Bulois, Michaël; Hivert, Pascal Sheets in symmetric Lie algebras and slice induction. (English) Zbl 1395.17012 Transform. Groups 21, No. 2, 355-375 (2016). Summary: 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 \(\mathrm{G}_{2}\). We characterize their singular loci and provide a nice desingularization lying in \(s\mathfrak{so}_{7}\). Cited in 1 Document MSC: 17B08 Coadjoint orbits; nilpotent varieties 17B20 Simple, semisimple, reductive (super)algebras Keywords:Slodowy slices; nilpotent orbits Software:SLA; GAP PDFBibTeX XMLCite \textit{M. Bulois} and \textit{P. Hivert}, Transform. Groups 21, No. 2, 355--375 (2016; Zbl 1395.17012) Full Text: DOI HAL References: [1] A. Altman, S Kleiman, Introduction to Grothendieck Duality Theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin, 1970. · Zbl 0215.37201 [2] 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. · Zbl 1110.81022 [3] W. Borho, Über Schichten halbeinfacher Lie-Algebren, Invent. Math. 65 (1981), 283-317. · Zbl 0484.17004 [4] W. Borho, H. Kraft, Über Bahnen und deren Deformationen bei linear Aktionen reduktiver Gruppen, Comment. Math. Helv. 54 (1979), 61-104. · Zbl 0395.14013 [5] A. Broer, Decomposition varieties in semisimple Lie algebras, Canad. J. Math. 50 (1998), 929-971. · Zbl 0928.17020 [6] 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. · Zbl 0940.17003 [7] M. Bulois, Sheets of symmetric Lie algebras and Slodowy slices, J. Lie Theory 21 (2011), 1-54. · Zbl 1226.14059 [8] 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. · Zbl 0639.17005 [9] W. Fulton, J. Harris, Representation Theory, Graduate Texts in Mathematics, Vol. 129, Springer-Verlag, Representation Theory, 1991. · Zbl 0744.22001 [10] The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.6.4 (2013), http://www.gap-system.org. [11] 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. [12] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York, 1977. Russian transl.: Р. Хартшорн, алгебраическая геометрия, Мир М., 1981. · Zbl 0367.14001 [13] 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. [14] 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. [15] 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. · Zbl 1148.17004 [16] 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. · Zbl 1154.68045 [17] B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753-809. · Zbl 0224.22013 [18] H. Kraft, Closures of conjugacy classes in G2, J. Algebra 126 (1989), 454_465. · Zbl 0693.17004 [19] 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. [20] M. Le Barbier Grünewald, The variety of reductions for a reductive symmetric pair, Transform. Groups 16 (2011), 1-26. · Zbl 1271.14071 [21] T. Levasseur, S. P. Smith, Primitive ideals and nilpotent orbits in typeG2, J. Algebra 114 (1988), 81-105. · Zbl 0644.17005 [22] G. Lusztig, N. Spaltenstein, Induced unipotent classes, J. London Math. Soc. 19 (1979), no. 2, 41_52. · Zbl 0407.20035 [23] T. Ohta, The singularities of the closure of nilpotent orbits in certain symmetric pairs, Tohoku Math. J. 38 (1986), 441-468. · Zbl 0654.22004 [24] T. Ohta, Induction of nilpotent orbits for real reductive groups and associated varieties of standard representations, Hiroshima Math. J. 29 (1999), 347-360. · Zbl 0956.17008 [25] D. I. Panyushev, O. Yakimova, Symmetric pairs and associated commuting varieties, Math. Proc. Cambr. Phil. Soc. 143 (2007), 307-321. · Zbl 1126.17010 [26] D. Peterson, Geometry of the Adjoint Representation of a Complex Semisimple Lie Algebra, PhD thesis, Harvard university (1978). [27] W. de Graaf, SLA—a GAP package, 0.13, 2013, http://www.science.unitn.unitn.it/ degraaf/sla.html. [28] P. Slodowy, Simple Singularities and Simple Algebraic Groups, Lecture Notes in Mathematics, Vol. 815, Springer-Verlag, Berlin, 1980. · Zbl 0441.14002 [29] P. Tauvel, R. W. T. Yu, Lie Algebras and Algebraic Groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. · Zbl 1068.17001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.