# Distinguished Cuspidal Representations over p-adic and Finite Fields

## TL;DR

This paper revisits distinguished tame supercuspidal representations of p-adic groups, simplifying Yu's construction and unifying the theory with finite field cases to enhance understanding of these representations.

## Contribution

It provides a simplified approach to Yu's construction and unifies the theory of distinguished cuspidal representations over p-adic and finite fields.

## Key findings

- Unified the theory of distinguished cuspidal representations across p-adic and finite fields.
- Simplified the construction of tame supercuspidal representations.
- Enhanced understanding of the structure of distinguished representations.

## Abstract

The author's work with Murnaghan on distinguished tame supercuspidal representations is re-examined using a simplified treatment of Jiu-Kang Yu's construction of tame supercuspidal representations of $p$-adic reductive groups. This leads to a unification of aspects of the theories of distinguished cuspidal representations over $p$-adic and finite fields.