The main problem that is solved in this paper has the following simple formulation (which is not used in its solution). The group $K={{O}_{p}}\left( C \right)\times {{O}_{q}}\left( C \right)$ acts on the space ${{M}_{p,\,q}}\,\text{of}\,p\,\times \,q$ complex matrices by $\left( a,b \right)\cdot x=ax{{b}^{-1}}$, and so does its identity component ${{K}^{0}}=\text{S}{{\text{O}}_{p}}\left( \text{C} \right)\times \text{S}{{\text{O}}_{\text{q}}}\left( \text{C} \right)$. 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, $\bar{\mathcal{O}}$, of a nilpotent $K$-orbit (resp. ${{K}^{0}}$-orbit) $\mathcal{O}$ in ${{M}_{p,q}}$ is a union of $\mathcal{O}$ and some nilpotent $K$-orbits (resp. ${{K}^{0}}$-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 $\min \left( p,\,q \right)\le 7$. In this paper we prove the conjecture. The proof is based on a study of two prehomogeneous vector spaces attached to $\mathcal{O}$ 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 $\mathfrak{s}\mathfrak{o}\left( p,\,q \right)$ under the adjoint action of the identity component of the real orthogonal group $\text{O}\left( p,\,q \right)$.