Automorphic Representations

There exists a satisfactory theory of automorphic representations of GL ( n ). One may associate an automorphic representation of GL (1) with a Dirichlet character, and one may associate an automorphic representation of GL (2) with a modular form or Maass form. Thus, automorphic representations are attached to classical objects of fundamental importance in number theory. We will discuss Tate's (1950) thesis, which is essentially the theory of automorphic representations of GL (1), before defining the general notion of an automorphic representation on GL ( n ). Insofar as we can in a brief account, we will (at least in Section 3.3) give statements that are valid for GL ( n ), but proofs that are valid only for n = 2. In places, we will give proofs that are only complete when the ground field is ℚ, though the only serious obstacle to filling these gaps is our omission in Chapter 2 of the representation theory of GL (2, ℂ). In Section 3.2, we will extend our discussion of the “classical” problem of decomposing L 2 (Γ\ PGL (2, ℝ) + ) to the case when the quotient is noncompact – for example, when Γ = SL (2, ℤ) or one of its congruence subgroups.