Skip to main content Accessibility help
×
Home

NILPOTENT SUBSPACES AND NILPOTENT ORBITS

  • DMITRI I. PANYUSHEV (a1) and OKSANA S. YAKIMOVA (a2)

Abstract

Let $G$ be a semisimple complex algebraic group with Lie algebra $\mathfrak{g}$ . For a nilpotent $G$ -orbit ${\mathcal{O}}\subset \mathfrak{g}$ , let $d_{{\mathcal{O}}}$ denote the maximal dimension of a subspace $V\subset \mathfrak{g}$ that is contained in the closure of ${\mathcal{O}}$ . In this note, we prove that $d_{{\mathcal{O}}}\leq {\textstyle \frac{1}{2}}\dim {\mathcal{O}}$ and this upper bound is attained if and only if ${\mathcal{O}}$ is a Richardson orbit. Furthermore, if $V$ is $B$ -stable and $\dim V={\textstyle \frac{1}{2}}\dim {\mathcal{O}}$ , then $V$ is the nilradical of a polarisation of ${\mathcal{O}}$ . Every nilpotent orbit closure has a distinguished $B$ -stable subspace constructed via an $\mathfrak{sl}_{2}$ -triple, which is called the Dynkin ideal. We then characterise the nilpotent orbits ${\mathcal{O}}$ such that the Dynkin ideal (1) has the minimal dimension among all $B$ -stable subspaces $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$ , or (2) is the only $B$ -stable subspace $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$ .

Copyright

Corresponding author

Footnotes

Hide All

The research of the first author was carried out at the IITP R.A.S. at the expense of the Russian Foundation for Sciences (project no. 14-50-00150). The second author is partially supported by the DFG priority programme SPP 1388 ‘Darstellungstheorie’ and by the Graduiertenkolleg GRK 1523 ‘Quanten- und Gravitationsfelder’.

Footnotes

References

Hide All
[1] Cellini, P. and Papi, P., ‘ad-nilpotent ideals of a Borel subalgebra’, J. Algebra 225 (2000), 130141.
[2] Cellini, P. and Papi, P., ‘ad-nilpotent ideals of a Borel subalgebra II’, J. Algebra 258 (2002), 112121.
[3] Collingwood, D. and McGovern, W., Nilpotent Orbits in Semisimple Lie Algebras, Mathematics Series (Van Nostrand Reinhold, 1993).
[4] Elashvili, A. G., ‘The centralizers of nilpotent elements in semisimple Lie algebras’, Tr. Razmadze Mat. Inst. (Tbilisi) 46 (1975), 109132 (in Russian); MR0393148.
[5] Fang, C., ‘Ad-nilpotent ideals of minimal dimension’, J. Algebra 403 (2014), 517543.
[6] Gerstenhaber, M., ‘On nilalgebras and linear varieties of nilpotent matrices, IV’, Ann. of Math. (2) 75 (1962), 382418.
[7] Joseph, A., ‘On the variety of a highest weight module’, J. Algebra 88 (1984), 238278.
[8] Kawanaka, N., ‘Generalized Gelfand–Graev representations of exceptional simple groups over a finite field I’, Invent. Math. 84 (1986), 575616.
[9] Kempken, G., ‘Induced conjugacy classes in classical Lie algebras’, Abh. Math. Semin. Univ. Hambg. 53 (1983), 5383.
[10] Panyushev, D., ‘Complexity and nilpotent orbits’, Manuscripta Math. 83 (1994), 223237.
[11] Panyushev, D. and Röhrle, G., ‘On spherical ideals of Borel subalgebras’, Arch. Math. 84 (2005), 225232.
[12] Sommers, E., ‘Equivalence classes of ideals in the nilradical of a Borel subalgebra’, Nagoya Math. J. 183 (2006), 161185.
[13] Spaltenstein, N., ‘On the fixed point set of a unipotent element on the variety of Borel subgroups’, Topology 16 (1977), 203204.
[14] Spaltenstein, N., Classes Unipotentes et Sous-groupes de Borel, Lecture Notes in Mathematics, 946 (Springer, Berlin–Heidelberg–New York, 1982).
[15] Steinberg, R., ‘On the desingularization of the unipotent variety’, Invent. Math. 36 (1976), 209224.
[16] Vinberg, E. B., Gorbatsevich, V. V. and Onishchik, A. L., Gruppy i algebry Li 3, Sovremennye Problemy Matematiki. Fundamental’nye Napravleniya, 41 (VINITI, Moskva, 1990), (in Russian); English translation: V. V. Gorbatsevich, A. L. Onishchik and E. B. Vinberg, Lie Groups and Lie Algebras III (Encyclopaedia of Mathematical Sciences, 41) (Springer, Berlin, 1994).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

NILPOTENT SUBSPACES AND NILPOTENT ORBITS

  • DMITRI I. PANYUSHEV (a1) and OKSANA S. YAKIMOVA (a2)

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed