Automorphisms, root systems, and compactifications of homogeneous varieties

Abstract

Let G be a complex semisimple group and let H C G be the group of fixed points of an involutive automorphism of G. Then X = G/H is called a symmetric variety. In [CP], De Concini and Procesi have constructed an equivariant compactification X which has a number of remarkable properties, some of them being: i) The boundary is the union of divisors D1,... , Dr. ii) There are exactly 2r orbits. Their closures are the intersections Di1 n .nDi (even schematically). In particular, there is only one closed orbit. iii) In case G is of adjoint type, all orbit closures are smooth. It is called the wonderful embedding of X or a complete symmetric variety and is the foundation for most deeper results about X. Independently, Luna and Vust developed in [LV] a general theory of equivariant compactifications of homogeneous varieties under a connected reductive group G. In particular, they realized the reason which makes symmetric varieties behave so nicely: A Borel subgroup B has an open dense orbit in G/H. Varieties with this property are called spherical. Luna and Vust were able to describe all equivariant compactifications of them in terms of combinatorial data, very similar to torus embeddings which are actually a special case. They obtained in particular that every spherical embedding has only finitely many orbits. Nevertheless, the reason for the existence of a compactification with properties i)-iii) remained mysterious. Then Brion and Pauer established a relation with the automorphism group. They proved in [BP]: A spherical variety X = G/H possesses an equivariant compactification with exactly one closed orbit if and only if AutG X = NG (H)/H is finite. In this case there is a unique one which dominates all others: the wonderful compactification X. They also showed that the orbits of X correspond to the faces of a strictly convex polyhedral cone Z. Then properties i) and ii) above are equivalent to Z being simplicial. This fact is much deeper and was proved by Brion in [Brl]. In fact he showed much more. Let F be the set of characters of B which are the characters of a rational B-eigenfunction on X. This is a finitely generated free abelian group. Then the cone Z is a subset of the real vector space Hom(F, R). Brion showed that there is a finite reflection group Wx acting on F such that Z is one of its Weyl chambers. In case of a symmetric variety, Wx is its little Weyl group.