let $m(n)$ be the algebra (viewed as both a lie and an associative algebra) of $n\times n$ matrices over $\mathbb{c}$. let $p(n)$ denote the algebra of polynomials on $m(n)$. the associative commutator on the universal enveloping algebra induces a poisson structure on $p(n)$. let $j(n)$ be the commutative poisson subalgebra of $p(n)$ generated by the invariants $p(m)^{gl(m)} \text{ for } m=1,\cdots , n$. $j(n)$ gives rise to a commutative lie algebra of vector fields on $m(n)$; $v=\{ \xi_{f}| f\in j(n)\}$. these fields integrate to an action of a commutative, simply connected complex analytic group $a\simeq\mathbb{c} ^{\frac{(n-1)n}{2}}$ on $m(n)$. note that the dimension of this group is exactly half the dimension of the generic coadjoint orbits. moreover, on the most generic orbits of $a$, the commutative lie algebra $v$ is an algebra of symplectic vector fields of exactly half the dimension of the generic coadjoint orbits. we will discuss the orbit structure of the action of $a$ on $m(n)$. we will give a description of the work of kostant-wallach in the most generic case in such a form that can be used to establish a formalism for dealing with less generic orbits.