  We consider the conjugation-action of an arbitrary upper-block parabolic
subgroup of ${\rm GL}_n(\mathbf{C})$, especially of the Borel subgroup $B$ and
of the standard unipotent subgroup $U$ of the latter on the nilpotent cone of
complex nilpotent matrices. We obtain generic normal forms of the orbits and
describe generating (semi-) invariants for the Borel semi-invariant ring as
well as for the $U$-invariant ring. The latter is described in more detail in
terms of algebraic quotients by a special toric variety closely related. The
study of a GIT-quotient for the Borel-action is initiated.
