Michel Brion

Quelques proprietes des espaces homogenes spheriques

Let G be a connected, reductive, algebraic group on an algebraically closed field k of characteristic zero. Let H be aspherical subgroup of G, i.e. H is a closed subgroup of G such that every Borel subgroup of G operates on G/H with an open orbit.It is shown that for a spherical subgroup H, the homogeneous space G/H is a deformation of a homogeneous space G/H0, where H0 contains a maximal unipotent subgroup of G (such a H0 is spherical). It is also shown that every Borel subgroup of G has a finite number of orbits in G/H.

Arrangement of hyperplanes, II: The Szenes formula and Eisenstein series

A motivation for computing such sums comes from the work of E. Witten [4]. In the special case where αj are the positive roots of a compact connected Lie group G, each of these roots being repeated with multiplicity 2 g − 2, Witten expressed the symplectic volume of the space of homomorphisms of the fundamental group of a Riemann surface of genus g into G, in terms of these sums. In [2], L. Jeffrey and F. Kirwan proved a special case of the Szenes formula leading to the explicit computation of this symplectic volume, when G is SU(n). Our interest in such series comes from a different motivation. Let us consider first the 1-dimensional case. By the Poisson formula, for Re (z) > 0, the convergent series ∑∞ m=1me−mz is also equal to ∑ n∈Z1/(z+2iπn). Similarly, sums of products

Spherical Varieties an Introduction

When one studies complex algebraic homogeneous spaces it is natural to begin with the ones which are complete (i.e. compact) varieties. They are the “generalized flag manifolds”. Their occurence in many problems of representation theory, algebraic geometry, … make them an important class of algebraic varieties. In order to study a noncompact homogeneous space G/H, it is equally natural to compactify it, i.e. to embed it (in a G- equivariant way) as a dense open set of a complete G-variety. A general theory of embeddings of homogeneous spaces has been developed by Luna and Vust [LV]. It works especially well in the so-called spherical case: G is reductive connected and a Borei subgroup of G has a dense orbit in G/H. (This class includes complete homogeneous spaces as well as algebraic tori and symmetric spaces). A nice feature of a spherical homogeneous space is that any embedding of it (called a spherical variety) contains only finitely many G-orbits, and these are themselves spherical. So we can hope to describe these embeddings by combinatorial invariants, and to study their geometry. I intend to present here some results and questions on the geometry (see [LV], [BLV], [BP], [Lun] for a classification of embeddings).