I am trying to get acquainted with various infinite dimensional representations of Lie groups. So a general reference would be appreciated. Right now I am trying to figure out the following question.

Consider a real Lie group $G$ and its maximal compact subgroup $K$. Let $P$ be a parabolic subgroup of the complexification of $G$ such that its Levi factor is a complexification of lie algebra of $K$. What is the relation between category of $(\mathfrak{g},K)$-modules and the parabolic category $\mathcal{O}_P$?

Is there a canonical way to extend the representation of $K$ on the $(\mathfrak{g},K)$-module to a representation of $P$ at least in some special cases (for example in hermitian symmetric setting)?

Of course one can drop the condition on $K$ being the maximal compact subgroup and ask basically the same thing.