Representations associated to involutions on algebraic groups and modular Harish-Chandra modules

In this article we consider an extension of Harish-Chandra modules for real Lie groups to the setting of algebraic groups over an algebraically closed field k of positive characteristic p>2. Let G be a connected, semisimple, simply connected algebraic group over k, defined and split over Fp, with Lie algebra g=Lie(G), 1≠θ∈Aut(G) an involution, K=Gθ the θ-fixed points, and Gr the rth Frobenius kernel of G,r⩾1. We first classify the irreducible KGr-modules and their injective envelopes. Then, we classify the irreducible finite dimensional 'modular Harish-Chandra modules' by showing they are exactly the irreducible KG1-modules for the infinitesimal thickening KG1, so in particular they are restricted as g-modules.