Smooth representations of p-adic groups arise in number theory mainly through the study of automorphic representations, and thus in the end, for example, from modular forms. We saw in the first lecture by Matt Emerton that a modular form, thought of as function on the set of lattices with level N structure, we obtain a function in C(GL2(Z)\GL2(R) × GL2(Z/N),C) satisfying certain differential eq...

Let G be a reductive p-adic group and let θ be an automorphism of G of order at most two. Let π be an irreducible smooth representation of G that is taken to its dual by θ. The space V of π then carries a non-zero bilinear form ( , ) ( unique up to scaling) with the invariance property (π(g)v, π(g)w) = (v, w), for g ∈ G and v, w ∈ V . The form is easily seen to be symmetric or skew-symmetric an...