On the little Weyl group of a real spherical space

TL;DR

This paper explores the geometric construction of the little Weyl group associated with a real spherical space, revealing its structure as a finite reflection group generated by cone walls.

Contribution

It provides a new geometric method to construct the little Weyl group of a real spherical space, building on recent theoretical developments.

Findings

The little Weyl group is a finite reflection group.

The construction is based on analyzing limits of conjugates of Lie subalgebras.

The approach links the Weyl group structure to the geometry of the compression cone.

Abstract

In the present paper we further the study of the compression cone of a real spherical homogeneous space $Z=G/H$. In particular we provide a geometric construction of the little Weyl group of $Z$ introduced recently by Knop and Kr\"otz. Our technique is based on a fine analysis of limits of conjugates of the subalgebra $\mathrm{Lie}(H)$ along one-parameter subgroups in the Grassmannian of subspaces of $\mathrm{Lie}(G)$. The little Weyl group is obtained as a finite reflection group generated by the reflections in the walls of the compression cone.