Consider a connected reductive algebraic group $G$ over an algebraically closed field $k$, a Borel subgroup $B\subset G$, and a closed subgroup $H\subset G$. The homogeneous space $G/H$ is spherical if $B$ acts on it with a dense orbit. Examples include flag varieties ($H$ is parabolic in $G$) and symmetric spaces ($H$ is the fixed point subgroup of an involutive automorhpism of $G$).
More genreally, define a spherical variety as a normal algebraic variety with an action of $G$ and a dense orbit of $B$. This rather abstract notion uncovers a very rich geometry, which is only partially understood. It combines features of flag varieties (e.g. the Bruhat decomposition and the Borel-Weil-Bott theorem) and of symmetric spaces (the little Weyl group and its role in equivariant embeddings; the Harish-Chandra isomorphism). Ideas and methods from the theory of toric varieties play a role, too.
Finally, spherical varieties are a test case for studying actions of reductive groups. Namely, several phenomena, first discovered for spherical varieties, have been generalized by F. Knop to arbitrary varieties with reductive group actions. Knop's work relies on the study of the moment map.
In this talk, we survey two aspects of spherical homogeneous spaces: finiteness properties, and structure of embeddings.
1. Finiteness properties of spherical varieties
Define an embedding of the homogeneous space $G/H$ as a normal algebraic variety $X$ with an action of $G$, and a dense $G$-orbit isomorphic to $G/H$. Then any embedding of a spherical homogeneous space contains only finitely many orbits of $G$, and even of $B$ 9see [Ma] for a simple proof of the last point). But any non-spherical homogeneous space has an embedding with infinitely many $G$-orbits.
For a spherical $G/H$, the (finite) set of double classes $B\sbs G/H$ is endowed with the action of a certain monoid derived from the Weyl group of $G$. In the case of a symmetric space, this action has been studied by R. Richardson and T. Springer, who described the Bruhat order on $B\sbs G/H$; see [RS]. On the other hand, an action of the Weyl group of $G$ on $B\sbs G/H$ has been defined by Knop, building on work of Lusztig and Vogan (see [K6]).
Another finiteness property is a qualitative generalization of the Borel-Weil-Bott theorem; it holds in characteristic zero. Observe that, for any spherical variety $X$ and any $G$-line bundle $L$ over $X$, the space of sections $H^0(X,L)$ is a multiplicity-free $G$-module. Moreover, such a property characterizes spherical varieties among quasi-projective $G$-varieties (see [Bi]). More generally, for any spherical variety $X$, there exists a constant $C(X)$ such that the multiplicities of all cohomology spaces of any $G$-vector bundle $E$ are at most $C(X)$ rank$(E)$ (see [B3]). In particular, the cohomology spaces of $G$-line bundles on a given spherical variety have uniformly bounded multiplicities; however, they are not multiplicity-free in general.
Finally, other finiteness properties occur when studying spherical varieties from the point of view of birational geometry; in particular, the ``cone of curves'' of a complete spherical variety can be described in a precise way, see [B2] and [B4].
2. Structure of embeddings of spherical homogeneous spaces
A theory of embeddings of an arbitrary homogeneous space $G/H$ has been set up by D. Luna and T. Vust. A basic role is played by the set ${\scr V}$ of all $G$-invariant discrete valuations of the function field of $G/H$. In the spherical case, ${\scr V}$ can be identified with a convex polyhedral cone. If moreover the index of $H$ in its normalizer $N_G(H)$ is finite, then ${\scr V}$ is a fundamental domain of a finite reflection group $W_{G/H}$; see [B1], and [K2] for a generalization. In this case, by the theory of Luna-Vust, there is a canonical embedding $X$ of $G/H$; namely, among all complete embeddings with only one closed orbit, there is one which dominates all others.
For a symmetric space $G/H$, these objects turn out to be already familiar. Namely $W_{G/H}$ is the little Weyl group; moreover, $X$ is the "wonderful compactification" constructed and studied by C. DeConcini and C. Procesi. In particular, if $k$ has characteristic zero and if $H=N_G(H)$, then $X$ is smooth, and points of $X$ are limits of $G$-conjugates of the Lie algebra of $H$. Similar results, with much more technical proofs, hold for all spherical homogeneous spaces $G/H$ with $H=N_G(H)$; see [B1] and [K5]. The geometry of the canonical embedding is used by D. Luna to classify spherical homogeneous spaces by combinatorial invariants, which generalize the Satake diagrams of symmetric spaces (work in progress, 1994).
There is a surprising connection between the little Weyl group $W_{G/H}$ of a spherical homogeneous space, and the action of the big Weyl group $W$ on the set of $B$-orbits in $G/H$. Namely, denote by $\Omega$ the open $B$-orbit in $G/H$, and by $P$ the set of all $g\in G$ such that $g\cdot\Omega=\Omega$ (a parabolic subgroup of $G$). Then the $W$-isotropy subgroup of $\Omega$ is the semi-direct product of $W_{G/H}$ with the Weyl group of $P$ (see [K6]).
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