Cuspidal representations of $GL(n,F)$ distinguished by a maximal Levi subgroup, with $F$ a non-archimedean local field

TL;DR

This paper characterizes when cuspidal representations of GL(n,F) over a non-archimedean local field are distinguished by a maximal Levi subgroup, showing such representations only occur when n is even and the subgroup is a product of two GL(n/2,F).

Contribution

It proves a precise criterion for the distinguishedness of cuspidal representations of GL(n,F) by a maximal Levi subgroup, specifically identifying the structure of the subgroup and the parity of n.

Findings

Cuspidal representations are distinguished only when n is even.

The maximal Levi subgroup must be isomorphic to GL(n/2,F)×GL(n/2,F).

Distinguishedness imposes strict structural conditions on the representation and subgroup.

Abstract

Let $\rho$ is a cuspidal representation of $GL(n,F)$, with $F$ a non archimedean local field, and $H$ a maximal Levi subgroup of $GL(n,F)$. We show that if $\rho$ is $H$-distinguished, then $n$ is even, and $H\simeq GL(n/2,F)\times GL(n/2,F)$.