Integration questions in separably good characteristics

TL;DR

This paper investigates the integration of p-nilpotent elements and subalgebras in reductive groups over fields of separably good characteristic, providing counterexamples and criteria for systematic integration.

Contribution

It extends the concept of infinitesimal saturation and generalizes Deligne's theorem to analyze integration of p-nil subalgebras in reductive groups.

Findings

Counterexamples of integration of p-nil subalgebras

Criteria for integrating certain p-nil subalgebras

Extension of Deligne's theorem on infinitesimal saturation

Abstract

Let G be a reductive group over an algebraically closed field k of separably good characteristic p>0 for G. Under these assumptions a Springer isomorphism from the reduced nilpotent scheme of the Lie algebra of G to the reduced unipotent scheme of G always exists. This allows to integrate any p-nilpotent element of Lie(G) into a unipotent element of G. One should wonder whether such a punctual integration can lead to a systematic integration of p-nil subalgebras of Lie(G). We provide counterexamples of the existence of such an integration in general as well as criteria to integrate some p-nil subalgebras of Lie(G) (that are maximal in a certain sense). This requires to generalise the notion of infinitesimal saturation first introduced by P. Deligne and to extend one of his theorem on infinitesimally saturated subgroups of G to the previously mentioned framework.