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Journal of Lie Theory 27 (2017), No. 1, 001--042
Copyright Heldermann Verlag 2017

Nilpotent Orbits: Finiteness, Separability and Howe's Conjecture

Julius Witte
Radboud Universiteit Nijmegen, Heyendaalseweg 135, Nijmegen -- 6525AJ, The Netherlands

This paper is about nilpotent orbits of reductive groups over local non-Archimedean fields. We will try to identify for which groups there are only finitely many nilpotent orbits, for which groups the nilpotent orbits are separable and for which groups Howe's conjecture holds. For split reductive groups we get a classification in terms of the root data and the characteristic of the underlying local field. For this classification the proof of the failure of Howe's conjecture for split reductive groups for which the characteristic of the field is bad and the proof of Howe's conjecture for the projective linear group are the key results. For general reductive groups we get some partial results, among which there is a proof of Howe's conjecture for groups for which all nilpotent orbits are separable.

Keywords: Nilpotent orbits, Howe's conjecture, reductive groups over local non-Archimedean fields.

MSC: 20G25, 22E50, 17B45

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