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 F-p, with Lie algebra g = Lie(G), 1 not equal theta is an element of Aut(G) an involution, K = G(0) the theta-fixed points, and G(r) the rth Frobenius kernel of G, r >= 1. We first classify the irreducible KG(r)-modules and their injective envelopes. Then, we classify the irreducible finite dimensional 'modular Harish-Chandra modules' by showing they are exactly the irreducible KG(1)-modules for the infinitesimal thickening KG(1), so in particular they are restricted as g-modules. (c) 2004 Elsevier B.V. All rights reserved.