4 - Representations of GL(2) Over a p-adic Field

Summary

In this chapter, we will discuss the representation theory of GL(2) over a p-adic field, a cornerstone of the modern theory of automorphic forms, emphasizing techniques that are applicable to GL(n).

We begin by looking at the representation theory of GL(2) over a finite field, where we encounter an essential tool, Mackey theory, which is the calculus of intertwining operators between induced representations. We will see that the irreducible representations of G = GL(2, F) when F is finite are roughly parametrized by the characters of maximal tori in G. The representations parametrized by maximal split tori are induced representations, those parametrized by nonsplit tori must be constructed by some other method. A convenient method of accomplishing this is afforded by the Weil representation, which we study in detail.

In the rest of the chapter, we will adapt the results of Section 4.1 to representations of GL(2, F) where F is local. In Section 4.2, we introduce the categories of smooth and admissible representations and establish their basic properties. In Section 4.3, we introduce some tools, sheaves, and distributions, which are needed to extend Mackey theory to locally compact groups, as in Bruhat's thesis (1956, 1961). We follow Bernstein and Zelevinsky (1976) in emphasizing these tools. In Section 4.4, we prove the uniqueness of Whittaker models, a fundamental result that was applied in Section 3.5 to the multiplicity one theorem for automorphic forms and to the construction of L-functions. We also introduce the Jacquet module, a functor adjoint to parabolic induction, and establish its basic properties.