The spectral theory of automorphic forms was developed to a large extent by Maass, Roelcke, and Selberg without the benefit of the insights of representation theory. The first work to recognize the connection between representation theory and automorphic forms was the paper of Gelfand and Fomin (1952), but it was not until the 1960s that the systematic introduction of representation theory into the study of automorphic forms commenced in earnest.

In this chapter, we will study the connection between the representation theory of GL(2, ℝ) and automorphic forms on the Poincaré upper half plane in a classical setting. We will concentrate on the spectral theory of compact quotients and return to the noncompact case in Chapter 3.

In Sections 2.1 and 2.7, we will discuss the relationship between the spectral problem for compact quotients of the upper half plane. Section 2.1 introduces the problem, and Section 2.7 summarizes the implications of the results obtained in the intervening sections.

Section 2.2 gives various foundational results from Lie theory such as the construction of the universal enveloping algebra U(gℂ), and describes its center Ƶ when g is the Lie algebra of GL(2, ℝ). We interpret these as rings of differential operators and realize the Laplace–Beltrami operator as an element of ℤ.

In Section 2.3, we show that the spectrum of the Laplacian on a compact quotient L2(Γ\PGL(2, ℝ)+) is discrete and that the Laplacian admits an extension to a self-adjoint operator.