Valuations des espaces homogènes sphériques. (Valuations of spherical homogeneous spaces).

*(French)*Zbl 0627.14038Let G be a reductive algebraic group over an algebraically closed field of characteristic zero, and let H be a closed algebraic subgroup. An ‘imbedding’ of the homogeneous space G/H is an irreducible algebraic variety X on which G acts regularly and into which G/H can be equivariantly imbedded as an open subset. Of special interest is the case where the homogeneous space G/H is ‘spherical’, that is, a Borel subgroup of G has an open orbit in G/H. In this situation, the ‘simple imbeddings’ play an important role. These are smooth spaces X on which G acts with exactly two orbits: the open orbit G/H and its complement, a closed orbit G/H’. The simple imbeddings of a spherical homogeneous space G/H are parametrized by the set \(\nu\) (G/H) of G-invariant, normalized, discrete valuations of the field of rational functions on G/H [see D. Luna and T. Vust, Comment. Math. Helv. 58, 186-245 (1983; Zbl 0545.14010)]. The set \(\nu\) (G/H) can be identified with the set of integral points in a convex rational cone \({\mathcal C}(G/H)\) contained in a finite dimensional vector space over \({\mathbb{Q}}\). The main result of the present paper is the following: If X is a simple imbedding of G/H, with closed orbit G/H’, and if v is the valuation associated to X, then G/H’ is a spherical homogeneous space and the convex cone \({\mathcal C}(G/H')\) can be realized as the quotient of \({\mathcal C}(G/H)\) by the line generated by v.

Reviewer: D.M.Snow