A WEAKLY GEOMETRICAL GELFAND MODEL FOR GL (N, Q) AND A REALIZATION OF THE GELFAND CHARACTER OF A FINITE-GROUP

We construct a weakly geometrical Gel'fand model for G(n) = GL (n, q), by proving that the representation series T(k) (n) of the Gel'fand model constructed by Klyachko is isomorpbic to a quotient of two natural representations of G(n). We show that the Gel'fand character chi(G) of a finite group G can be realized as a ''twisted trace'' through the function tau from G to C defined by tau(g)=tr(rho(g)-degrees T)(g is-an-element-of G), where T is an involutive automorphism of L2(G) and rho is the right regular representation of G. For the particular case G = G(n) the automorphism T comes from the transpose, recovering the realization of the Gel'fand character obtained for G(n) by Gow.