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 $L_i\mathcal{ R}$ 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 $L_i\mathcal{ R}$ takes unipotent representations of $G$ to unipotent representations of $G_{ \mathbb{ R}}$. Taking the alternating sum of $L_i\mathcal{ R}$, we get a well-defined homomorphism on the level of characters. We compute this homomorphism in the case when $G_{ \mathbb{ R}}$ is split.
Categories: math.RT