Springer Isomorphisms In Characteristic $p$

TL;DR

This paper constructs a special Springer isomorphism for simple algebraic groups over fields of characteristic p, which aligns with a canonical isomorphism on unipotent radicals, answering longstanding questions in the field.

Contribution

It establishes the existence of a particularly nice Springer isomorphism that restricts to a canonical isomorphism on certain unipotent radicals for groups in very good characteristic.

Findings

Existence of a Springer isomorphism with special properties

Explicit description using Artin-Hasse exponential for classical groups

Answers to questions posed by McNinch and Carlson et al.

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p$, and assume that $p$ is a very good prime for $G$. Let $P$ be a parabolic subgroup whose unipotent radical $U_P$ has nilpotence class less than $p$. We show that there exists a particularly nice Springer isomorphism for $G$ which restricts to a certain canonical isomorphism $\text{Lie}(U_P) \xrightarrow{\sim} U_P$ defined by J.-P. Serre. This answers a question raised both by G. McNinch in \cite{M2}, and by J. Carlson \textit{et. al} in \cite{CLN}. For the groups $SL_n, SO_n$, and $Sp_{2n}$, viewed in the usual way as subgroups of $GL_n$ or $GL_{2n}$, such a Springer isomorphism can be given explicitly by the Artin-Hasse exponential series.