Lifting semisimple characters of $p$-adic types from fixed-point subgroups

TL;DR

This paper proves that semisimple characters of fixed-point subgroups in $p$-adic groups can be lifted to the larger group, providing explicit methods for such lifts using advanced techniques in representation theory.

Contribution

It introduces a method to lift semisimple characters from fixed-point subgroups to the entire $p$-adic group, expanding understanding of character theory in this context.

Findings

Every semisimple character of a type for $G^\Gamma$ arises from a semisimple character of $G$.

Explicit lifting of Kim--Yu data from $G^\Gamma$ to $G$ is achieved.

Defines a lift of Howe factorization for characters of maximal tori.

Abstract

Given a $p$-adic group $G=\mathbf{G}(F)$ and a finite group $\Gamma\subset\mathrm{Aut}_F(\mathbf{G})$ such that the fixed-point subgroup $\mathbf{G}^\Gamma$ is reductive, we show that every semisimple character (in the sense of Bushnell and Kutzko) of a type for $G^\Gamma = \mathbf{G}^\Gamma(F)$ arises as the restriction of a semisimple character of a type for $G$. We achieve this by explicitly lifting the truncated Kim--Yu datum (or character-datum) that parametrizes the semisimple character for $G^\Gamma$ to a character-datum that parametrizes a semisimple character for $G$. Our proof, which is of independent interest, uses state-of-the-art techniques and, as a special case, defines a lift of a Howe factorization of a character of a maximal torus of $G^\Gamma$.