The weight in a Serre-type conjecture for tame n-dimensional Galois representations

We formulate a conjecture generalising the weight in Serre's Conjecture to n-dimensional representations rho : Gal( Q/Q ) → GLn( Fp ) that are tamely ramified at p. A weight in this context is an irreducible representation of GLn( Fp ) over Fp . The conjecture describes the predicted set of weights in terms of the reduction modulo p of a Deligne-Lusztig representation of GLn( Fp ) which only depends on the restriction of rho to the inertia subgroup at p.; When n = 3 a weight conjecture had already been made by Ash, Doud, Pollack and Sinnott. The advantage of our conjecture is that it is more conceptual. It moreover predicts more weights for many representations rho. We give computational examples which strongly suggest the existence of these extra weights.; When n = 4 we obtain some theoretical evidence by considering automorphic inductions of Hecke characters over non Galois quartic CM fields.; Finally we show that the recent conjecture of Buzzard, Diamond and Jarvis on the weights associated to rho : Gal(K/K ) → GL2( Fp ), where K is a totally real number field unramified at p, is related in an analogous way to the reduction modulo p of Deligne-Lusztig representations if rho is tamely ramified at p. This improves on a result of Diamond.