  We study cohomological induction for a pair $(\frak g,\frak k)$, $\frak g$
being an infinite dimensional locally reductive Lie algebra and $\frak k
\subset\frak g$ being of the form $\frak k_0 + C_\gg(\frak k_0)$, where $\frak
k_0\subset\frak g$ is a finite dimensional reductive in $\frak g$ subalgebra
and $C_{\gg} (\frak k_0)$ is the centralizer of $\frak k_0$ in $\frak g$. We
prove a general non-vanishing and $\frak k$-finiteness theorem for the output.
This yields in particular simple $(\frak g,\frak k)$-modules of finite type
over $\frak k$ 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 $\frak g$ is a root-reductive or
diagonal locally simple Lie algebra.
