The Isotropy Representation of a Real Flag Manifold: Split Real Forms

TL;DR

This paper investigates the isotropy representation of real flag manifolds linked to split real forms of complex simple Lie algebras, revealing non-uniqueness in their decomposition into invariant subspaces.

Contribution

It characterizes the invariant irreducible subspaces for each Dynkin diagram and highlights the non-uniqueness of decomposition in real flag manifolds.

Findings

Invariant subspaces are described for each Dynkin diagram.

Decomposition into irreducible components is not always unique.

Some cases have infinitely many invariant subspaces.

Abstract

We study the isotropy representation of real flag manifolds associated to simple Lie algebras that are split real forms of complex simple Lie algebras. For each Dynkin diagram the invariant irreducible subspaces for the compact part of the isotropy subgroup are described. Contrary to the complex flag manifolds the decomposition into irreducible components is not in general unique, since there are cases with infinitely many invariant subspaces.