Conjugation on varieties of nilpotent matrices
We consider the conjugation-action of an arbitrary upper-block parabolic subgroup P of the general linear group (over 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 systems of representatives for the orbits in all finite cases. Furthermore, we give a set-theoretic description of their closures and examine them for minimal degenerations in case x=2. Concerning the action on the nilpotent cone, we obtain a generic P-normal form of the orbits which yields generic normal forms for the actions of the Borel subgroup B of upper-triangular matrices and of the standard unipotent subgroup U of B. We describe generating (semi-) invariants for the Borel semi-invariant ring as well as for the U-invariant ring. The latter is examined in more detail in terms of algebraic quotients by a toric variety closely related.