Alcoves associated to special fibers of local models

By Thomas J. Haines, B. Ngô. 2001

51 citations recorded by the source corpus.

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 An, we produce counterexamples showing the situation is not as nice in general. We also recover a theorem of Deodhar characterizing the Bruhat order on Sn. The local geometry can be approached using the Rapoport-Zink local model M/j, (13). This protective Op -scheme is a local model for the singularities in the special fiber of Sh (G, X9 K)9 but it is defined in terms of the pair (G, p) 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?f of M?. To fix ideas, consider the case where G is split. It turns out that M^f 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 M?f has a stratification indexed by a finite set Perm (p) of the extended affine Weyl group W(G) for G. Equivalently, Perm (p) can be regarded as a finite set of alcoves in the affine Coxeter complex determined by a splitting for G. Let Wo denote the finite Weyl group of G. For each translation ? in the Wo-orbit

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/91b44428844de69eacb5db4a034075473e5c9738

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Citations in the corpus

Introduction to Shimura Varieties with Bad Reduction of Parahoric Type (2004, background): “The strata in the special fiber of M loc = M−w0� are naturally indexed by the set Perm(−w0� ), which agrees with Adm(−w0� ) by the following non-trivial combinatorial theorem due to Kottwitz and Rapoport. Theorem 4.2 ([KR]; see also [ HN2 ]).”
Local models of Shimura varieties, I. Geometry and combinatorics (2010, methodology): “Then in (i), we have Admf ({µ}) = Permf ({µ}) for any {µ} and any f when R involves only type A; this was proved in the case of minuscule {µ} in [KR, 9.6], and the general case is an immediate consequence of Görtz’s result [Gö4, Cor. 9] (which itself makes crucial use of the cited result in the Iw”