On Rational Nilpotent Orbits of $SL_{n}$ and $Sp_{2n}$ over a Local Non-Archimedean Field

TL;DR

This paper explores the classification of rational nilpotent orbits in split classical groups over local non-Archimedean fields, connecting partition parametrizations with Bruhat-Tits building methods to deepen understanding of their structure.

Contribution

It establishes a link between partition-based parametrizations and Bruhat-Tits building techniques for classifying nilpotent orbits in $SL_n$ and $Sp_{2n}$ over local fields.

Findings

Unified framework for orbit classification

Connection between partition parametrization and Bruhat-Tits theory

Enhanced understanding of nilpotent orbit structure in classical groups

Abstract

We relate the partition-type parametrization of rational (arithmetic) nilpotent adjoint orbits of the split classical groups $SL_n$ and $Sp_{2n}$ over local non-Archimedean fields with a parametrization, introduced by DeBacker in 2002, which uses the associated Bruhat-Tits building to relate the question to one over the residue field.