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We study cohomological induction for a pair ,  being an infinitedimensional locally reductive Lie algebra and  being of the form , where  is a finite-dimensional reductive in  subalgebra and  is the centralizer of  in . We prove a general nonvanishing and -finiteness theorem for the output. This yields, in particular, simple -modules of finite type over k which are analogs of the fundamental series of generalized Harish-Chandra modules constructed in [PZ1] and [PZ2]. We study explicit versions of the construction when  is a root-reductive or diagonal locally simple Lie algebra.

A Construction of Generalized Harish-Chandra Modules for Locally Reductive Lie Algebras