The Luna-Vust Theory of Spherical Embeddings

Abstract

The work of Krämer, Luna, Vust, Brion and others [Krä,LV,BLV,BP] established the importance of a very distinguished class of homogeneous varieties G/H, those which are now called spherical. Such varieties are homogeneous for a connected reductive group G and are characterized by many equivalent properties, the most important being (see [BLV]):-- Any Borel subgroup B of G has an open orbit in G/H.-- Every equivariant completion of G/H contains only finitely many orbits.-- For every irreducible G-module V and any character O / of H dim {v 2 V | hv = O/(h)v for all h 2 H} < = 1. Because of the last property spherical varieties are sometimes called "multiplicity free". This class of varieties includes some of the most important spaces, e. g.-- Tori, i. e. G is a torus and H is trivial;-- Symmetric varieties, i. e. H is the set of fixed points of an automorphism of G of order two.-- Horospherical varieties, i. e. H contains a maximal unipotent subgroup of G (for