CHAPTER 1

Eigenfunctions of the hyperbolic Laplace operator

This chapter has a preliminary character. It discusses concepts and results

needed in the next chapters. In Sections 1–3 we recall results concerning eigenfunc-

tions of the Laplace operator and principal series representations that we treat in

more detail in [4]. The averaging operators in Section 4 form another important

tool used in these notes.

1. Eigenfunctions on the hyperbolic plane

Maass forms are functions on the hyperbolic plane that satisfy Δu = λsu and

are invariant under a group of transformations. We define in this subsection the

space of all such eigenfunctions of the Laplace operator and introduce several related

spaces. An important result is Theorem 1.1, which plays for eigenfunctions of Δ

the role of Cauchy’s theorem for holomorphic functions.

1.1. The hyperbolic plane. By H we denote the hyperbolic plane. We

use two realizations as a subset of PC.

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The first is the upper half-plane model

H = {z = x+iy : y 0}, the other the disk model D = {w ∈ C : |w| 1}. In the

upper half-plane model, geodesics are Euclidean vertical half-lines and Euclidean

half-circles with their center on the real axis. In the disk model, geodesics are given

by Euclidean circles intersecting the boundary ∂ D =

S1

= {ξ ∈ C : |ξ| = 1}

orthogonally and Euclidean lines through 0. The real projective line PR

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= R ∪{∞}

is the boundary of the upper half-plane. See Table 1.1 for a further comparison

between both models.

• The space of eigenfunctions. By Es we denote the space of solutions of

(1.1) Δu = λs u in H, λs = s(1 − s) .

The Laplace operator Δ =

−y2∂x 2

−

y2∂y 2

is an elliptic differential operator with

real-analytic coeﬃcients. Hence all elements of Es are real-analytic functions. This

operator commutes with the action of the group G (on the right) given by

(u | g)(z) = u(gz) .

(We will use z to denote the coordinate in both H and D when we make statements

applying to both models of H.) Obviously, Es = E1−s. If U is an open subset of

H, we denote by Es(U) the space of solutions of Δu = λsu on U, thus defining

Es as a sheaf on H. We will refer to elements of Es = Es(H) and of Es(U) as

λs-eigenfunctions of Δ.

So Es denotes a sheaf as well as the space of global sections of that sheaf. For

other sheaves we will allow ourselves a similar ambiguity.

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