Abstract
In this paper, we study sheets of symmetric Lie algebras through their Slodowy slices. In particular, we introduce a notion of slice induction of nilpotent orbits which coincides with the parabolic induction in the Lie algebra case. We also study in more detail the sheets of the non-trivial symmetric Lie algebra of type G2. We characterize their singular loci and provide a nice desingularization lying in so 7.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Altman, S Kleiman, Introduction to Grothendieck Duality Theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin, 1970.
Л. В. Антонян, О классификации однородных элементов ℤ2-градуированных алгебр Ли, Вестн. МГУ, Сер. мат., меx (1982), no. 2, 29–34. Engl. transl.: L. V. Antonyan, On classification of homogeneous elements of ℤ2 -graded semisimple Lie algebras, Moscow Univ. Math. Bull. 37 (1982), no. 2, 36–43.
W. Borho, Über Schichten halbeinfacher Lie-Algebren, Invent. Math. 65 (1981), 283–317.
W. Borho, H. Kraft, Über Bahnen und deren Deformationen bei linear Aktionen reduktiver Gruppen, Comment. Math. Helv. 54 (1979), 61–104.
A. Broer, Decomposition varieties in semisimple Lie algebras, Canad. J. Math. 50 (1998), 929–971.
A. Broer, Lectures on decomposition classes, in: Representation Theories and Algebraic Geometry (Montreal, PQ, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 514, Kluwer, Dordrecht, 1998, 39–83.
M. Bulois, Sheets of symmetric Lie algebras and Slodowy slices, J. Lie Theory 21 (2011), 1–54.
D. Z. Djokovic, Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers, J. Algebra 112 (1988), 503–524.
W. Fulton, J. Harris, Representation Theory, Graduate Texts in Mathematics, Vol. 129, Springer-Verlag, Representation Theory, 1991.
The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.6.4 (2013), http://www.gap-system.org.
R. Goodman, N. R. Wallach, An algebraic group approach to compact symmetric spaces, http://www.math.rutgers.edu/~goodman/pub/symspace.pdf, in: Representations and Invariants of the Classical Groups, Encyclopedia of Mathematics and its Applications, Vol. 68, Cambridge University Press, 1998.
R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York, 1977. Russian transl.: Р. Хартшорн, алгебраическая геометрия, Мир М., 1981.
P. Hivert, Nappes Sous-Régulières et Équations de Certaines Compactifications Magnifiques, PhD thesis, Université de Versailles (2010), http://tel.archives-ouvertes.fr/docs/00/56/45/94/PDF/these.pdf.
A. E. Im Hof, The Sheets of Classical Lie Algebra, PhD thesis, University of Basel (2005), http://pages.unibas.ch/diss/2005/DissB 7184.pdf.
S. G. Jackson, A. G. Noel, Prehomogeneous spaces associated with nilpotent orbits in simple real Lie algebras E6(6) and E6(−26) and their relative invariants, Experiment. Math. 15 (2006), no. 4, 455–469.
П. И. Кацыло, Сечения пластов в редуктивной алгебре Ли, Изв. АН СССР Сер. Мат. 46 (1982), вьш. 3, 477–486. Engl. transl.: P. I. Katsylo, Sections of sheets in a reductive algebraic Lie algebra, Math. USSR-Izv. 20 (1983), 449–458.
B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809.
H. Kraft, Closures of conjugacy classes in G2, J. Algebra 126 (1989), 454_465.
J. Landsberg, L. Manivel, Representation theory and projective geometry, in: Algebraic Transformation Groups and Algebraic Varieties, Encyclopaedia of Mathematical Sciences, Vol. 132, Subseries Invariant Theory and Algebraic Transformation Groups, Vol. III, Springer, Berlin, 2004, pp. 131–167.
M. Le Barbier Grünewald, The variety of reductions for a reductive symmetric pair, Transform. Groups 16 (2011), 1–26.
T. Levasseur, S. P. Smith, Primitive ideals and nilpotent orbits in type G 2, J. Algebra 114 (1988), 81–105.
G. Lusztig, N. Spaltenstein, Induced unipotent classes, J. London Math. Soc. 19 (1979), no. 2, 41_52.
T. Ohta, The singularities of the closure of nilpotent orbits in certain symmetric pairs, Tohoku Math. J. 38 (1986), 441–468.
T. Ohta, Induction of nilpotent orbits for real reductive groups and associated varieties of standard representations, Hiroshima Math. J. 29 (1999), 347–360.
D. I. Panyushev, O. Yakimova, Symmetric pairs and associated commuting varieties, Math. Proc. Cambr. Phil. Soc. 143 (2007), 307–321.
D. Peterson, Geometry of the Adjoint Representation of a Complex Semisimple Lie Algebra, PhD thesis, Harvard university (1978).
W. de Graaf, SLA—a GAP package, 0.13, 2013, http://www.science.unitn.unitn.it/~degraaf/sla.html.
P. Slodowy, Simple Singularities and Simple Algebraic Groups, Lecture Notes in Mathematics, Vol. 815, Springer-Verlag, Berlin, 1980.
P. Tauvel, R. W. T. Yu, Lie Algebras and Algebraic Groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
BULOIS, M., HIVERT, P. SHEETS IN SYMMETRIC LIE ALGEBRAS AND SLICE INDUCTION. Transformation Groups 21, 355–375 (2016). https://doi.org/10.1007/s00031-015-9355-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-015-9355-4