Exponentiation of commuting nilpotent varieties

TL;DR

This paper establishes a canonical bijection between commuting nilpotent elements in the Lie algebra and infinitesimal one-parameter subgroups in algebraic groups, generalizing exponential maps in positive characteristic.

Contribution

It proves the existence of a canonical exponential map for connected reductive groups in pretty good characteristic, linking nilpotent elements and subgroup varieties.

Findings

Bijection between commuting nilpotent elements and infinitesimal subgroups

Existence of a canonical exponential map for reductive groups

Answers to questions posed by Suslin, Friedlander, and Bendel

Abstract

Let $H$ be a linear algebraic group over an algebraically closed field of characteristic $p>0$. We prove that any "exponential map" for $H$ induces a bijection between the variety of $r$-tuples of commuting $[p]$-nilpotent elements in $Lie(H)$ and the variety of height $r$ infinitesimal one-parameter subgroups of $H$. In particular, we show that for a connected reductive group $G$ in pretty good characteristic, there is a canonical exponential map for $G$ and hence a canonical bijection between the aforementioned varieties, answering in this case questions raised both implicitly and explicitly by Suslin, Friedlander, and Bendel.