Restricting Representations from a Complex Group to a Real Form

TL;DR

This paper develops a sequence of functors that restrict complex group representations to real forms, preserving key properties and unipotent representations, and computes their effect on characters, especially for split real forms.

Contribution

It introduces new functors for restricting complex group representations to real forms, analyzing their properties and effects on unipotent representations and characters.

Findings

Functors preserve admissibility and finite length.

Unipotent representations are mapped to unipotent representations.

Explicit character homomorphism computed for split real forms.

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 $L_i\mathcal{R}$ 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.