Restricting admissible representations to fixed-point subgroups

Abstract

Given a $p$-adic group $G$ equipped with an action of a finite group $\Gamma \subset {\mathrm{Aut}}_F(\mathbf{G})$, and a reductive fixed-point subgroup $G^{\Gamma }$, we establish a relationship between constructions of types for these two groups due to Yu, Kim--Yu and Fintzen, and generalizes the relationship between general linear and classical groups observed by Stevens. As an application, given a parabolically induced or supercuspidal representation $\pi$ of $G$, we explicitly identify a number of the inertial equivalence classes occurring in the representation $\pi {|}_{G^{[\Gamma ]}}$.

17 pages; Lemma 3.1 corrected; details added to include a missing case