# Quaternion distinguished generic representations of $\mathrm{GL}_{2n}$

## TL;DR

This paper classifies generic representations of isplaystyle GL_{2m}(E) distinguished by GL_m(D) over a quadratic extension, linking representation theory with base change lifts and quaternion algebra structures.

## Contribution

It provides a classification of GL_m(D)-distinguished generic representations of GL_{2m}(E) using Zelevinsky classification and relates it to base change lifts from unitary groups.

## Key findings

- Classification of distinguished representations via Zelevinsky parameters.
- A sufficient condition for base change lift representations to be distinguished.
- Connection between quaternion algebra embeddings and representation distinction.

## Abstract

Let $E/F$ be a quadratic extension of non-Archimedean local fields of characteristic 0. Let $D$ be the unique quaternion division algebra over $F$ and fix an embedding of $E$ to $D$. Then, $\mathrm{GL}_m(D)$ can be regarded as a subgroup of $\mathrm{GL}_{2m}(E)$. Using the method of Matringe, we classify irreducible generic $\mathrm{GL}_m(D)$-distinguished representations of $\mathrm{GL}_{2m}(E)$ in terms of Zelevinsky classification. Rewriting the classification in terms of corresponding representations of the Weil-Deligne group of $E$, we prove a sufficient condition for a generic representation in the image of the unstable base change lift from the unitary group $\mathrm{U}_{2m}$ to be $\mathrm{GL}_m(D)$-distinguished.