## Tohoku Mathematical Journal

### The closure ordering of adjoint nilpotent orbits in $\germ s\germ o(p,q)$

#### Abstract

Let ${\mathcal O}$ be a nilpotent orbit in ${\mathfrak so}(p,q)$ under the adjoint action of the full orthogonal group ${\rm O}(p,q)$. Then the closure of ${\mathcal O}$ (with respect to the Euclidean topology) is a union of ${\mathcal O}$ and some nilpotent ${\rm O}(p,q)$-orbits of smaller dimensions. In an earlier work, the first author has determined which nilpotent ${\rm O}(p,q)$-orbits belong to this closure. The same problem for the action of the identity component ${\rm SO}(p,q)^0$ of ${\rm O}(p,q)$ on ${\mathfrak so}(p,q)$ is much harder and we propose a conjecture describing the closures of the nilpotent ${\rm SO}(p,q)^0$-orbits. The conjecture is proved when $\min(p,q)\le7$. Our method is indirect because we use the Kostant-Sekiguchi correspondence to translate the problem to that of describing the closures of the unstable orbits for the action of the complex group ${\rm SO}_p({\bf C})\times{\rm SO}_q({\bf C})$ on the space $M_{p,q}$ of complex $p\times q$ matrices with the action given by $(a,b)\cdot x=axb^{-1}$. The fact that the Kostant--Sekiguchi correspondence preserves the closure relation has been proved recently by Barbasch and Sepanski.

#### Article information

Source
Tohoku Math. J. (2) Volume 53, Number 3 (2001), 395-442.

Dates
First available in Project Euclid: 3 May 2007

https://projecteuclid.org/euclid.tmj/1178207418

Digital Object Identifier
doi:10.2748/tmj/1178207418

Mathematical Reviews number (MathSciNet)
MR2002e:17027

Zentralblatt MATH identifier
0995.17007

Subjects
Primary: 17B45: Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx]
Secondary: 20G05: Representation theory

#### Citation

Đoković, Dragomir Ž.; Lemire, Nicole; Sekiguchi, Jiro. The closure ordering of adjoint nilpotent orbits in $\germ s\germ o(p,q)$. Tohoku Math. J. (2) 53 (2001), no. 3, 395--442. doi:10.2748/tmj/1178207418. https://projecteuclid.org/euclid.tmj/1178207418

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