1. Towards a symplectic version of the Chevalley restriction theorem
- Author
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Ronan Terpereau, Manfred Lehn, Christian Lehn, Michael Bulois, Algèbre, géométrie, logique ( AGL ), Institut Camille Jordan [Villeurbanne] ( ICJ ), École Centrale de Lyon ( ECL ), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon-Institut National des Sciences Appliquées de Lyon ( INSA Lyon ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -Centre National de la Recherche Scientifique ( CNRS ) -École Centrale de Lyon ( ECL ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -Centre National de la Recherche Scientifique ( CNRS ), Fakultät für Mathematik der Technischen Universität Chemnitz, Fakultaet fuer Mathematik der Technischen Universitaet Chemnitz, Institut für Mathematik ( MI ), Johannes Gutenberg - Universität Mainz ( JGU ), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Max Planck Institute for Mathematics ( Bonn ), The second-named author gratefully acknowledges the support by the DFG through the research grant Le 3093/2-1. The thirdnamed author is supported by the SFB Transregio 45 'Periods, Moduli Spaces and Arithmetic of Algebraic Varieties'. The fourth-named author is grateful to the Max-Planck-Institut für Mathematik of Bonn for the warm hospitality and support provided during the writing of this paper. Part ofthe genesis of this paper occurred during the half semester Algebraic Groups and Representations supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program 'Investissements d'Avenir' (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR)., ANR-11-IDEX-0007-02/10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon ( 2011 ), ANR-15-CE40-0012,GeoLie,GeoLie, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), Chemnitz University of Technology / Technische Universität Chemnitz, Institut für Mathematik (MI), Johannes Gutenberg - Universität Mainz (JGU), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Max Planck Institute for Mathematics (MPIM), Max-Planck-Gesellschaft, ANR-10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon(2010), and ANR-15-CE40-0012,GéoLie,Méthodes géométriques en théorie de Lie(2015)
- Subjects
Polar representation ,Symplectic variety ,[ MATH.MATH-SG ] Mathematics [math]/Symplectic Geometry [math.SG] ,01 natural sciences ,Combinatorics ,Mathematics - Algebraic Geometry ,symbols.namesake ,Morphism ,0103 physical sciences ,FOS: Mathematics ,14L30, 53D20, 20G05, 20G20, 13A50 ,Representation Theory (math.RT) ,0101 mathematics ,MSC: 14L30, 53D20, 20G05, 20G20, 13A50 ,Mathematics::Representation Theory ,Symplectic reduction ,Algebraic Geometry (math.AG) ,Moment map ,Mathematics ,Weyl group ,Algebra and Number Theory ,Conjecture ,[MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT] ,Reductive group ,010102 general mathematics ,[MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG] ,[ MATH.MATH-RT ] Mathematics [math]/Representation Theory [math.RT] ,[ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG] ,Mathematics - Symplectic Geometry ,symbols ,Symplectic Geometry (math.SG) ,010307 mathematical physics ,Isomorphism ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Mathematics - Representation Theory ,Symplectic geometry - 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 $\mathfrak c \oplus {\mathfrak c}^*/W \to V\oplus V^*/\!\!/\!\!/ G$. 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 certain further examples., Comment: 18 pages, final version
- Published
- 2017