Cohomology of real Lie algebras

Abstract

We show how to describe the cohomology of a nilpotent part of some parabolic subalgebra of a semisimple Lie algebra with values in its irreducible representation. The situation in the complex case is well--known, the Kostant's result (see below) gives an explicit description of a representation of a proper reductive subalgebra on the space of the complex cohomology. The aim of this work is to read the structure of the real cohomology from the structure of the complex one. We will use the notation of Dynkin and Satake diagrams for the description of semisimple and parabolic real and complex Lie algebras and their representations.