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

We study cohomological induction for a pair (g, k), g being an infinite dimensional locally reductive Lie algebra and k ⊂ g being of the form k0 + Cg(k0), where k0 ⊂ g is a finite dimensional reductive in g subalgebra and Cg(k0) is the centralizer of k0 in g. We prove a general non-vanishing and k-finiteness theorem for the output. This yields in particular simple (g, k)-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 g is a root-reductive or diagonal locally simple Lie algebra. (2000 MSC): Primary 17B10, Secondary 17B55