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
$\text{O}\left( p,\,q \right)$
.