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

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 $\text{GL}(n, \text{F})$.