The Steinberg Lattice of a Finite Chevalley Group and its Modular Reduction

Let p be a prime and let q = p^a, where a is a positive integer.Let G 7equals; G(F_q) be a Chevalley group over F_q, with associatedsystem of roots and Weyl group W. Steinberg showed in 1957that G has an irreducible complex representation whose degreeequals the p-part of |G| 11]. This representation, now knownas the Steinberg representation, has remarkable properties,which reflect the structure of G, and there have been many researchpapers devoted to its study. The module constructed in 11]is in fact a right ideal in the integral group ring ZG of G,and is thus a ZG-lattice, which we propose to call the Steinberglattice of G. It should be noted that lattices not integrallyisomorphic to the Steinberg lattice may also afford the Steinbergrepresentation, and such lattices may differ considerably intheir properties compared with the Steinberg lattice.