Week 2: Betti Cohomology of Shimura Varieties — the Matsushima Formula

Rough exposition of the goal of this course. Hecke actions on Shimura varieties and their cohomology groups. The Matsushima formula in the classical setting; relative Lie algebra cohomology, spectral decomposition on L(Γ\G). The Matsushima formula in the adelic setting — automorphic representations appear in the Betti cohomology of Shimura varieties. Admissible (g, U)-modules and cohomological representations, Hodge theory. Back to GL2-case; discussion of the Eichler-Shimura isomorphism. Lecture 4 (Sep. 22, 2008) – continued 1. Where we are going We defined the automorphic representations of GL2(A∞) as the representations appearing in the space of automorphic forms Ak, or Ak, which were in turn the space of sections of certain holomorphic line bundle Lk on the compactifications of a modular curve XU := GL2(Q)\GL2(A)/(U · U∞) for some open compact subgroup U ⊂ GL2(A∞) (i.e. the subgroups conjugate to a finite index subgroup U ⊂ GL2(Ẑ)). The goal of our lecture is to associate a 2-dimensional `-adic Galois representation to each cuspidal automorphic representation of GL2(A∞) (although we will turn to compact unitary Shimura curves over CM fields at some point). 1.1. Hecke action. How do we find the Galois representations? They do not seem to appear in the space Ak of automorphic forms. But being representations, automorphic representations are more mobile than automorphic forms – they can appear in different vector spaces with a left GL2(A∞)-action. In fact, we will look for various cohomology groups associated to XU , because they will give rise to vector spaces on which GL2(A∞) acts (in a smooth admissible way), and will have more structures as well. This happens because the spaces XU have the right action of GL2(A∞), in a certain sense. Let us make it precise. Let us denote by X the inverse system of (non-connected) Riemann surfaces {XU}U for each open compact subgroup U ⊂ GL2(A∞) (which eventually turns out to be algebraic varieties, when we give the moduli interpretation which descends to Q), with the canonical