Unipotent elements and generalized exponential maps

TL;DR

This paper introduces generalized exponential maps for algebraic groups over fields of positive characteristic, providing explicit constructions, parameterizations, and applications to unipotent element analysis.

Contribution

It formally defines and constructs generalized exponential maps for all root systems, offering a uniform approach and complete parameterization, with applications to the saturation problem.

Findings

Explicit construction of generalized exponential maps for all root systems

Uniform approach to the saturation problem for unipotent elements

New computations for Lie algebra and algebraic group structures

Abstract

Let $G$ be a simple and simply connected algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p>0$. Assume that $p$ is good for the root system of $G$ and that the covering map $G_{sc} \rightarrow G$ is separable. In previous work we proved the existence of a (not necessarily unique) Springer isomorphism for $G$ that behaved like the exponential map on the resticted nullcone of $G$. In the present paper we give a formal definition of these maps, which we call `generalized exponential maps.' We provide an explicit and uniform construction of such maps for all root systems, demonstrate their existence over $\mathbb{Z}_{(p)}$, and give a complete parameterization of all such maps. One application is that this gives a uniform approach to dealing with the "saturation problem" for a unipotent element $u$ in $G$, providing a new proof of the known result that $u$ lies inside a subgroup of $C_G(u)$ that is isomorphic to a truncated Witt group. We also develop a number of other explicit and new computations for $\mathfrak{g}$ and for $G$. This paper grew out of an attempt to answer a series of questions posed to us by P. Deligne, who also contributed several of the new ideas that appear here.