A construction of generalized Harish-Chandra modules for locally reductive Lie algebras

Abstract

We study cohomological induction for a pair $(\mathfrak{g},\mathfrak{k})$, $\mathfrak{g}$ being an infinite dimensional locally reductive Lie algebra and $\mathfrak{k}\subset \mathfrak{g}$ being of the form ${\mathfrak{k}}_0+C_{\gg }({\mathfrak{k}}_0)$, where ${\mathfrak{k}}_0\subset \mathfrak{g}$ is a finite dimensional reductive in $\mathfrak{g}$ subalgebra and $C_{\gg }({\mathfrak{k}}_0)$ is the centralizer of ${\mathfrak{k}}_0$ in $\mathfrak{g}$. We prove a general non-vanishing and $\mathfrak{k}$-finiteness theorem for the output. This yields in particular simple $(\mathfrak{g},\mathfrak{k})$-modules of finite type over $\mathfrak{k}$ which are analogs of the fundamental series of generalized Harish-Chandra modules constructed in \cite{PZ1} and \cite{PZ2}. We study explicit versions of the construction when $\mathfrak{g}$ is a root-reductive or diagonal locally simple Lie algebra.