Let $R$ be a reductive, finite-dimensional Lie algebra over a field of characteristic 0, and let $V$ be a semisimple $R$-module (also finite dimensional). I have seen a reference to the fact that $H^i(R,V) \cong H^i(R,V^R)$, which seems like a stronger version of Theorem 10 of [Hochschild and Serre's paper](https://www.jstor.org/stable/1969740?seq=9). The only reference I can find is to an out-of-print book in Romanian, "Introducer in Coomologia Algebrelor Lie," by Verona. Is there an English reference for this fact? Thanks!



Lie algebra cohomology: $H^i(R,V)=H^i(R,V^R)$ with $R$ reductive and $V$ an $R$-module
(This is meant to be a comment, but I don't have enough reputations) If I'm not mistaken, this follows almost directly from the paper you cited. By semisimplicity of $V$, it suffices to prove for irreducible $V$. If $V$ is not the trivial module then $V^R =0$ and the statement follows from Theorem 10 of the paper. If $V$ is trivial then the statement is obvious.


