Local unitary periods and relative discrete series

TL;DR

This paper constructs specific representations in the discrete spectrum of a $p$-adic symmetric space involving unitary groups and general linear groups over quadratic extensions, advancing understanding of automorphic representations.

Contribution

It introduces a method to explicitly construct representations in the discrete spectrum of certain $p$-adic symmetric spaces involving unitary groups.

Findings

Constructed representations in the discrete spectrum of $H \backslash G$

Established connections between unitary periods and discrete series

Enhanced understanding of automorphic representations over $p$-adic fields

Abstract

Let $F$ be a $p$-adic field ($p\neq 2$), let $E$ be a quadratic Galois extension of $F$, and let $n \geq 2$. We construct representations in the discrete spectrum of the $p$-adic symmetric space $H \backslash G$, where $G = \mathbf{GL}_{2n}(E)$ and $H = \mathbf{U}_{E/F}(F)$ is a quasi-split unitary group over $F$.