  We consider the conjugation-action of an arbitrary upper-block parabolic
subgroup of GL_n(C) on the variety of x-nilpotent complex matrices. We obtain a
criterion as to whether the action admits a finite number of orbits and specify
a system of representatives for the orbits in the finite case of 2-nilpotent
matrices. Furthermore, we give a set-theoretic description of their closures
and specify the minimal degenerations in detail for the action of the Borel
subgroup. Concerning the action on the nilpotent cone, we obtain a generic
normal form of the orbits which yields a U-normal form as well, here U is the
standard unipotent subgroup. We 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.
