| Let (A) be a complex simple Lie algebra, k the finite field of order q, and G(,k) a simply connected Chevalley group over (A) and k. Let G(,k) (--->) Aut(,(//C)) I be the Steinberg representation. It can be shown that I has a k-form that is isomorphic to the highest weight module V(,k) = V(,Z ) (CRTIMES)(,Z ) k for (lamda) = (q - 1)(rho).;The construction of V('(lamda)) depends on the theory of affine Lie algebras. If A is the affine Cartan matrix corresponding to A, Kac and Moody construct an infinite-dimensional complex Lie algebra (A). Kac defines infinite-dimensional highest weight modules V('(lamda)) for (A). Garland constructs (,k)(A), a k-form of (A), and V('(lamda)), a k-form of V('(lamda)) and a (,k)(A)-module. In general, V('(lamda)) is a module for a central extension of G(, ), and if (lamda) = (q - 1)(rho), V('(lamda)) is a G(, )-module.;Iwahori and Matsumoto show G(, ) has a Tits system (G(, ), B, N, S). Borel defines the special representation G(, ) (--->) Aut(,(//C)) I as the (infinite-dimensional) representation induced from a certain one-dimensional representation of the Hecke algebra of G(, ). If (//C)(G(, )/B) is the free vector space on the family of right B-cosets of G(, ), with G(, ) acting by left multiplication, I can also be characterized as a quotient module of (//C)(G(, )/B).;The k-form I(,k) is a quotient of k(G(, )/B), and the homomorphism (psi): I(,k) (--->) V('(lamda)) is given by {gB} (--->) g(.)v(,0), v(,0) a highest weight vector. I prove (psi) is an isomorphism of I(,k) onto a dense submodule of V('(lamda)). If A = A(,1), I also show Im (psi) is not the whole of V('(lamda)).;This thesis proves a similar result for Chevalley groups over the local field of formal power series over k. If G(, ) is a simply connected Chevalley group over (A) and , Borel defines the special representation G(, ) (--->) Aut(,(//C)) I. This is a unitary representation of the locally compact group G(, ), analogous to the Steinberg representation of a finite Chevalley group. Garland, using Kac's highest weight modules V('(lamda)) and an analogue of the Kostant basis theorem for the universal enveloping algebra of (A), constructs modular highest weight representations G(, ) (--->) Aut(,k) V('(lamda)). I show I has a k-form I(,k) isomorphic to a submodule of V(,k) for (lamda) = (q - 1)(rho). This submodule is dense under a suitably defined topology.;The proof fails if q = 2 and the root system of A has roots of different lengths, and if q = 3 and A = G(,2). |
