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 ${\mathrm{H}}^{\bullet }(\mathfrak{u},V)$ and the Lie algebra homology space ${\mathrm{H}}_{\bullet }(\mathfrak{u},V)$ are both Hausdorff, where $\mathfrak{u}$ is the nilpotent radical of a parabolic subalgebra of the complexified Lie algebra $\mathfrak{g}$ of $G$.