Homotopy type of the nilpotent orbits in classical Lie algebras

TL;DR

This paper extends the understanding of the homotopy types of nilpotent orbits in all real simple classical Lie algebras, completing previous classifications by considering cases with semisimple maximal compact subgroups.

Contribution

It provides a comprehensive description of the homotopy types of nilpotent orbits in all real simple classical Lie algebras, including previously unaddressed cases.

Findings

Explicit descriptions of homotopy types for all real simple classical Lie algebras.

Extension of previous results to cases with semisimple maximal compact subgroups.

Completes the classification of homotopy types of nilpotent orbits in classical Lie algebras.

Abstract

In the paper "The Second cohomology of nilpotent orbits in classical Lie algebras, Kyoto J. Math. 60 (2020), no. 2, 717-799" by I. Biswas, P. Chatterjee and C. Maity homotopy types of nilpotent orbits are explicitly described in the case of real simple classical Lie algebras for which any maximal compact subgroup in the associated adjoint group is not semisimple. In this paper we extend the above description of homotopy type of nilpotent orbits to the remaining cases of real simple classical Lie algebras for which any maximal compact subgroup in the associated adjoint group is semisimple.