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Towards a symplectic version of the Chevalley restriction theorem

Part of: Algebraic groups Symplectic geometry, contact geometry Linear algebraic groups and related topics General commutative ring theory

Published online by Cambridge University Press:  02 March 2017

Michael Bulois ,
Christian Lehn ,
Manfred Lehn  and
Ronan Terpereau
Michael Bulois
Affiliation:
Univ. Lyon, UJM-Saint-Etienne, CNRS UMR 5208, Institut Camille Jordan, 42023 Saint-Etienne, France email michael.bulois@univ-st-etienne.fr
Christian Lehn
Affiliation:
Fakultät für Mathematik, Technische Universität Chemnitz, Reichenhainer Straße 39, 09126 Chemnitz, Germany email christian.lehn@mathematik.tu-chemnitz.de
Manfred Lehn
Affiliation:
Institut für Mathematik, Johannes Gutenberg–Universität Mainz, 55099 Mainz, Germany email lehn@mathematik.uni-mainz.de
Ronan Terpereau
Affiliation:
Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany email ronan.terpereau@u-bourgogne.fr Current address:Institut de Mathématiques de Bourgogne - UMR 5584 du CNRS Université de Bourgogne, 9 avenue Alain Savary, BP 47870 - 21078 DIJON Cedex, France
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Abstract

If $(G,V)$ is a polar representation with Cartan subspace $\mathfrak{c}$ and Weyl group $W$, it is shown that there is a natural morphism of Poisson schemes $\mathfrak{c}\oplus \mathfrak{c}^{\ast }/W\rightarrow V\oplus V^{\ast }/\!\!/\!\!/G$. This morphism is conjectured to be an isomorphism of the underlying reduced varieties if$(G,V)$ is visible. The conjecture is proved for visible stable locally free polar representations and some other examples.

Keywords

polar representationreductive groupsymplectic reductionsymplectic variety

MSC classification

Primary: 14L30: Group actions on varieties or schemes (quotients) 53D20: Momentum maps; symplectic reduction
Secondary: 20G05: Representation theory 20G20: Linear algebraic groups over the reals, the complexes, the quaternions 13A50: Actions of groups on commutative rings; invariant theory
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 3 , March 2017 , pp. 647 - 666
Copyright
© The Authors 2017 

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