The closure ordering of adjoint nilpotent orbits in so(p,q)

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.

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