Abstract
The aim of this paper is to describe the irregular locus of the commuting variety of a reductive symmetric Lie algebra. More precisely, we want to enlighten a remark of V. L. Popov. In one of his papers, the irregular locus of the commuting variety of any reductive Lie algebra is described and its codimension is computed. This provides a bound for the codimension of the singular locus of this commuting variety. V. L. Popov also suggests that his arguments and methods are suitable for obtaining analogous results in the symmetric setting. We show that some difficulties arise in this case and we obtain some results on the irregular locus of the component of maximal dimension of the “symmetric commuting variety”. As a by-product, we study some pairs of commuting elements specific to the symmetric case, that we call rigid pairs. These pairs allow us to find all symmetric Lie algebras whose commuting variety is reducible.
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Л. B. Aнтонян, О клaссuфuкацuu оƌнороƌных элеменmов Z 2-ƨраƌyuрованных nолуросmых алƨебр Лu, Becтн. Mоск. унив., cep. I, мaт., мeх. 37 (1982), 29–34. Engl. transl.: L. V. Antonyan, On the classification of homogeneous elements of \( {{\mathbb {Z}}_2} \) -graded semisimple Lie algebras, Mosc. Univ. Math. Bull. 37 (1982), no. 2, 36–43.
S. Araki, On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ. 13 (1962), 1–34.
P. Bala, R. W. Carter, Classes of unipotent elements in simple algebraic groups II, Math. Proc. Cambridge Philos. Soc. 80 (1976), 1–18.
W. Borho, Über Schichten halbeinfacher Lie-Algebren, Invent. Math. 65 (1981), 283–317.
M. Bulois, Composantes irréductibles de la variété commutante nilpotente d’une algèbre de Lie symétrique semi-simple, Ann. Inst. Fourier 59 (2009), 37–80.
M. Bulois, Sheets of symmetric Lie algebras and Slodowy slices, J. Lie Theory 21 (2011), 1–54.
D. H. Collingwood, W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold, New York, 1993.
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.
D. Z. Djokovic, Classification of nilpotent elements in simple real Lie algebras E6(6) and E6(−26) and description of their centralizers, J. Algebra 116 (1988), 196–207.
A. G. Elashvili, D. Panyushev, Towards a classification of principal nilpotent pairs, [Gi, Appendix].
A. G. Elashvili, D. Panyushev, A classification of the principal nilpotent pairs in simple Lie algebras and related problems, J. London Math. Soc. (2) 63 (2001), 299–318.
The GAP Group, GAP—Groups, Algorithms, and Programming, version 4:4:12, 2008, http://www.gap-system.org .
V. Ginzburg, Principal nilpotent pairs in a semisimple Lie algebra I, Invent. Math. 140 (2000), 511–561.
R. Goodman, N. R. Wallach, An algebraic group approach to compact symmetric spaces, http://www.math.rutgers.edu/~goodman/pub/symspace.pdf .
R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York, 1977. Russia transl.: P. Xapтсхоpнб, Aлƨебраuческая ƨeoмеmрuя, Mиp, M., 1981.
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics, Vol. 80, Academic Press, New York, 1978.
S. G. Jackson, A. G. Noel, Prehomogeneous spaces associated with nilpotent orbits, http://www.math.umb.edu/~anoel/publications/tables/ .
B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809.
G. Lusztig, N. Spaltenstein, Induced unipotent classes, J. London Math. Soc. (2) 19 (1979), 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, The closure of nilpotent orbits in the classical symmetric pairs and their singularities, Tohoku Math. J. 43 (1991), 161–211.
D. I. Panyushev, The Jacobian modules of a representation of a Lie algebra and geometry of commuting varieties, Compositio Math. 94 (1994), 181–199.
D. I. Panyushev, Nilpotent pairs in semisimple Lie algebras and their characteristics, Internat. Math. Res. Notices 2000 , 1–21.
D. I. Panyushev, Nilpotent pairs, dual pairs and sheets, J. Algebra 240 (2001), 635–664.
Д. И. Панюшев, О неnpuвоƋuмocmu коммymamopных мноƨообpa3uŭ, свя3анных с uнвопюцuямu npocmых апƨебp Лu, Φункц. aнaлиз и его пpилож. 38 (2004), no. 1, 47‐55, 95. Engl. transl.: D. I. Panyushev, On the irreducibility of commuting varieties associated with involutions of simple Lie algebras, Func. Anal. Appl. 38 (2004), 38–44.
D. I. Panyushev, O. Yakimova, Symmetric pairs and associated commuting varieties, Math. Proc. Cambridge Philos. Soc. 143 (2007), 307–321.
V. L. Popov, Irregular and singular loci of commuting varieties, Transform. Groups 13 (2008), 819–837.
V. L. Popov, E. A. Tevelev, Self-dual projective algebraic varieties associated with symmetric spaces, in: V. Popov, ed., Algebraic Transformation Groups and Algebraic Varieties, Encyclopedia of Mathematical Sciences, Vol. 132, Series Invariant Theory and Transformation Groups, Vol. III, Springer, Berlin, 2004, pp. 131–167.
A. Premet, Nilpotent commuting varieties of reductive Lie algebras, Invent. Math. 154 (2003), 653–683.
R. W. Richardson, Commuting varieties of semisimple Lie algebras and algebraic groups, Compositio Math. 38 (1979), 311–327.
H. Sabourin, R. W. T. Yu, On the irreducibility of the commuting variety of the symmetric pair \( \left( {{\mathfrak{s}}{{\mathfrak{o}}_{p + 2}},{\mathfrak{s}}{{\mathfrak{o}}_p} \times {\mathfrak{s}}{{\mathfrak{o}}_2}} \right) \), J. Lie Theory 16 (2006), 57–65.
R. Steinberg, Torsion in reductive groups, Adv. Math. 15 (1995), 63–92.
P. Tauvel, R. W. T. Yu, Lie Algebras and Algebraic Groups, Springer Monographs in Mathematics, Springer, Berlin, 2005.
R. W. T. Yu, Centralizers of distinguished nilpotent pairs and related problems, J. Algebra 252 (2002), 167–194.
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Bulois, M. Irregular locus of the commuting variety of reductive symmetric lie algebras and rigid pairs. Transformation Groups 16, 1027–1061 (2011). https://doi.org/10.1007/s00031-011-9162-5
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DOI: https://doi.org/10.1007/s00031-011-9162-5