About Knop's Action of the Weyl Group on the Set of Orbits of a Spherical Subgroup in the Flag Manifold

Abstract

Let G be a connected reductive algebraic group defined on an algebraically closed field k of characteristic different from 2. Let B denote the flag variety of G. Let H be a spherical subgroup of G. F. Knop defined an action of the Weyl group W of G on the finite set of the H-orbits in B. Here, we define an invariant, namely the type, separating the orbits of W .

On the Geometry of Spherical Varieties

“…Knop in [91] defines an action of the Weyl group W of G on B(X) and recovers the little Weyl group (see Subsection 3.4.1) as a subgroup of the stabiliser in W of X seen as an element of B(X). N. Ressayre gives in [131] invariants characterising the orbits of the Weyl group action on Γ(X). In [132] he gives a classification and several equivalent definitions of the spherical varieties for which W acts transitively on the B-orbits within the same G-orbit.…”

Orbits of strongly solvable spherical subgroups on the flag variety

“…Proof. As a consequence of the main theorem in [30], D δ(Ψ Φ + (w)) ⊂ δ(Ψ ∩ Φ + (w)) and the latter has cardinality at most l(w), therefore |D M| = l(w) if and only if Φ + (w) ⊂ Ψ and D δ(Ψ Φ + (w)) = δ(Φ + (w)) has cardinality l(w). Assuming ii), the claim follows by noticing that the equality D δ(Ψ Φ + (w)) = δ(Φ + (w)) is equivalent to iii), whereas the equality |δ(Φ + (w))| = l(w) is equivalent to iv).…”

“…Proof. As a consequence of the main theorem in [30], every closed B-orbit has minimal rank. Therefore every closed B-orbit in G/H is of the shape O w,∅ for some w ∈ W .…”