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Stratifications with respect to actions of real reductive groups

Part of:Complex spaces with a group of automorphismsAutomorphic functions

Published online by Cambridge University Press:  01 January 2008

Peter Heinzner,
Gerald W. Schwarz and
Henrik Stötzel
Peter Heinzner
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstrasse 150, D-44780 Bochum, Germany (email: heinzner@cplx.rub.de, henrik.stoetzel@rub.de)
Gerald W. Schwarz
Affiliation:
Department of Mathematics, Brandeis University, PO Box 549110, Waltham, MA 02454-9110, USA (email: schwarz@brandeis.edu)
Henrik Stötzel
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstrasse 150, D-44780 Bochum, Germany (email: heinzner@cplx.rub.de, henrik.stoetzel@rub.de)
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Abstract

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We study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of G extends holomorphically to an action of the complexified group $G^{\mathbb {C}}$ and that with respect to a compatible maximal compact subgroup U of $G^{\mathbb {C}}$ the action on Z is Hamiltonian. There is a corresponding gradient map $\mu _{\mathfrak {p}}\colon X\to \mathfrak {p}^*$ where $\mathfrak {g}=\mathfrak {k}\oplus \mathfrak {p}$ is a Cartan decomposition of $\mathfrak {g}$. We obtain a Morse-like function $\eta _{\mathfrak {p}}:=\Vert \mu _{\mathfrak {p}}\Vert ^2$ on X. Associated with critical points of $\eta _{\mathfrak {p}}$ are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds Sβ of X which are called pre-strata. In cases where $\mu _{\mathfrak {p}}$ is proper, the pre-strata form a decomposition of X and in cases where X is compact they are the strata of a Morse-type stratification of X. Our results are generalizations of results of Kirwan obtained in the case where $G=U^{\mathbb {C}}$ and X=Z is compact.

Keywords

Cartan decompositionHamiltonian actionmoment mapstratification

MSC classification

Secondary:32M05: Complex Lie groups, automorphism groups acting on complex spaces

Information

Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 1 , January 2008, pp. 163 - 185
Copyright
Copyright © Foundation Compositio Mathematica 2008

References

The first author is partially supported by the Sonderforschungsbereich SFB/TR12 of the Deutsche Forschungsgemeinschaft. The second author is partially supported by NSA Grant H98230-06-1-0023. The third author is supported by the Sonderforschungsbereich SFB/TR12 of the Deutsche Forschungsgemeinschaft.