Restricting Representations from a Complex Reductive Group to a Real Form

Abstract

Abstract Let $G$ be a complex connected reductive algebraic group and let $G_{{\mathbb {R}}}$ be a real form of $G$. We construct a sequence of functors $L_{n}\mathcal {R}$ from admissible (resp. finite-length) representations of $G$ to admissible (resp. finite-length) representations of $G_{{\mathbb {R}}}$. We establish many basic properties of these functors, including their behavior with respect to infinitesimal character, associated variety, and restriction to a maximal compact subgroup. We deduce that each $L_{n}\mathcal {R}$ takes unipotent representations of $G$ to unipotent representations of $G_{{\mathbb {R}}}$. Taking the alternating sum of $L_{n}\mathcal {R}$, we get a well-defined homomorphism on the level of characters. We compute this homomorphism in the case when $G_{{\mathbb {R}}}$ is split.