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

Abstract

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.