# Linear periods and distinguished local parameters

## TL;DR

This paper confirms a conjecture relating to the Langlands parameters of specific representations in the discrete spectrum of a p-adic symmetric space, advancing understanding in the area of automorphic forms and representation theory.

## Contribution

It verifies a conjecture by Sakellaridis and Venkatesh on the Langlands parameters of certain discrete spectrum representations for a specific p-adic symmetric space.

## Key findings

- Confirmed the conjecture for the symmetric space $H ackslash G$ with $G=GL_{2n}(F)$ and $H=GL_n(F) imes GL_n(F)$.
- Identified the Langlands parameters of the relevant representations.
- Enhanced understanding of the spectral decomposition in p-adic symmetric spaces.

## Abstract

Let $F$ be a nonarchimedean local field of characteristic zero and odd residual characteristic. Let $X$ be the $p$-adic symmetric space $X = H \backslash G$, where $G = \mathbf{GL}_{2n}(F)$ and $H = \mathbf{GL}_n(F) \times \mathbf{GL}_n(F)$. We verify a conjecture of Sakellaridis and Venkatesh on the Langlands parameters of certain representations in the discrete spectrum of $X$.