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

We study cohomological induction for a pair $ {left( {mathfrak{g},mathfrak{k}} right)} $ , $ mathfrak{g} $ being an infinitedimensional locally reductive Lie algebra and $ mathfrak{k} subset mathfrak{g} $ being of the form $ mathfrak{k}_{0} subset C_{mathfrak{g}} {left( {mathfrak{k}_{0} } right)} $ , where $ mathfrak{k}_{0} subset mathfrak{g} $ is a finite-dimensional reductive in $ mathfrak{g} $ subalgebra and $ C_{mathfrak{g}} {left( {mathfrak{k}_{0} } right)} $ is the centralizer of $ mathfrak{k}_{0} $ in $ mathfrak{g} $ . We prove a general nonvanishing and $ mathfrak{k} $ -finiteness theorem for the output. This yields, in particular, simple $ {left( {mathfrak{g},mathfrak{k}} right)} $ -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 $ mathfrak{g} $ is a root-reductive or diagonal locally simple Lie algebra.