Twisting and Mixing

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.