Spherical Varieties

Abstract

Although the theory developed in the previous chapters applies to arbitrary homogeneous spaces of reductive groups, and even to more general group actions, it acquires its most complete and elegant form for spherical homogeneous spaces and their equivariant embeddings, called spherical varieties. A justification of the fact that spherical homogeneous spaces are a significant mathematical object is that they arise naturally in various fields, such as embedding theory, representation theory, symplectic geometry, etc. In {\textsection}25 we collect various characterizations of spherical spaces, the most important being: the existence of an open B-orbit, the ``multiplicity-free'' property for spaces of global sections of line bundles, commutativity of invariant differential operators and of invariant functions on the cotangent bundle with respect to the Poisson bracket.