Actions of relative Weyl groups I, II

Actions of relative Weyl group I

We construct a new isomorphism between the endomorphism algebra of an induced cuspidal character sheaf and the group algebra of the relative Weyl group involved. We show it differs from Lusztig one by a linear character, and we relate this linear character to some stabilizers. Some consequences on characteristic functions of character sheaves are obtained. In the forthcoming part II, we will apply these methods to obtain a precision on Digne, Lehrer and Michel's theorem on Lusztig restriction of Gel'fand-Graev characters.

限制Weyl群在表示的$L$-不变向量上的作用

Action of the restricted Weyl group on the $L$-invariant vectors of a representation

本文是一项研究简报，综述了针对实李群$G$，确定限制Weyl群最长元$w_0$在$L$-不变向量子空间上作用非平凡的表示集合的相关近期工作。

Action of Weyl group on zero weight space

For any simple complex Lie group we classify irreducible finite-dimensional representations $ρ$ for which the longest element $w_0$ of the Weyl group acts nontrivially on the zero weight space. Among irreducible representations that have zero among their weights, $w_0$ acts by $\pm$Id if and only if the highest weight of $ρ$ is a multiple of a fundamental weight, with a coefficient less than a bound that depends on the group and on the fundamental weight.

Strongly and Weyl transitive group actions on buildings arising from Chevalley groups

Let K be a field and g(K) a Chevalley group (scheme) over K. Let (B,N) be the standard spherical BN-pair in g(K), with T=B\cap N and Weyl group W=N/T. We prove that there exist non-trivial elements w\in W such that all representatives of w in N have finite order. This allows us to exhibit examples of subgroups of g(Q_p) that act Weyl transitively but not strongly transitively on the affine building Delta associated with g(Q_p). Such examples were previously known only in the case when g(Q_p)=SL_2(Q_p) and Delta is a tree.

Schur--Weyl duality over commutative rings

The classical case of Schur--Weyl duality states that the actions of the group algebras of $GL_n$ and $S_d$ on the $d^{th}$-tensor power of a free module of finite rank centralize each other. We show that Schur--Weyl duality holds for commutative rings where enough scalars can be chosen whose non-zero differences are invertible. This implies all the known cases of Schur--Weyl duality so far. We also show that Schur--Weyl duality fails for $\mathbb{Z}$ and for any finite field when $d$ is sufficiently large.

About Knop's action of the Weyl group on the set of the set of orbits of a spherical subgroup in the flag manifold

Let $G$ be a complex connected reductive algebraic group. Let $G/B$ denote the flag variety of $G$. Let $H$ be an algebraic subgroup of $G$ such that the set ${\bf H}(G/B)$ of the $H$-orbits in $G/B$ is finite ; $H$ is said to be {\it spherical}. These orbits are of importance in representation theory and in the geometry of the $G$-equivariant embeddings of $G/H$. In 1995, F. Knop has defined an action of the Weyl group $W$ of $G$ on ${\bf H}(G/B)$. The aim of this note is to construct natural invariants separating the $W$-orbits of Knop's action.