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

We study cohomological induction for a pair $(\mathfrak{g},\mathfrak{k})$, $\mathfrak{g}$ being an infinite dimensional locally reductive Lie algebra and $\frak k \subset\frak g$beingoftheform$\frak k_0 + C_\gg(\frak k_0)$,where$\frak k_0\subset\frak g$isafinitedimensionalreductivein$\frak g$ 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.