SL_{2n}^{Θ}, and a corresponding group GSp_{2n}. The automorphism Θ preserves the standard “upper triangular” Borel subgroup and the diagonal torus in GL_{2n}, hence induces an automorphism, also denoted Θ, of the root system R for GL_{2n}.

The root system R^{[Θ]} of Steinberg is the root system of the group GSp_{2n} determined by its standard “upper triangular” Borel subgroup and diagonal torus.

Fix a dominant cocharacter μ = (μ_{1}, …, μ_{n}, ν_{n}, …, ν_{1}) of GSp_{2n}. We can interpret the set Perm(μ) ∩ W̃(GSp_{2n}) in terms of lattice chains. Let k denote an algebraic closure of 𝔽_{p}, and let V_{i} = t^{−1}k[[t]]^{i} ⊕ k[[t]]^{2n−i}, for 0 ≤ i ≤ 2n; extend by periodicity to get the “standard” infinite lattice chain V_{•}. Consider the following set of periodic lattice chains

⋯ ⊂ L_{−1} ⊂ L_{0} ⊂ ⋯ ⊂ L_{2n} = t^{−1}L_{0} ⊂ …

consisting of k[[t]]-lattices in k((t))^{2n} with the following properties:

• inv(L_{i}, V_{i}) ⪯ μ, for every i ∈ ℤ,

• L_{i}^{⊥} = t^{−c(μ)}L_{−i}, for every i ∈ ℤ. Here inv(L, L′) denotes the standard notion of invariant between two k[[t]]-lattices,

and ⪯ denotes the usual partial order on dominant coweights for GL_{2n}. Moreover, ⊥ is defined using the symplectic form (x, y) ↦ x^{t}J̃y on k((t))^{2n}, and c(μ) is the common value for μ_{i} + ν_{i}, 1 ≤ i ≤ n.

This is precisely the set of k-points M_{μ}(k) for the local model M_{μ} considered in [7]. It carries an action of the standard Iwahori subgroup I_{k} of GSp_{2n}(k[[t]]), namely the stabilizer of the standard lattice chain V_{•}.

The I_{k}-orbits are clearly parametrized by the set Perm(μ) ∩ W̃(GSp(2n)). The content of Proposition 9.7 is therefore that the strata in the special fiber of the model M_{μ} are indexed by the set of μ-admissible elements of W̃(GSp_{2n}). As indicated by Görtz [5],[6], this makes it very likely that the models M_{μ} from [7] are flat.

10. Admissible and permissible sets for GSp_{2n}

In light of Theorem 3, it makes sense to ask when the equality Adm(μ) = Perm(μ) holds for a dominant coweight μ of the root system for GSp_{2n}. This root system arises as the “fixed-points” under an automorphism Θ of that of GL_{2n}, as in section 9. By Proposition 9.7, we obviously have the following criterion:

Adm^{Θ}(μ) = Perm^{Θ}(μ) ⟺ Perm^{Θ}(μ) ⊂ Perm(μ) ∩ W̃^{[Θ]}.

Using this we will deduce that the equality Adm^{Θ}(μ) = Perm^{Θ}(μ) holds whenever μ is a sum of minuscule coweights of GSp_{2n}.

Define X^{∗} = X_{∗} = ℤ^{2n} and equip V = ℝ^{2n} with the standard inner product ( , ). Let R = Ř = {e_{i} − e_{j} | 1 ≤ i < j ≤ 2n}, where the e_{i} are the standard basis vectors. Let Π = {e_{i} − e_{i+1} | 1 ≤ i < 2n}. Then R = (X^{∗}, X_{∗}, R, Ř, Π) is the root system for GL_{2n}. The finite Weyl group W_{0} = S_{2n} acts on X_{∗} = ℤ^{2n} by permuting the coordinates, and the extended affine Weyl group is W̃ = ℤ^{2n} ⋊ S_{2n}.

Let Θ denote the automorphism of X_{∗} = ℤ^{2n} defined by

Θ(x_{1}, …, x_{2n}) = (−x_{2n}, …, −x_{1}).

We can then form the root system R^{[Θ]} as in section 9. It is easy to check that R^{[Θ]} is the root system of GSp_{2n}, so all the results of section 9 are in force in studying the latter.

Theorem 10.1. Let μ be a sum of minuscule coweights for GSp_{2n}. Then

Adm(μ) = Perm(μ).

Proof. According to Proposition 9.7, we only need to show

Perm^{Θ}(μ) ⊂ Perm(μ) ∩ W̃^{[Θ]}.

The arguments we use are very close to those in Lemma 12.4 of [10].

We will need some notation. For a vector v ∈ ℤ^{2n}, let v(m) denote its mth entry, so that v = (v(1), …, v(2n)). Let r = −Θ, that is, r is the automorphism of ℝ^{2n} which reverses the order of the coordinate entries. If v, v′ ∈ ℝ^{2n}, we write v ≤ v′ if v(m) ≤ v′(m) for each m. For c ∈ ℤ, let c = (c^{2n}) = (c, …, c) ∈ ℤ^{2n}.

Let ω_{i} = (1^{i}, 0^{2n−i}) for 1 ≤ i ≤ 2n. The vectors ω_{i}, 1 ≤ i ≤ 2n − 1, together with 0 serve as “vertices” of the base alcove A_{0} for GL_{2n}.

The only minuscule coweights are the vectors ν in ℤ^{2n} of the form ν = (1^{n}, 0^{n})+c or ν = c, for c ∈ ℤ. Therefore we may write μ = (a^{n}, b^{n}), where a, b ∈ ℤ and a ≥ b. For simplicity we discuss only the case where b = 0, the more general case being similar.

We choose a coordinate system (x_{1}, …, x_{n}, y_{n}, …, y_{1}) on V = ℝ^{2n}. Then X_{∗}^{[Θ]} = {(x_{1}, …, x_{n}, y_{n}, …, y_{1}) ∈ ℤ^{2n} | x_{1} + y_{1} = ⋯ = x_{n} + y_{n}}. The finite Weyl group W_{0}^{[Θ]} is W_{n} = (ℤ/2ℤ)^{n} ⋊ S_{n}. The element e_{i} ∈ (ℤ/2ℤ)^{n} acts by switching x_{i} and y_{i}, and an element σ ∈ S_{n} acts by simultaneously permuting the x_{i}’s and the y_{i}’s. Let Conv^{Θ}(μ) denote the convex hull in ℝ^{2n} of the set W_{n}(μ). We have

Conv(μ) = {(x_{1}, …, x_{n}, y_{n}, …, y_{1}) ∈ ℝ^{2n} | 0 ≤ x_{i}, y_{i} ≤ a, ∀i, and ∑_{i} x_{i} + y_{i} = na}
= {v ∈ ℝ^{2n} | 0 ≤ v ≤ a and ∑_{m} v(m) = na},

and

Conv^{Θ}(μ) = {(x_{1}, …, x_{n}, y_{n}, …, y_{1}) ∈ ℝ^{2n} | 0 ≤ x_{i}, y_{i} ≤ a and x_{i} + y_{i} = a, ∀i}
= {v ∈ ℝ^{2n} | 0 ≤ v ≤ a and v + r(v) = a}.

The vectors η_{i} := (ω_{i} + ω_{2n−i})/2, 1 ≤ i ≤ n, together with 0, serve as “vertices” for the base alcove A′_{0}. We now fix x ∈ W̃^{[Θ]}, and let v_{i} := x(ω_{i}), for 1 ≤ i ≤ 2n − 1. One can easily verify that x(η_{i}) − η_{i} = (v_{i} − ω_{i} + v_{2n−i} − ω_{2n−i})/2, for 1 ≤ i ≤ n.

Now suppose x = t_{λ}w ∈ Perm^{Θ}(μ), i.e., λ = x(0) − 0 and x(η_{i}) − η_{i} belong to Conv^{Θ}(μ) for 1 ≤ i ≤ n. Therefore

0 ≤ λ ≤ a (10.1) λ + r(λ) = a (10.2) 0 ≤ v_{i} − ω_{i} + v_{2n−i} − ω_{2n−i} ≤ 2a (10.3) v_{i} − ω_{i} + v_{2n−i} − ω_{2n−i} + r(v_{i} − ω_{i} + v_{2n−i} − ω_{2n−i}) = 2a, (10.4)

the last two equations holding for all i = 1, …, n. We also have

v_{i} + r(v_{2n−i}) = a + 1 (10.5) ω_{i} + r(ω_{2n−i}) = 1, (10.6) for all i = 1, …, 2n (for (10.5), use (10.2) and (10.6) together with the observation that w ∈ W_{0}^{Θ} ⇒ w commutes with r).

