Categorical representations of algebraic groups in positive characteristic

In this paper we present a formalism of categorical representation theory for affine group schemes of finite type over fields (both in the weak and strong settings) which applies to arbitrary base fields. This is based on the development of an appropriate $\infty$-categorical setting for the study of representations and Harish-Chandra (bi)modules of such group schemes, which constitute the building blocks and basic examples of such structures. We also study categorical traces in this context, and show in particular that the categorical trace of the identity morphism, resp.~Frobenius morphism, on the appropriate $\infty$-category of representations of a connected reductive algebraic group $G$ over an algebraically closed field $k$ of positive characteristic identifies (under suitable assumptions) with the $\infty$-category of Ind-coherent sheaves on the adjoint quotient $G/G$, resp.~of the fixed points $G^{\mathrm{F}}$ of the Frobenius.