Twisting and Mixing

TL;DR

This paper develops a unified framework connecting twisted, mixed, and exotic pseudo-reductive groups through schemes, demonstrating how various known groups can be realized as rational points of these schemes.

Contribution

It introduces categories of twisted and mixed schemes, linking classical groups to these new structures and providing a geometric perspective on their construction.

Findings

Twisted Chevalley groups are rational points of twisted group schemes.

Mixed groups of Tits are rational points of mixed group schemes over mixed fields.

Exotic pseudo-reductive groups are Weil restrictions of mixed group schemes.

Abstract

We present a framework that connects three interesting classes of groups: the twisted groups (also known as Suzuki-Ree groups), the mixed groups and the exotic pseudo-reductive groups. For a given characteristic p, we construct categories of twisted and mixed schemes. Ordinary schemes are a full subcategory of the mixed schemes. Mixed schemes arise from a twisted scheme by base change, although not every mixed scheme arises this way. The group objects in these categories are called twisted and mixed group schemes. Our main theorems state: (1) The twisted Chevalley groups ${}^2\mathsf B_2$, ${}^2\mathsf G_2$ and ${}^2\mathsf F_4$ arise as rational points of twisted group schemes. (2) The mixed groups in the sense of Tits arise as rational points of mixed group schemes over mixed fields. (3) The exotic pseudo-reductive groups of Conrad, Gabber and Prasad are Weil restrictions of mixed group schemes.