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PARABOLIC CONJUGATION AND COMMUTING VARIETIES

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Abstract

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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Authors and Affiliations

  1. Ruhr-University Bochum, Faculty of Mathematics, Chair of Algebra, Universitätsstraße 150, D - 44780, Bochum, Germany

    M. BOOS

  2. Univ Lyon, Université Jean Monnet, Saint-Étienne, CNRS UMR 5208, Institut Camille Jordan, F-42023, Saint-Etienne, France

    M. BULOIS

Authors
  1. M. BOOS
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  2. M. BULOIS
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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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BOOS, M., BULOIS, M. PARABOLIC CONJUGATION AND COMMUTING VARIETIES. Transformation Groups 24, 951–986 (2019). https://doi.org/10.1007/s00031-018-9507-4

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