A Real Analog of Kostant's Version of the Bott-Borel-Weil Theorem

We show how to describe the cohomology of the nilradical of a parabolic subalgebra a semisimple Lie algebra with coefficients in an irreducible representation of g.The situation in the complex case is well-known, 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 determine 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.