Let $G$ be a reductive Lie group (over $\bf C$, say), $K$ a maximal compact subgroup and $\frak g$, $\frak k$ their Lie algebras.
It is standard that Lie algebra cohomology $H^n({\frak g}, V)$ of some complex valued $\frak g$-module $V$ can be expressed as $Hom_{D(\frak g)}({\bf C}, V[n])$, the Hom-group in the derived category of $\frak g$-modules.
What is the analogue expression for $({\frak g}, K)$-cohomology, i.e., the Hom's in which (triangulated or stable $\infty$-)category yield $H^n({\frak g}, K; V)$?
