Galois representations for non-self-dual automorphic forms

Abstract: I will report on recent joint work with Harris, Lan and Thorne, constructing l-adic representations for regular algebraic cuspidal automorphic representations of GL(n) over a number field, without any self-duality hypothesis. In this case it is believed (perhaps even proven) that the desired Galois representations do not occur in the l-adic cohomology of a Shimura variety. Rather we construct them as an l-adic limit of Galois representations which do occur in the cohomology of Shimura varieties.