We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of GL_n(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.
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