# Speh representations are relatively discrete

## TL;DR

This paper proves that Speh representations associated with discrete series of GL(n) are part of the discrete spectrum for a specific p-adic symmetric space involving symplectic and general linear groups.

## Contribution

It establishes the relative discreteness of Speh representations in the spectrum of a p-adic symmetric space, extending understanding of their spectral properties.

## Key findings

- Speh representations are relatively discrete in the specified symmetric space.
- The result applies to p-adic fields with odd residual characteristic.
- It advances the theory of automorphic representations and harmonic analysis on p-adic groups.

## Abstract

Let $F$ be a $p$-adic field of characteristic zero and odd residual characteristic. Let $\mathbf{Sp}_{2n}(F)$ denote the symplectic group defined over $F$, where $n\geq 2$. We prove that the Speh representations $\mathcal{U}(\delta,2)$, where $\delta$ is a discrete series representation of $\mathbf{GL}_n(F)$, lie in the discrete spectrum of the $p$-adic symmetric space $\mathbf{Sp}_{2n}(F) \backslash \mathbf{GL}_{2n}(F)$.