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We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of the general linear group on the variety of nilpotent matrices in its Lie algebra. Lie-theoretically, it is natural to wonder about the number of orbits of this action. We translate the setup to a representation-theoretic one and obtain a finiteness criterion which classifies all actions with only a finite number of orbits over an arbitrary infinite field. These results are applied to commuting varieties and nested punctual Hilbert schemes.

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Corresponding author

Correspondence to M. BULOIS.

Additional information

Supported by Ruhr-University Bochum; benefited from a one month “professeur invité” position at University of Saint-Étienne.

Supported by Université Jean Monnet, Labex MILYON/ANR-10-LABX-0070 and ANR Grant GeoLie/ANR-15-CE40-0012.

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