Representations of reductive groups distinguished by symmetric subgroups

TL;DR

This paper investigates the properties of representations of complex reductive groups that are distinguished by symmetric subgroups, establishing key symmetry relations and bounds on their multiplicities, and addressing a conjecture by Lapid.

Contribution

It proves that distinguished representations satisfy a specific symmetry under involution and provides bounds on their multiplicities, partially confirming Lapid's conjecture.

Findings

Established that distinguished representations satisfy $ heta$-twist symmetry.

Bounded the dimension of $H$-invariant linear forms by the double coset space size.

Provided partial confirmation of Lapid's conjecture on distinguished representations.

Abstract

Let $G$ be a complex reductive group and $H=G^{\theta}$ be its fixed point subgroup under a Galois involution $\theta$. We show that any $H$-distinguished representation $\pi$ (i.e $\mathrm{dim}_{\mathbb{C}}\left(\pi^{*}\right)^{H}\neq0$) satisfies: 1) $\pi^{\theta}\simeq\tilde{\pi}$, where $\tilde{\pi}$ is the contragredient representation and $\pi^{\theta}$ is the twist of $\pi$ under $\theta$. 2) $\mathrm{dim}_{\mathbb{C}}\left(\pi^{*}\right)^{H}\leq\left|B\backslash G/H\right|$, where $B$ is a Borel subgroup of $G$. By proving Statement 1), we give a partial answer to a conjecture by Lapid.