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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_{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. [ABSTRACT FROM AUTHOR]