Restricting admissible representations to fixed-point subgroups

Given a $p$-adic group $G$ equipped with an action of a finite group $Γ\subset\mathrm{Aut}_F(\mathbf{G})$, and a reductive fixed-point subgroup $G^Γ$, 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 $π$ of $G$, we explicitly identify a number of the inertial equivalence classes occurring in the representation $π|_{G^{[Γ]}}$.

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