Tensor Products of Finite and Infinite Dimensional Representations of Semisimple Lie Groups

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study of the category of Harish-Chandra modules over a real semisimple Lie algebra equipped with a Cartan involution. In this paper we prove some very general theorems about tensor products of finite dimensional modules with Harish-Chandra modules. These theorems yield new structural information about the category of Harish-Chandra modules, as well as new information on the classification, construction, and properties of the infinite dimensional irreducible Harish-Chandra modules. We study the Harish-Chandra modules associated to a connected semisimple Lie group G having finite center. We let q, be the Lie algebra of G,