Distinguished Simple Supercuspidal Representations of $p$-adic $\text{GL}(n)$

TL;DR

This paper characterizes when simple supercuspidal representations of $ ext{GL}(n, E)$ are distinguished by $ ext{GL}(n, F)$, using gamma factors and maximal simple types, providing new criteria for distinction.

Contribution

It offers new equivalent conditions for distinction of supercuspidal representations in terms of gamma factors and simple types, advancing understanding of their distinction criteria.

Findings

Provides conditions for a supercuspidal representation to be distinguished by $ ext{GL}(n, F)$.

Shows that gamma factors at 1/2 determine distinction for these representations.

Connects distinction to properties of twisted gamma factors and simple types.

Abstract

Let $\text{E}/\text{F}$ be a quadratic extension of non-Archimedean local fields with odd residual characteristic. In this paper, we give equivalent conditions for a simple supercuspidal representation $\pi$ of $\text{GL}(n, \text{E})$ to be distinguished by $\text{GL}(n, \text{F})$ in terms of its defining maximal simple type and twisted gamma factors. Furthermore, we prove that the collection of twisted gamma factors evaluated at $\frac{1}{2}$ between $\pi$ and all unitary, tamely ramified quasi-characters of $\text{E}^{\times}$ that are trivial on $\text{F}^{\times}$ is sufficient to determine whether $\pi$ is distinguished by $\text{GL}(n, \text{F})$.