Nilpotent orbits: finiteness, separability and Howe's conjecture

TL;DR

This paper investigates the properties of nilpotent orbits in reductive groups over local non-Archimedean fields, focusing on finiteness, separability, and Howe's conjecture, providing classifications and partial results.

Contribution

It offers a classification of nilpotent orbits for split reductive groups based on root data and field characteristic, and explores conditions for finiteness, separability, and Howe's conjecture.

Findings

Finiteness of nilpotent orbits characterized for certain groups

Separability of orbits established in specific cases

Conditions for Howe's conjecture to hold identified

Abstract

This paper is about nilpotent orbits of reductive groups over local non-Archimedean fields. In this paper 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 general reductive groups we get some partial results. For split reductive groups we get a classification in terms of the root data and the characteristic of the underlying local field.