The Closure Ordering of Nilpotent Orbits of the Complex Symmetric Pair (SO(p+q),SOp * SOq)

The main problem that is solved in this paper has the following simple formulation (which is not used in its solution). The group K =(O)p (C) * (O)q (C) acts on the space M(p,q) of p * q complex matrices by (a,b) . x = axb^(-1) , and so does its identity component K0 = SOp (C) * SOq (C). A K-orbit (or K^0-orbit) in M(p,q) is said to be nilpotent if its closure contains the zero matrix. The closure, O, of a nilpotent Korbit (resp.K^0-orbit) (O) in M(p,q) is a union of (O) and some nilpotent K-orbits (K^0-orbits) of smaller dimensions. The description of the closure of nilpotent K-orbits has been known for some time, but not so for the nilpotent K^0-orbits. A conjecture describing the closure of nilpotent K^0-orbits was proposed in [11] and verified when min(p,q) <= 7. In this paper we prove the conjecture. The proof is based on a study of two prehomogeneous vector spaces attached to (O) and determination of the basic relative invariants of these spaces. The above problem is equivalent to the problem of describing the closure of nilpotent orbits in the real Lie algebra so (p,q) under the adjoint action of the identity component of the real orthogonal group (O)(p,q).