Distinction for unipotent $p$-adic groups

TL;DR

This paper characterizes when irreducible representations of unipotent p-adic groups are distinguished by fixed points of an involution, establishing a criterion based on self-duality and a one-dimensional Hom space.

Contribution

It extends Benoist's real case arguments to p-adic unipotent groups, providing a criterion for distinguished representations and a bijection with representations of fixed point subgroups.

Findings

An irreducible representation is distinguished iff it is sigma-self-dual.

Hom space dimension for distinguished representations is one.

Establishes a bijection between irreducible representations of fixed point subgroup and distinguished irreducible representations.

Abstract

Let $F$ be a $p$-adic field and $\mathbf{U}$ be a unipotent group defined over $F$, and set $U=\mathbf{U}(F)$. Let $\sigma$ be an involution of $\mathbf{U}$ defined over $F$. Adapting the arguments of Yves Benoist in the real case, we prove the following result: an irreducible representation $\pi$ of $U$ is $U^{\sigma}$-distinguished if and only if it is $\sigma$-self-dual and in this case $\mathrm{Hom}_{U^\sigma}(\pi,\mathbb{C})$ has dimension one. When $\sigma$ is a Galois involution these results imply a bijective correspondence between the set $\mathrm{Irr}(U^\sigma)$ of isomorphism classes of irreducible representations of $U^\sigma$ and the set $\mathrm{Irr}_{U^\sigma-\mathrm{dist}}(U)$ of isomorphism classes of distinguished irreducible representations of $U$.