On the exactness of Kostant-Kirillov form and the second cohomology of nilpotent orbits

TL;DR

This paper provides a criterion for when the Kostant-Kirillov form on real semisimple Lie group orbits is exact and explicitly computes the second cohomology of nilpotent orbits in complex simple Lie algebras.

Contribution

It introduces a criterion for the exactness of the Kostant-Kirillov form and computes second cohomology for all nilpotent orbits in complex simple Lie algebras.

Findings

Criterion for exactness of Kostant-Kirillov form on real semisimple Lie group orbits

Explicit second cohomology calculations for nilpotent orbits in complex simple Lie algebras

Comprehensive classification results for nilpotent orbit cohomology

Abstract

We give a criterion for the Kostant-Kirillov form on an adjoint orbit in a real semisimple Lie group to be exact. We explicitly compute the second cohomology of all the nilpotent adjoint orbits in every complex simple Lie algebras.