Quotients of group completions by spherical subgroups

✍ Scribed by Nicolas Ressayre

• Book ID: 104140697
• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 393 KB
• Volume: 265
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00227-8

✦ Synopsis

Let G be a semi-simple algebraic group and let H be a spherical subgroup. The ground field k is algebraically closed and of characteristic zero. This article is concerned with projective embeddings Y of spherical homogeneous spaces G/H . Our approach in the study of such a variety Y is to realize them as quotients under the action of H of projective embeddings of G. First, we give a more precise sense to this project by defining the quotient of a G-variety by a spherical subgroup H . Then, we give a condition, in terms of G-invariant valuations, under which Y can be obtained by quotient of an embedding of G. Finally, if the index of H in its normalizer is finite, we show that an important class of embeddings of G/H (toroidal and liftable) geometric quotients of embeddings of G.