Hausdorffness of certain nilpotent cohomology spaces

Abstract

Let $(\pi ,V)$ be a smooth representation of a compact Lie group G on a quasi-complete locally convex complex topological vector space. We show that the Lie algebra cohomology space $H^{•}(u,V)$ and the Lie algebra homology space $H_{•}(u,V)$ are both Hausdorff, where $u$ is the nilpotent radical of a parabolic subalgebra of the complexified Lie algebra $g$ of G.