Automorphic functions on adele groups

Our approach to proving the Geometric Langlands conjecture is based on mapping the category of D-modules on Bun(G) to more accessible categories that have to do with the Borel subgroup of G. These categories are the global Whittaker and the principal series categories, and various versions thereof. When defining these categories, we will be guided by parallel with the classical theory of automorphic functions. Understanding these classical constructions is essential for having a good grasp of what we are doing in the geometric context.

Basically, you need to be familiar with the following notions (for the global field being a function field, which is a lot easier technically than a number field):

• • The adele group and the space of automorphic functions,

  • The space of Whittaker functions and the operation of taking the Whittaker coefficients of an automorphic function,

  • The operations of Constant Term and Eisenstein series.

A good source is Volume 6 of "Generalized functions", by Gelfand, Graev and Piatetskii-Shapiro.

Another good source is S. Gelbart's "Automorphic forms on adele groups".