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NILPOTENT SUBSPACES AND NILPOTENT ORBITS

Published online by Cambridge University Press:  30 May 2018

DMITRI I. PANYUSHEV*
Affiliation:
IITP of the R.A.S., Bolshoi Karetnyi per. 19, 127051 Moscow, Russia email [email protected]
OKSANA S. YAKIMOVA
Affiliation:
Institut für Mathematik, Friedrich-Schiller-Universität Jena, 07737 Jena, Germany email [email protected]
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Abstract

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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 $\mathfrak{c}$.

Type
Research Article
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

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’.

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