Alcoves Associated to Special Fibers of Local Models

Let G be a classical group over the p-adic field Qp, 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 Kp of the compact open subgroup K is an Iwahori subgroup of G(Qp) = G(Qp). We assume that the corresponding Shimura variety Sh(G, X,K) has a model over a p-adic number ring Op. 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 Y0(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 [12]). The local geometry can be approached using the Rapoport-Zink local model Mμ [13]. This projective Op-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 Mμ,Fp of Mμ. To fix ideas, consider the case where G is split. It turns out that Mμ,Fp can be regarded as a finite-dimensional union of Iwahori-orbits in the affine flag variety for G(Fp((t))), see [5] and [8]. It follows that Mμ,Fp 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 W0 denote the finite Weyl group of G. For each translation λ in the W0-orbit of μ there is an Iwahori-orbit in Mμ,Fp corresponding to λ; in other words, the element tλ 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 tλ, for some λ ∈ W0(μ). We are primarily concerned with the equality Adm(μ) = Perm(μ), which has important geometric content. For example, if the Op-scheme Mμ is flat, this equality automatically holds. On the other hand, the equality has been established combinatorially when G = GLn or GSp2n by Kottwitz and Rapoport [11], and this was exploited by U. Görtz in his proof of the flatness of the local models associated to ResE/FGLn, where E/F is an unramified extension of p-adic fields. In [11], 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