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Journal of Lie Theory 29 (2019), No. 4, 969--996
Copyright Heldermann Verlag 2019

Parabolic Orbits of 2-Nilpotent Elements for Classical Groups

Magdalena Boos
Faculty of Mathematics, Ruhr University, 44780 Bochum, Germany

Giovanni Cerulli Irelli
Department SBAI, Sapienza University, 00161 Rome, Italy

Francesco Esposito
Department of Mathematics, University of Padova, 35121 Padova, Italy

We consider the conjugation-action of the Borel subgroup of the symplectic or the orthogonal group on the variety of nilpotent complex elements of nilpotency degree 2 in its Lie algebra. We translate the setup to a representation-theoretic context in the language of a symmetric quiver algebra. This makes it possible to provide a parametrization of the orbits via a combinatorial tool that we call symplectic/orthogonal oriented link patterns. We deduce information about numerology. We then generalize these classifications to standard parabolic subgroups for all classical groups. Finally, our results are restricted to the nilradical.

Keywords: B-orbits, symmetric quiver, algebra with self-duality, combinatorial classification, Auslander-Reiten quiver.

MSC: 14R20, 16N40, 17B45, 16G20, 16G70.

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