Representations having vectors fixed by a Levi subgroup associated to a real form

Abstract

For any semisimple real Lie algebra gR, we classify the representations of gR that have at least one nonzero vector on which the centralizer of a Cartan subspace, also known as the centralizer of a maximal split torus, acts trivially. In the process, we revisit the notion of g-standard Young tableaux, introduced by Lakshmibai and studied by Littelmann, that provides a combinatorial model for the characters of the irreducible representations of any classical semisimple Lie algebra g. We construct a new version of these objects, which differs from the old one for g=so(2r) and seems, in some sense, simpler and more natural.