On the little Weyl group of a real spherical space

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 Krötz. Our technique is based on a fine analysis of limits of conjugates of the subalgebra Lie(H) along one-parameter subgroups in the Grassmannian of subspaces of Lie(G). The little Weyl group is obtained as a finite reflection group generated by the reflections in the walls of the compression cone. Contents 1 Introduction 2 Notation and assumptions 3 Adapted points 4 A description of h z 5 Limits of subspaces 6 The compression cone 7 Limit subalgebras and open P -orbits 8 Limits of h z 9 Adapted points in boundary degenerations 10 Admissible points 11 The little Weyl group 12 Spherical root system 13 Reduction to quasi-affine spaces Appendix: Faces of the compression cone * JJK was funded by the Deutsche Forschungsgemeinschaft grant 262362164

The most continuous part of the Plancherel decomposition for a real spherical space

“…Adapted points have several of the properties that are listed in the local structure theorem, Proposition 3.1. The following proposition is a combination of Proposition 3.6 and Remark 3.7 (b) in [38].…”

“…The main tool for our considerations is the limit subalgebra h z,X from (1.3). Previously we used an analysis of these limit subalgebras to give a construction of the little Weyl group in [38]. We heavily rely on the results from that article for the two main results in Section 3.…”

“…Following [38] we say that a point z ∈ Z is adapted (to the Langlands decomposition P = MAN) if the following two conditions are satisfied.…”

“…See Definition 3.3 and Remark 3.4 (b) in [38]. It follows from Proposition 3.1 that every open P -orbit in Z contains an adapted point.…”

“…The Lie subalgebra a ∩ h z is the same for all adapted points z by [38,Corollary 3.17]. We denote this subalgebra by a h and refer to the dimension of a/a h as the rank of Z.…”