Representations of reductive groups distinguished by symmetric subgroups

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 ${\dim }_{\mathbb{C}}{\left({\pi }^{\ast }\right)}^H\ne 0$) 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) ${\dim }_{\mathbb{C}}{\left({\pi }^{\ast }\right)}^H\le \left|B\backslash G/H\right|$, where $B$ is a Borel subgroup of $G$. By proving Statement 1), we give a partial answer to a conjecture by Lapid.

References in corpus (2)

• Schwartz functions on Nash manifolds
• A `relative' local Langlands correspondence

Cited by in corpus (2)

• Distinguished regular supercuspidal representations
• Bounds on multiplicities of symmetric pairs of finite groups