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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 $G_{\mathbb{R}}$. Weestablish many basic properties of these functors, including their behaviorwith respect to infinitesimal character, associated variety, and restriction toa maximal compact subgroup. We deduce that each $L_i\mathcal{R}$ takesunipotent representations of $G$ to unipotent representations of$G_{\mathbb{R}}$. Taking the alternating sum of $L_i\mathcal{R}$, we get awell-defined homomorphism on the level of characters. We compute thishomomorphism in the case when $G_{\mathbb{R}}$ is split.

Restricting Representations from a Complex Group to a Real Form