Symmetric spaces and skew multiplicity free representations

Abstract

Let $G$ be a non-compact connected reductive real Lie group and $P$ its parabolic subgroup. We give a list of equivalent conditions for $P$ to have an abelian unipotent radical. These conditions are related to skew multiplicity free and multiplicity free actions, and using the theory of Clifford algebra and the cubic Dirac operator we deduce formulas for the de Rham cohomology of compact symmetric spaces $G/P$.