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 image, with Lie algebra image, 1≠θset membership, variantAut(G) an involution, K=Gθ the θ-fixed points, and Gr the rth Frobenius kernel of G,rgreater-or-equal, slanted1. 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 image-modules.