×

Regular functions on spherical nilpotent orbits in complex symmetric pairs: exceptional cases. (English) Zbl 1460.14115

Let \(G\) be a connected simple complex algebraic group and let \(K\) be the fixed point subgroup of an involution of \(G\). Let \(\mathfrak{p}\) be the \(-1\)-eigenspace for \(\theta\) in the Lie algebra \(\mathfrak{g}\) of \(G\), which is stable under the adjoint action of \(K\). The objects under study are the \(K\)-orbits of nilpotent elements in \(\mathfrak{p}\) which are spherical (classified by D. R. King [J. Lie Theory 14, No. 2, 339–370 (2004; Zbl 1057.22020)]), and their closures.
In a series of three articles (the article under review, together with [P. Bravi and J. Gandini, Kyoto J. Math. 60, No. 2, 405–450 (2020; Zbl 1470.14105)] and [P. Bravi et al., Kyoto J. Math. 57, No. 4, 717–787 (2017; Zbl 1402.14068)]), with a case by case analysis, the authors determine the spherical systems encoding the orbits as spherical homogeneous spaces, then they use the theory of spherical varieties to determine if multiplication of sections of globally generated line bundles on a wonderful variety associated with such an orbit is surjective, finally, they use this to conclude on the normality of orbit closures. This article is the third and final of the series. It is not to be read without the first two as it makes use of notations and results from those. The first two papers in the series dealt with symmetric pairs under classical groups. The remaining exceptional cases are dealt with in the paper under review. In this situation, there is only one nilpotent orbit closure which is not normal (in the case \(G=G_2\) and \(K\) of type \(A_1\times A_1\)).

MSC:

14M27 Compactifications; symmetric and spherical varieties
20G05 Representation theory for linear algebraic groups

Software:

SLA; GAP
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] D. N. Ahiezer, Equivariant completions of homogeneous algebraic varieties by homogeneous divisors, Ann. Global Anal. Geom. 1 (1983), no. 1, 49-78. Zentralblatt MATH: 0537.14033
Digital Object Identifier: doi:10.1007/BF02329739
· Zbl 0537.14033
[2] N. Bourbaki, Éléments de mathématique, fasc. 34: Groupes et algèbres de Lie, chapitres 4-6, Actualités Sci. Indust. 1337, Hermann, Paris, 1968.
[3] P. Bravi, R. Chirivì, and J. Gandini, Regular functions on spherical nilpotent orbits in complex symmetric pairs: Classical non-Hermitian cases, Kyoto J. Math. 57 (2017), no. 4, 717-787. Zentralblatt MATH: 1402.14068
Digital Object Identifier: doi:10.1215/21562261-2017-0013
Project Euclid: euclid.kjm/1504080147
· Zbl 1402.14068
[4] P. Bravi and J. Gandini, Regular functions on spherical nilpotent orbits in complex symmetric pairs: Classical Hermitian cases, Kyoto J. Math. 60 (2020), no. 2, 405-450. Zentralblatt MATH: 07223240
Digital Object Identifier: doi:10.1215/21562261-2019-0039
Project Euclid: euclid.kjm/1581930017
· Zbl 1470.14105
[5] P. Bravi, J. Gandini, and A. Maffei, Projective normality of model varieties and related results, Represent. Theory 20 (2016), 39-93. Zentralblatt MATH: 1395.14039
Digital Object Identifier: doi:10.1090/ert/477
· Zbl 1395.14039
[6] P. Bravi and D. Luna, An introduction to wonderful varieties with many examples of type \({\normalfont{\mathsf{F}}}_4 \), J. Algebra 329 (2011), 4-51. Zentralblatt MATH: 1231.14040
Digital Object Identifier: doi:10.1016/j.jalgebra.2010.01.025
· Zbl 1231.14040
[7] P. Bravi and G. Pezzini, The spherical systems of the wonderful reductive subgroups, J. Lie Theory 25 (2015), no. 1, 105-123. Zentralblatt MATH: 1349.14164
· Zbl 1349.14164
[8] P. Bravi and G. Pezzini, Primitive wonderful varieties, Math. Z. 282 (2016), no. 3-4, 1067-1096. Zentralblatt MATH: 1356.14041
Digital Object Identifier: doi:10.1007/s00209-015-1578-5
· Zbl 1356.14041
[9] A. Broer, Decomposition varieties in semisimple Lie algebras, Canad. J. Math. 50 (1998), no. 5, 929-971. Zentralblatt MATH: 0928.17020
Digital Object Identifier: doi:10.4153/CJM-1998-048-6
· Zbl 0928.17020
[10] A. Broer, Normal nilpotent varieties in \(F_4\), J. Algebra 207 (1998), no. 2, 427-448. Zentralblatt MATH: 0918.17003
Digital Object Identifier: doi:10.1006/jabr.1998.7458
· Zbl 0918.17003
[11] R. Chirivî and A. Maffei, Projective normality of complete symmetric varieties, Duke Math. J. 122 (2004), no. 1, 93-123. Zentralblatt MATH: 1064.14058
Digital Object Identifier: doi:10.1215/S0012-7094-04-12213-4
Project Euclid: euclid.dmj/1080137203
· Zbl 1064.14058
[12] W. A. de Graaf, SLA - computing with Simple Lie Algebras: A GAP package, 2009, available at http://science.unitn.it/ degraaf/sla.html.
[13] W. A. de Graaf, Computing representatives of nilpotent orbits of \(\theta \)-groups. J. Symbolic Comput. 46 (2011), no. 4, 438-458. · Zbl 1222.17002
[14] D. Ž. 3pt0pt[4pt]3pt0.8ptDokovic, Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers, J. Algebra 112 (1988), no. 2, 503-524. Zentralblatt MATH: 0639.17005
Digital Object Identifier: doi:10.1016/0021-8693(88)90104-4
· Zbl 0639.17005
[15] D. Ž. 3pt0pt[4pt]3pt0.8ptDokovic, Classification of nilpotent elements in simple real Lie algebras \(E_{6(6)}\) and \(E_{6(-26)}\) and description of their centralizers, J. Algebra 116 (1988), no. 1, 196-207. Zentralblatt MATH: 0653.17004
Digital Object Identifier: doi:10.1016/0021-8693(88)90201-3
· Zbl 0653.17004
[16] B. Fu, D. Juteau, P. Levy, and E. Sommers, Generic singularities of nilpotent orbit closures, Adv. Math. 305 (2017), 1-77. Zentralblatt MATH: 1366.14007
Digital Object Identifier: doi:10.1016/j.aim.2016.09.010
· Zbl 1366.14007
[17] The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.5.4, 2012 (http://www.gap-system.org).
[18] D. R. King, Classification of spherical nilpotent orbits in complex symmetric space, J. Lie Theory 14 (2004), no. 2, 339-370. Zentralblatt MATH: 1057.22020
· Zbl 1057.22020
[19] D. I. Panyushev, Some amazing properties of spherical nilpotent orbits, Math. Z. 245 (2003), no. 3, 557-580. Zentralblatt MATH: 1101.17012
Digital Object Identifier: doi:10.1007/s00209-003-0555-6
· Zbl 1101.17012
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.