Minuscule alcoves for \(\text{GL}_n\) and \(\text{GSP}_{2n}\) (Q1589733)

The aim of this paper is to study the extended affine Weyl group \(\widetilde{W}\) for \(\text{GL}_n\). Let \(\mu\) be a minuscule coweight for either \(\text{GL}_N\) or \(\text{GSp}_{2n}\) and regard \(\mu\) as an element \(t_\mu\) in the extended affine Weyl group \(\widetilde{W}\). Say an element \(x\in\widetilde{W}\) is \(\mu\)-admissible if there exists \(\mu'\) in the Weyl group orbit of \(\mu\) such that \(x\leq t_{\mu'}\) in the Bruhat order on \(\widetilde{W}\). The main result Theorem 3.5(3) is that \(x\in\widetilde{W}\) is \(\mu\)-admissible if and only if it is \(\mu\)-permissible, where \(\mu\)-permissibility is defined using inequalities arising naturally in the study of bad reduction of Shimura varieties. Moreover, the authors show Theorem 4.5(3) as a completely analogous result for the group \(G = \text{GSp}_{2n}\), of symplectic similitudes.