For enhanced groups from GL_n and module M, the nilpotent cone has finitely many orbits under uG-action precisely when M is 1-dimensional, natural, dual (n>2), or small irreducible for n=2; orbits classified via enhanced partitions with closures via enhanced flag varieties.
Shoji, Geometry of orbits and Springer correspondence , Societe Mathematigue de France Asterisque 168 (1988), 61-140
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On enhanced reductive groups (II): Finiteness of nilpotent orbits under enhanced group action and their closures
For enhanced groups from GL_n and module M, the nilpotent cone has finitely many orbits under uG-action precisely when M is 1-dimensional, natural, dual (n>2), or small irreducible for n=2; orbits classified via enhanced partitions with closures via enhanced flag varieties.