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Slodowy slices and universal Poisson deformations

Part of: Local theory Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 November 2011

M. Lehn ,
Y. Namikawa  and
Ch. Sorger
M. Lehn
Affiliation:
Fachbereich Physik, Mathematik u. Informatik, Johannes Gutenberg–Universität Mainz, D-55099 Mainz, Germany (email: lehn@mathematik.uni-mainz.de)
Y. Namikawa
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kitashirakawa-Oiwakecho, Kyoto, 606-8502, Japan (email: namikawa@math.kyoto-u.ac.jp)
Ch. Sorger
Affiliation:
Laboratoire de Mathématiques Jean Leray (UMR 6629 du CNRS), Université de Nantes, 2, Rue de la Houssinière, BP 92208, F-44322 Nantes Cedex 03, France (email: christoph.sorger@univ-nantes.fr)
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Abstract

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We classify the nilpotent orbits in a simple Lie algebra for which the restriction of the adjoint quotient map to a Slodowy slice is the universal Poisson deformation of its central fibre. This generalises work of Brieskorn and Slodowy on subregular orbits. In particular, we find in this way new singular symplectic hypersurfaces of dimension four and six.

Keywords

nilpotent orbitssymplectic singularitiessymplectic hypersurfacesPoisson deformations

MSC classification

Secondary: 14B07: Deformations of singularities 17B45: Lie algebras of linear algebraic groups 17B63: Poisson algebras

Information

Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 1 , January 2012 , pp. 121 - 144
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[BC76]Bala, P. and Carter, R. W., Classes of unipotent elements in simple algebraic groups, Math. Proc. Cambridge Philos. Soc. 79 (1976), 401–425 and 80 (1976), 1–17.CrossRefGoogle Scholar
[Bea00]Beauville, A., Symplectic singularities, Invent. Math. 139 (2000), 541–549.CrossRefGoogle Scholar
[Bou81]Bourbaki, N., Groupes et Algèbres de Lie (Masson, Paris, 1981).Google Scholar
[BMO11]Braverman, A., Maulik, D. and Okounkov, A., Quantum cohomology of the Springer resolution, Adv. Math. 227 (2011), 421–458.CrossRefGoogle Scholar
[Bri71]Brieskorn, E., Singular elements of semisimple algebraic groups, in Actes Congrès Intern. Math., Nice, 1970 (Gauthier Villars, Paris, 1971), vol. 2, 279–284.Google Scholar
[CM93]Collingwood, D. H. and McGovern, W. M., Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series (Van Nostrand Reinhold, New York, NY, 1993).Google Scholar
[CLP88]De Concini, C., Lusztig, G. and Procesi, C., Homology of the zero-set of a nilpotent vector field on a flag manifold, J. Amer. Math. Soc. 1 (1988), 15–34.CrossRefGoogle Scholar
[FH91]Fulton, W. and Harris, J., Representation theory, Graduate Texts in Mathematics, vol. 129 (Springer, Berlin, 1991).Google Scholar
[GG02]Gan, W. L. and Ginzburg, V., Quantization of slodowy slices, Int. Math. Res. Not. IMRN (2002), 243–255.CrossRefGoogle Scholar
[GK04]Ginzburg, V. and Kaledin, D., Poisson deformations of symplectic quotient singularities, Adv. Math. 186 (2004), 1–57.CrossRefGoogle Scholar
[GPS01]Greuel, G.-M., Pfister, G. and Schönemann, H., Singular 3-0-4, A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern (2001), http://www.singular.uni-kl.de.Google Scholar
[Gro68]Grothendieck, A., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, in Séminaire de Géométrie Algébrique du Bois-Marie 1962 (SGA 2) (North-Holland, Amsterdam, 1968).Google Scholar
[HL91]Hamm, H. and Lê, D. T., Rectified homotopical depth and Grothendieck conjectures, in Grothendieck Festschrift II, ed. Cartier, P. (Birkhäuser, Basel, 1991), 311–351.Google Scholar
[Kos63]Kostant, B., Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.CrossRefGoogle Scholar
[KP82]Kraft, H. and Procesi, C., On the geometry of conjugacy classes in classical groups, Comment. Math. Helv. 57 (1982), 539–602.CrossRefGoogle Scholar
[Nam08a]Namikawa, Y., Birational geometry and deformations of nilpotent orbits, Duke Math. J. 143 (2008), 375–405.CrossRefGoogle Scholar
[Nam08b]Namikawa, Y., Flops and Poisson deformations of symplectic varieties, Publ. Res. Inst. Math. Sci. 44 (2008), 259–314.CrossRefGoogle Scholar
[Nam10]Namikawa, Y., Poisson deformations of affine symplectic varieties II, Kyoto J. Math. 50 (2010), 727–752.CrossRefGoogle Scholar
[Nam11]Namikawa, Y., Poisson deformations of affine symplectic varieties, Duke Math. J. 156 (2011), 51–85.CrossRefGoogle Scholar
[Pre02]Premet, A., Special transverse slices and their enveloping algebras, Adv. Math. 170 (2002), 1–55.CrossRefGoogle Scholar
[Slo80a]Slodowy, P., Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, vol. 815 (Springer, New York, 1980).CrossRefGoogle Scholar
[Slo80b]Slodowy, P., Four lectures on simple groups and singularities, Communications of the Mathematical Institute (Mathematical Institute, Rijksuniversiteit Utrecht 11, Utrecht, 1980).CrossRefGoogle Scholar
[Spa82]Spaltenstein, N., Classes unipotentes et Sous-Groupes de Borel, Lecture Notes in Mathematics, vol. 946 (Springer, New York, 1982).CrossRefGoogle Scholar
[Wei83]Weinstein, A., The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), 523–557.CrossRefGoogle Scholar
[Yam95]Yamada, H., Lie group theoretical construction of period mapping, Math. Z. 220 (1995), 231–255.CrossRefGoogle Scholar