The local structure theorem for real spherical varieties. (9th July 2015)

• Record Type: Journal Article
• Authors: Knop, Friedrich; Krötz, Bernhard; Schlichtkrull, Henrik
• Is Part Of: Compositio mathematica. Volume 151:Number 11(2015)
• Page Start: 2145
• Page End: 2159
• Publication Date: 2015-07-09
• Subjects: 14L30, -- 14M17, -- 14M27, -- 22F30 (primary); spherical varieties, -- homogeneous spaces, -- real reductive groups; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X15007307
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : Let $G$ be an algebraic real reductive group and $Z$ a real spherical $G$ -variety, that is, it admits an open orbit for a minimal parabolic subgroup $P$ . We prove a local structure theorem for $Z$ . In the simplest case where $Z$ is homogeneous, the theorem provides an isomorphism of the open $P$ -orbit with a bundle $Q\times _{L}S$ . Here $Q$ is a parabolic subgroup with Levi decomposition $L\ltimes U$, and $S$ is a homogeneous space for a quotient $D=L/L_{n}$ of $L$, where $L_{n}\subseteq L$ is normal, such that $D$ is compact modulo center.