Perfecting group schemes

TL;DR

This paper systematically studies the perfection of affine group schemes over fields of positive characteristic, classifying reductive groups and linking their representations to topological classifying spaces.

Contribution

It provides an intrinsic classification of perfect reductive groups and establishes a correspondence with classifying spaces of compact Lie groups, advancing understanding of their structure and representations.

Findings

Classification of perfections of reductive groups

Bijection with classifying spaces of Lie groups

Highest weight classification of simple modules

Abstract

We initiate a systematic study of the perfection of affine group schemes of finite type over fields of positive characteristic. The main result intrinsically characterises and classifies the perfections of reductive groups, and obtains a bijection with the set of classifying spaces of compact connected Lie groups topologically localised away from the characteristic. We also study the representations of perfectly reductive groups. We establish a highest weight classification of simple modules, the decomposition into blocks, and relate extension groups to those of the underlying abstract group.