Minimal reduction type in classical cases

TL;DR

This paper proves Yun's minimal reduction conjecture for all classical groups, showing that the minimal reduction set for certain elements is a single nilpotent orbit, and provides an explicit method for determination.

Contribution

It extends Yun's conjecture proof to types B and D, completing the classical groups case, and offers an explicit procedure for computing minimal reductions.

Findings

Confirmed Yun's minimal reduction conjecture for all classical groups.

Established that the minimal reduction set is a single nilpotent orbit for relevant elements.

Provided an explicit method to determine the minimal reduction set.

Abstract

We prove Yun's minimal reduction conjecture for all classical groups. More precisely, for any topologically nilpotent regular semisimple element $\gamma$, we show that the associated minimal reduction set $\mathrm{RT}_{\mathrm{min}}(\gamma)$ consists of a single nilpotent orbit. This result confirms and extends Yun's earlier work in types A and C, and resolves the remaining cases in types B and D. Moreover, we provide an explicit and effective procedure for determining $\mathrm{RT}_{\mathrm{min}}(\gamma)$.