We need to show that x ∈ Perm(μ), i.e., λ and x(ω_{i}) − ω_{i} belong to Conv(μ) for 1 ≤ i ≤ 2n − 1. Thus we need to show

0 ≤ λ ≤ a (10.7)

∑_{m} λ(m) = na (10.8)

0 ≤ v_{i} − ω_{i} ≤ a (10.9)

∑_{m} v_{i}(m) − ω_{i}(m) = na, (10.10)

the last two equations holding for all i = 1, …, 2n − 1. It is easy to check that (10.7), (10.8), and (10.10) are satisfied. It is enough to verify (10.9) for i such that 1 ≤ i ≤ n: applying r to (10.9) and using the identities in (10.5-10.6) yields (10.9) with 2n − i replacing i. Henceforth we fix i in this range. From (10.5-10.6) we can easily verify that a − r(v_{i} − ω_{i}) = v_{2n−i} − ω_{2n−i}. Therefore (10.2) and (10.3), respectively, yield

a ≤ v_{i} + r(v_{i}) ≤ a + 1 (10.11) −a ≤ v_{i} − ω_{i} − r(v_{i} − ω_{i}) ≤ a. (10.12)

Let u = v_{i}(m)−ω_{i}(m) and v = r(v_{i})(m)−r(ω_{i})(m) for any m in the range 1 ≤ m ≤ 2n. Then (10.11) and (10.12) yield

a ≤ u + v + δ ≤ a + 1 (10.13) −a ≤ u − v ≤ a, (10.14)

where δ = ω_{i}(m) + r(ω_{i})(m). Since 1 ≤ i ≤ n, the term δ is either 0 or 1. Adding (10.13) and (10.14) gives

(10.15)  0 ≤ 2u + δ ≤ 2a + 1

and thus 0 ≤ u = v_{i}(m) − ω_{i}(m) ≤ a. Since this inequality holds for every m, we have 0 ≤ v_{i} − ω_{i} ≤ a, as desired.

□

11. Remarks on PSO(2n + 1)

The results we obtained in the symplectic case depend in a crucial way on the fact that the restriction of the Bruhat order of W̃(GL(2n)) to W̃(GSp(2n)) is the Bruhat order of the latter group. This inheritance property is no longer true for the odd orthogonal group. We are grateful to Kottwitz for his help in preparing this section.

The group PSO(2n + 1) whose root system is B_{n} can be defined as the fixed point subgroup of an involution of PGL(2n + 1). This induces an involution Θ on the root system A_{2n}. However the fixed point root system of Θ in the sense of Steinberg, see [14] or section 9, is not B_{n} but rather C_{n}. We have the natural inclusions

W_{aff}(B_{n}) ⊂ W_{aff}(C_{n}) ⊂ W_{aff}(A_{2n}).

According to [14] and [10], the Bruhat order of W_{aff}(C_{n}) is inherited from W_{aff}(A_{2n}). We will show by the following examples that the Bruhat order of W_{aff}(B_{n}) is not.

Let denote s_{0}, …, s_{n} the simple reflections of W_{aff}(B_{n}) and s′_{0}, …, s′_{n} those of W_{aff}(C_{n}). The inclusion W_{aff}(B_{n}) ⊂ W_{aff}(C_{n}) is given by s_{i} ↦ s′_{i} for i ≠ 0 and s_{0} ↦ s′_{0}s′_{1}s′_{0}. The elements s_{0} and s_{1} are not related in the Bruhat order on W_{aff}(B_{n}) but their images s′_{0}s′_{1}s′_{0} and s′_{1} in W_{aff}(C_{n}) are related. See figure 3 below.

The extended affine Weyl group W̃(B_{n}) = W_{aff}(B_{n}) ⋊ {1, τ} is canonically identified with W_{aff}(C_{n}) by τ ↦ s′_{0}. One can show by similar examples that the extended Bruhat order on the coset W_{aff}(B_{n})τ is also not inherited from W_{aff}(C_{n}).

