\'Etale geometry of closures of Jordan classes

TL;DR

This paper extends a local geometric analysis method for points in the closure of Jordan classes from complex algebraic groups to algebraically closed fields of characteristic p, with specific restrictions on p.

Contribution

It adapts the existing complex case method to positive characteristic fields, providing conditions under which the geometric properties can be studied similarly.

Findings

Method successfully extended to characteristic p under certain restrictions.

Provides a framework for analyzing Jordan class closures in positive characteristic.

Establishes conditions on p for the method to be valid.

Abstract

Let $G$ be a connected reductive algebraic group with simply connected derived subgroup. Over the complex numbers there exists a local method to study the geometric properties of a point $g$ in the closure of a Jordan class of $G$ in terms of Jordan classes of a maximal rank reductive subgroup $M \leq G$ depending on the point $g$, and further to the closures of certain decomposition classes in Lie($M$). We adapt this method to the case of an algebraically closed field of characteristic $p$, and we give sufficient restrictions on $p$ for it to hold.