# Self-dual cuspidal representations

## TL;DR

This paper characterizes when finite groups of Lie type have irreducible, cuspidal, self-dual representations, and discusses implications for related p-adic groups, advancing understanding of their representation theory.

## Contribution

It precisely determines conditions for the existence of self-dual cuspidal representations in finite groups of Lie type under certain assumptions.

## Key findings

- Identifies when $G()$ admits self-dual cuspidal representations.
- Provides criteria for Deligne-Lusztig type representations.
- Outlines implications for self-dual supercuspidal representations of p-adic groups.

## Abstract

Let $G$ be a connected reductive group over a finite field $\mathfrak{f}$ of order $q$. When $q$ is small, we make further assumptions on $G$. Then we determine precisely when $G(\mathfrak{f})$ admits irreducible, cuspidal representations that are self-dual, of Deligne-Lusztig type, or both. Finally, we outline some consequences for the existence of self-dual supercuspidal representations of reductive $p$-adic groups.