Isotropy representation of flag manifolds

Abstract. Flag manifolds of a classical compact Lie group G considered up to a dieomorphism are described in terms of painted Dynkin diagrams. The explicit decomposition of the isotropy representation into irreducible components is given. 1. Flag manifolds and painted Dynkin graphs We describe ag manifold of a compact semisimple Lie group G up to some equivalence relation in terms of painted Dynkin graph. De nition. (1) Flag manifold of a compact semisimple Lie group G is a quotient M = G=K of G over a subgroup K which is the centralizer of an one-parametric subgroup exp th of G, or, in other words, the homogeneous manifold G=K which is G-dieomorphic to the adjoint orbit AdGh of an element h of the Lie algebra g = LieG. (2) Two ag manifold M = G=K; M 0 =