Restricting admissible representations to fixed-point subgroups

Abstract

Given a p-adic group $G=\mathbf{G}(F)$ and a finite group $\Gamma \subset {\text{Aut}}_F(\mathbf{G})$ such that the fixed-point subgroup ${\mathbf{G}}^{\Gamma }$ is reductive, we establish a relationship between constructions of types for G and $G^{\Gamma }$ due to Yu, Kim–Yu and Fintzen, which generalizes the relationship between those of general linear and classical groups originally observed by Stevens. As an application, given an irreducible representation π of G, we explicitly identify a number of the inertial equivalence classes occurring in the representation $\pi {|}_{G^{[\Gamma ]}}$.

Keywords:

• Fixed-point subgroups
• p-adic groups
• theory of types

2020 Mathematics Subject Classification:

• 22E50

Acknowledgments

The second author’s research is supported by a Discovery Grant from NSERC Canada.

Additional information

Funding

The second author’s research is supported by a Discovery Grant from NSERC Canada.