Representations having vectors fixed by a Levi subgroup

Abstract

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.

63 pages, 8 tables, 3 figures. Made some minor edits (including referee suggestions), added journal reference and added "version information" paragraph

References in corpus (4)

• Proper affine actions: a sufficient criterion
• Proper affine actions for right-angled Coxeter groups
• Action of Weyl group on zero weight space
• Construction of Milnorian representations

Cited by in corpus (1)

• Action of $w_0$ on $V^L$ for orthogonal and exceptional groups