The second cohomology groups of nilpotent orbits in classical Lie algebras

TL;DR

This paper explicitly computes the second and first de Rham cohomology groups of nilpotent orbits in non-compact classical Lie algebras, revealing new insights into their topological structure and generalizing previous theorems.

Contribution

It provides explicit calculations of cohomology groups for nilpotent orbits and introduces a generalized description of cohomology of homogeneous spaces, extending prior results.

Findings

Second cohomology groups computed for nilpotent orbits in non-compact classical Lie algebras

First cohomology groups shown to be zero for all nilpotent orbits in complex simple Lie algebras

A generalized theorem describing cohomology of homogeneous spaces is established

Abstract

The second de Rham cohomology groups of nilpotent orbits in non-compact real forms of classical complex simple Lie algebras are explicitly computed. Furthermore, the first de Rham cohomology groups of nilpotent orbits in non-compact classical simple Lie algebras are computed; they are proven to be zero for nilpotent orbits in all the complex simple Lie algebras. A key component in these computations is a description of the second and first cohomology groups of homogeneous spaces of general connected Lie groups which is obtained here. This description, which generalizes a previous theorem of the first two authors, may be of independent interest.