# Relative discrete series representations for two quotients of $p$-adic $\mathbf{GL}_n$

## TL;DR

This paper explicitly constructs discrete spectrum representations for two specific $p$-adic symmetric spaces involving $ ext{GL}_n$, using a symmetric space version of Casselman's Criterion, advancing understanding of harmonic analysis on these spaces.

## Contribution

It provides an explicit construction of discrete series representations for two $p$-adic symmetric spaces, applying a symmetric space adaptation of Casselman's Criterion.

## Key findings

- Explicit discrete series representations constructed for $ ext{GL}_n(F) imes ext{GL}_n(F) ackslash ext{GL}_{2n}(F)$
- Explicit discrete series representations constructed for $ ext{GL}_n(F) ackslash ext{GL}_n(E)$
- Application of symmetric space Casselman's Criterion to prove square integrability

## Abstract

We provide an explicit construction of representations in the discrete spectrum of two $p$-adic symmetric spaces. We consider $\mathbf{GL}_n(F) \times \mathbf{GL}_n(F) \backslash \mathbf{GL}_{2n}(F)$ and $\mathbf{GL}_n(F) \backslash \mathbf{GL}_n(E)$, where $E$ is a quadratic Galois extension of a nonarchimedean local field $F$ of characteristic zero and odd residual characteristic. The proof of the main result involves an application of a symmetric space version of Casselman's Criterion for square integrability due to Kato and Takano.