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Tohoku Mathematical Journal
2001
September
SECOND SERIES VOL. 53, NO. 3
Tohoku Math. J.
53 (2001), 395442

Title
THE CLOSURE ORDERING OF ADJOINT NILPOTENT ORBITS IN $\mathfrak{so}(p, q)$
Author
Dragomir Ž. Đoković, Nicole Lemire and Jiro Sekiguchi
(Received August 31, 2001, revised June 12, 2000) 
Abstract.
Let $\mathcal{O}$ be a nilpotent orbit in $\mathfrak{so} (p,q)$ under the adjoint action of the full orthogonal group $\mathrm{O} (p,q)$. Then the closure of $\mathcal{O}$ (with respect to the Euclidean topology) is a union of $\mathcal{O}$ and some nilpotent $\mathrm{O} (p,q)$orbits of smaller dimensions. In an earlier work, the first author has determined which nilpotent $\mathrm{O} (p,q)$orbits belong to this closure. The same problem for the action of the identity component $\mathrm{SO} (p,q)^0$ of $\mathrm{O} (p,q)$ on $\mathfrak{so} (p,q)$ is much harder and we propose a conjecture describing the closures of the nilpotent $\mathrm{SO} (p,q)^0$orbits. The conjecture is proved when $\min(p,q) \le 7$.
Our method is indirect because we use the KostantSekiguchi correspondence to translate the problem to that of describing the closures of the unstable orbits for the action of the complex group $\mathrm{SO}_p (\boldsymbol{C}) \times\mathrm{SO}_q (\boldsymbol{C})$ 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 KostantSekiguchi correspondence preserves the closure relation has been proved recently by Barbasch and Sepanski.
2000 Mathematics Subject Classification.
Primary 17B45; Secondary 17B20, 17B40.
Key words and phrases.
Nilpotent adjoint orbits, standard triples, KostantSekiguchi
correspondence.


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