The second cohomology groups of nilpotent orbits in classical Lie algebras

Abstract

The second de Rham cohomology groups of nilpotent orbits in noncompact real forms of classical complex simple Lie algebras are explicitly computed. Furthermore, the first de Rham cohomology groups of nilpotent orbits in noncompact classical simple Lie algebras are computed, and we prove them 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.