Let μ ∈ ℤ^{n} be a dominant coweight of type B_{n}. Concerning the analog of Theorem 4 for PSO(2n + 1), we do not know whether the equality

Adm^{B_{n}}(μ) = Perm^{B_{n}}(μ)

holds for μ a sum of minuscule coweights. However, it is not hard to see that the analog of Proposition 5 is false. Example. Let n = 2 and μ = (1, 0). Then Adm^{B_{n}}(μ) consists of 13 elements. But

Perm^{A_{2n}}(μ) ∩ W̃(B_{n}) = Perm^{A_{2n}}(μ) ∩ W_{aff}(C_{n}) = Adm^{C_{n}}(μ)

consists of 19 elements. The final equality above follows from Proposition 9.7.

Figure 3. The alcoves on the right are related in the Bruhat order while the corresponding alcoves on the left are not.

12. Acknowledgments

We thank R. Kottwitz for encouragement, helpful comments, and a careful reading of the paper. We are also grateful to M. Rapoport for communicating his guess that Theorem 4 might hold, and for his continued interest in this work. We thank W. Casselman for his comments on the paper. We are very grateful to U. Görtz for his help in drawing pictures using graphics from Casselman’s home page.

This article was written during the Semestre Hecke workshop on automorphic forms at the Institut Henri Poincaré in Paris, held in the spring semester of 2000. The first author thanks the IHP for its hospitality. His research is partially supported by an NSF-NATO postdoctoral fellowship, an NSERC research grant, a Connaught New Staff grant, and NSF grant DMS 97-29992. The second author would like to thank the department of mathematics of the University of Toronto for a fruitful visit in February 2000 during which this research in collaboration could begin.

References

[1] I.N. Bernstein, I.M. Gelfand, and S.I. Gelfand, Differential operators on the base affine space and a study of 𝔤-modules, Prepr. 77, IPM Akad. Nauk SSSR (1972) (English translation in: Lie groups and their representations. Proc. Summer School in Group Representations. Bolyai Janos Math. Soc., Budapest 1971, pp.21-64, New York: Halsted (1975)).

[2] I.N. Bernstein, I.M. Gelfand and S.I. Gelfand, Schubert cells and cohomology of the spaces G/P, Uspekhi Mat. Nauk. 28 (1973), 3-26; English translation, Russ. Math. Surv. 28 (1973), 1-26.

[3] V. Deodhar, Some characterizations of the Bruhat ordering on a Coxeter group and determination of the relative Möbius function, Invent. Math. 39 (1977), 187-198.

[4] V. Deodhar, On Bruhat ordering and weight-lattice ordering for a Weyl group, Indag. Math. 40 (1978), 423-435.

[5] U. Görtz, On the flatness of certain Shimura varieties of PEL-type, to appear Math. Ann.

[6] U. Görtz, forthcoming paper.

[7] T. Haines, B.C. Ngô, Nearby cycles for local models of some Shimura varieties, preprint (2000), to appear, Compositio Math.

[8] J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, no. 29, Cambridge Univ. Press (1990).

[9] N. Iwahori, H. Matsumoto, On Some Bruhat Decomposition and the Structure of the Hecke Rings of p-adic Chevalley Groups, Publ. Math. IHES, No. 25, (1965) 5-48.

[10] R. Kottwitz, M. Rapoport, Minuscule Alcoves for Gl_{n} and GSp_{2n}, manuscripta math. 102, 403-428 (2000).

[11] M. Rapoport, On the Bad Reduction of Shimura Varieties, Automorphic Forms, Shimura Varieties and L-functions, part II, Perspectives in Mathematics, vol. 11, Academic Press, San Diego, CA, 1990, 253-321.

[12] M. Rapoport, Th. Zink.Period spaces for p-divisible groups. Annals of Math. Studies 144, Princeton Univ. Press 1996

[13] T.A. Springer, Reductive groups, in Automorphic forms, Representations, and L-functions, part I, Proc. of the Symp. in Pure Math., vol. XXXIII, Amer. Math. Soc. 1977.

[14] R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. 80 (1968), 1-108.

University of Toronto Department of Mathematics 100 St. George Street Toronto, ON M5S 3G3, Canada email:haines@math.toronto.edu

CNRS, Université de Paris Nord Département de mathématiques, 93430 Villetaneuse, FRANCE email: ngo@math.univ-paris13.fr