Alcoves associated to special fibers of local models.

*(English)*Zbl 1047.20037Let \(G\) 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 \(\mu\).

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 \(\mu\).

Reviewer: Shi Jian-yi (Shanghai)

##### 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 |