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The local structure theorem for real spherical varieties

Part of: Noncompact transformation groups Special varieties Algebraic groups

Published online by Cambridge University Press:  09 July 2015

Friedrich Knop ,
Bernhard Krötz  and
Henrik Schlichtkrull
Friedrich Knop
Affiliation:
FAU Erlangen-Nürnberg, Department Mathematik, Cauerstraße 11, D-91058 Erlangen, Germany email friedrich.knop@fau.de
Bernhard Krötz
Affiliation:
Universität Paderborn, Institut für Mathematik, Warburger Straße 100, D-33098 Paderborn, Germany email bkroetz@gmx.de
Henrik Schlichtkrull
Affiliation:
University of Copenhagen, Department of Mathematics, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark email schlicht@math.ku.dk
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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.

Keywords

spherical varietieshomogeneous spacesreal reductive groups

MSC classification

Primary: 14L30: Group actions on varieties or schemes (quotients) 14M17: Homogeneous spaces and generalizations 14M27: Compactifications; symmetric and spherical varieties 22F30: Homogeneous spaces

Information

Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 11 , November 2015 , pp. 2145 - 2159
Copyright
© The Authors 2015 

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