Distinguished Representations of $\mathrm{GL}_{n}(\mathbb{F}_{q})$

TL;DR

This paper investigates the properties of distinguished representations of the general linear group over finite fields, establishing conditions under which these representations are self-dual, extending known results from local fields to finite fields.

Contribution

The paper proves that irreducible representations of GL_n over finite fields with an H-invariant functional are self-dual, extending previous results from p-adic fields to finite fields.

Findings

Irreducible representations with H-invariant functionals are self-dual over finite fields.

Extension of known self-duality results from local fields to finite fields.

Provides new insights into the structure of distinguished representations over finite fields.

Abstract

Let $\mathrm{G} = \mathrm{Gl}_{n}(K)$, and $\mathrm{H} = \mathrm{G}^{\sigma}$ for $\sigma$ an involution of the form $g\rightarrow aga^{-1}$, It is known that for $K =\mathbb{Q}_q$ any irreducible representation of $\mathrm{G}$ with an $\mathrm{H}$ invariant functional, is self dual, we prove an analogous result for $\mathbb{F}_{q}$.