Representations having vectors fixed by a Levi subgroup

For any semisimple real Lie algebra $\mathfrak{g}_\mathbb{R}$, we classify the representations of $\mathfrak{g}_\mathbb{R}$ 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 $\mathfrak{g}$-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 $\mathfrak{g}$. We construct a new version of these objects, which differs from the old one for $\mathfrak{g} = \mathfrak{so}(2r)$ and seems, in some sense, simpler and more natural.