The theory of automorphic forms and the global Langlands program have been very active research areas for the past 30 years. Significant progress has been achieved by developing intricate geometric methods, but most results to date are restricted to general linear groups (and general unitary groups).
In this talk I will present new results about the representation theory of p-adic groups and demonstrate how these can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group.
This is joint work with Sug Woo Shin.