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

Let O be a nilpotent orbit in so(p, q) under the adjoint action of the full orthogonal group O(p, q). Then the closure of O (with respect to the Eucli dean topology) is a union of O and some nilpotent O(p, q)-orbits of smaller dimensions. In an earlier work, the first author has determined which nilp otent O(p, q)-orbits belong to this closure. The same problem for the actio n of the identity component SO(p, q)(0) of O(p, q) on so(p, q) is much hard er and we propose a conjecture describing the closures of the nilpotent SO( p, q)(0)-orbits. The conjecture is proved when min (p, q) less than or equa l to 7. Out 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 SOp(C) x SOq(C) on the space M- p,M-q of complex p x q matrices with the action given by (a, b) . x = axb(- 1). The fact that the Kostant-Sekiguchi correspondence preserves the closur e relation has been proved recently by Barbasch and Sepanski.