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

Abstract

In this note, we present some results that partially answer the following question. Let $G$ be a simple real Lie group; what is the set of representations $V$ of $G$ in which the longest element $w_0$ of the restricted Weyl group $W$ acts nontrivially on the subspace $V^L$ of $V$ formed by vectors that are invariant by $L$, the centralizer of a maximal split torus of $G$? We give a conjectural answer to that question, as well as the experimental results that back this conjecture, when $G$ is either an orthogonal group (real form of ${\mathrm{SO}}_n(\mathbb{C})$ for some $n$) or an exceptional group.

8 pages. arXiv admin note: substantial text overlap with arXiv:2002.09378

References in corpus (4)

• Proper affine actions: a sufficient criterion
• Action of Weyl group on zero weight space
• Construction of Milnorian representations
• Representations having vectors fixed by a Levi subgroup