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Alcoves associated to special fibers of local models. (English) Zbl 1047.20037
Let $$G$$ be a classical group over the $$p$$-adic field $$\mathbb{Q}_p$$, and let $$\mu$$ be a dominant minuscule coweight of $$G$$. The special fiber $$M_{\mu,\overline\mathbb{F}_p}$$ of the Rapoport-Zink local model $$M_\mu$$ has a stratification indexed by a finite subset $$\text{Perm}(\mu)$$ of the extended affine Weyl group $$\widetilde W(G)$$ for $$G$$. Let $$W_0$$ denote the finite Weyl group of $$G$$. For each translation $$\lambda$$ in the $$W_0$$-orbit $$W_0(\mu)$$ of $$\mu$$, the element $$t_\lambda$$ in $$\widetilde W(G)$$ is contained in $$\text{Perm}(\mu)$$. Let $$\text{Adm}(\mu)$$ denote the subset of $$\text{Perm}(\mu)$$ indexing those strata which lie in the closure of the stratum indexed by $$t_\lambda$$ for some $$\lambda\in W_0(\mu)$$.
The paper under review is concerned with the equality $$\text{Adm}(\mu)=\text{Perm}(\mu)$$. The main result of the paper is to prove that if the root system for $$G$$ has type $$A_{n-1}$$, then the equality $$\text{Adm}(\mu)=\text{Perm}(\mu)$$ holds for every dominant coweight $$\mu$$. The authors also show that the equality $$\text{Adm}(\mu)=\text{Perm}(\mu)$$ holds if $$\mu$$ is a sum of minuscule coweights for the symplectic group $$G=\text{GSp}_{2n}$$. On the other hand, the authors show that if the root system for $$G$$ is irreducible, of rank $$\geqslant 4$$ and not of type $$A_{n-1}$$, then $$\text{Adm}(\mu)\neq\text{Perm}(\mu)$$ for every sufficiently regular dominant coweight $$\mu$$.

##### MSC:
 20G05 Representation theory for linear algebraic groups 20F55 Reflection and Coxeter groups (group-theoretic aspects) 14G35 Modular and Shimura varieties 11G18 Arithmetic aspects of modular and Shimura varieties 22E35 Analysis on $$p$$-adic Lie groups
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