Hausdorffness of certain nilpotent cohomology spaces

TL;DR

This paper proves that certain nilpotent Lie algebra cohomology and homology spaces associated with smooth representations of compact Lie groups are Hausdorff, ensuring well-behaved topological properties in these algebraic structures.

Contribution

It establishes the Hausdorff property for Lie algebra cohomology and homology spaces in the context of smooth group representations, a result not previously confirmed.

Findings

Cohomology spaces are Hausdorff.

Homology spaces are Hausdorff.

Results apply to nilpotent radicals of parabolic subalgebras.

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 $G$.