In this part of the book we study Fourier analysis on finite nonabelian groups and we should expect them to be able to be a bit more obnoxious before breakfast than their nonabelian relatives. We will mostly deal with specific groups of 2 × 2 or 3 × 3 matrices over finite fields. Before meeting these groups for breakfast, the reader will hopefully have tasted some undergraduate algebra books such as Gallian [1990].

Brauer [1963] said: “Groups are the mathematical concept with which we describe symmetry. One of the outstanding achievements of Greek mathematics is the discovery of the 5 regular polyhedra, the Platonic solids. Each of them is closely associated with a finite group.” And we will find that these polyhedra have much to do with some of the Cayley graphs that we associate to the affine group of 2 × 2 invertible matrices over a finite field with lower row (0 1). Brauer goes on to say: “We have to confess that it took mathematicians 2000 years to achieve a mathematical formulation of the group concept. In its abstract form, it was given by Cayley, after permutation groups had been used by Lagrange, Cauchy, Abel, Galois, and others in their work on the solution of equations by radicals.”

We have found many reasons to study the representations of GL(2, q). In Chapter 13, for example, we discussed the molecule buckminsterfullerene or C60 and noted that an understanding of the spectral lines of this molecule requires a knowledge of the representations of A5 ≅ PSL(2, 5). In the last chapter we found that an understanding of the representations of GL(2, q) seems necessary to bound the spherical functions on the finite upper half plane. Also the Ramanujan graphs of Lubotzky, Phillips, and Sarnak [1988] are Cayley graphs for either PGL(2, q) or PSL(2, q), with prime q, using generating sets with p + 1 elements (p denoting a different prime). See also Lubotzky [1994, pp. 96 ff] and Sarnak [1990, pp. 73 ff]. Finally, the representations of SL(2, q) are needed in the paper of Lafferty and Rockmore [1992] where spectra of degree four Cayley graphs of SL(2, q) are studied.

In this chapter we want to consider finite analogues of the symmetric spaces G/K mentioned above. These are related to the concept of Gelfand pair (G, K). See also Gelfand [1988], Helgason [1968, 1984], Selberg [1989], Terras [1985 and 1988], and Vilenkin [1968]. In particular, we will study a finite analogue of the Poincaré upper half plane H ≅ G/K, with G = SL(2, ℝ) and K = SO(2). We will replace ℝ with the finite field q. Such quotients have been considered by Angel et al. [1992], Celniker et al. [1993], Soto-Andrade [1987], and Terras [1991, 1996]. Other quotients G/B, for B the Borel subgroup of upper triangular matrices, have been considered by Krieg [1990] and Stanton [1990], for example.

One goal is to obtain a Fourier transform on K\G/K called the spherical transform (see Corollary 1 of Theorem 2 in Chapter 20 and formula (15) of Chapter 23). The spherical transform has many of the properties of the Fourier transform on abelian groups since it is complex valued rather than matrix valued.

This chapter gives a way of constructing representations of finite groups from those of subgroups called the method of induced representations. In 1898 Frobenius invented the method of induction by writing down the formula for the character of the induced representation of a finite group. In 1927 Speiser gave formulas for the matrix entries. Wigner obtained induced representations of the (infinite) Lorentz group from the subgroup of translations in 1939. In 1940 Weil showed how to do induction for compact groups. In the 1950s Mackey [1976], [1978a,b] developed the theory for noncompact, locally compact groups. Around the same time Selberg invented his trace formula and applied it to the group G = SL(2, ℝ) and various discrete subgroups acting on G/K, K = SO(2). See Terras [1985, Vol. I]. We will see in Chapter 22 that the finite analogue of Selberg's trace formula will allow us to prove some of the results such as the Frobenius reciprocity law in an elegant way (see Chapter 22 and Arthur [1989]).

In the previous chapter we saw that for every compact operator T on a Banach space X, the space can almost be written as a direct sum of generalized eigenspaces of T. If we assume that X is not merely a Banach space, but a Hilbert space, and T is not only compact but compact and normal, then such a decomposition is indeed possible – in fact, there is a decomposition with even better properties. Such a decomposition will be provided by the spectral theorem for compact normal operators: a complete and very simple description of compact normal operators. Thus with the study of a compact normal operator on a Hilbert space we arrive in the promised land: everything fits, everything works out beautifully, there are no blemishes. This is the best of all possible worlds.

We shall give two proofs of the spectral theorem, claiming the existence of the desired decomposition. In the first proof we shall make use of some substantial results from previous chapters, including one of the important results concerning the spectrum of a compact operator. The second proof is self-contained: we shall replace the results of the earlier chapters by easier direct arguments concerning Hilbert spaces and normal operators.

To start with, we collect a number of basic facts concerning normal operators in the following lemma. Most of these facts have already been proved, but for the sake of convenience we prove them again.

Lemma 1. Let T ∈ B(H) be a normal operator. Then the following assertions hold.