Central to the study of the Dirichlet space is the concept of logarithmic capacity. This assigns to subsets of the unit circle a notion of size which is closely linked to various aspects of functions in D, notably boundary behavior, zeros, multipliers and cyclicity. It will thus be a recurring theme throughout the book.

In this chapter we present a brief but self-contained account of capacity. We do so in an abstract setting, replacing the unit circle by a general compact metric space, and the logarithmic kernel by an arbitrary decreasing function. This entails no extra work, and has the advantage that it covers not only logarithmic capacity but also certain other capacities such as Riesz capacities, which arise naturally in the context of the spaces Dα.

Potentials, energy and capacity

Throughout the chapter, we fix a compact metric space (X, d) and a continuous decreasing function K : (0, ∞) → [0, ∞). The function K is called a kernel (though it has nothing to do with reproducing kernels). We extend the definition of K to 0 by defining K(0) := limt→0+K(t). It may well happen that K(0) = ∞, and in fact this is the case for most interesting kernels, though we do not insist upon it. However, in order to avoid trivialities, we do assume that K ≢ 0.

Definition 2.1.1 Let μ be a finite positive Borel measure on X.

The aim of these lecture notes is to present an introduction to the representation theory of wreath products of finite groups and to harmonic analysis on the corresponding homogeneous spaces.

The exposition is completely self-contained. The only requirements are the fundamentals of the representation theory of finite groups, for which we refer the possibly inexperienced reader to the monographs by Serre [67], Simon [68], Sternberg [73] and to our recent books [11, 15].

The first chapter constitutes an introduction to the theory of induced representations. It focuses on two main topics, namely harmonic analysis on homogeneous spaces which decompose with multiplicity, and Clifford theory. The latter is developed with the aim of presenting a general formulation of the little group method. The exposition is based on our papers [12, 13, 64].

The second chapter is the core of the monograph. We develop the representation theory of wreath products of finite groups following, in part, the approach by James and Kerber [38] and Huppert [35] and developing our research expository paper [14]. Our approach is both analytical and geometrical. In particular, we interpret the exponentiation and composition actions in terms of actions on suitable finite rooted trees and describe the group of automorphisms of a finite rooted tree as the iterated wreath product of symmetric groups.

We explicitly describe the conjugacy classes of wreath products and the corresponding parameterization of irreducible representations.

In this chapter we discuss the notion of an induced representation and the structure of the commutant of a representation, and we present a new approach to Clifford theory. We assume the reader to be familiar with the basic rudiments of the representation theory of finite groups. We refer to the monographs by Bump [7], Fulton and Harris [29], Isaacs [36], Serre [67], Simon [68] and Sternberg [73] as basic references; see also our monograph [15].

In Section 1.1 we present the main properties of induction, focusing on the Frobenius character formula and Frobenius reciprocity. Then, in Section 1.2, we discuss several aspects of Frobenius reciprocity for a permutation representation; in particular, we show that the spherical Fourier transform provides an explicit isomorphism between the commutant of a permutation representation and the algebra of bi-K-invariant functions. In the last part of the section we examine the particular case of a multiplicity free permutation representation, which yields the notion of a Gelfand pair. Finally, in Section 1.3, we present an exposition of Clifford theory, which provides a powerful tool for relating the representation theory of a group G and the representation theory of a normal subgroup N ≤ G.

Induced representations

The presentation of this section was inspired by the books by Bump [7], Serre [67] and Sternberg [73] and by our research-expository paper [12].

1.1.1 Definitions

Let G be a finite group. Let K be a subgroup of G and (ρ, W) a representation of K.

In this chapter, which constitutes the core of the book, we develop the representation theory of wreath products. Our exposition is inspired by the monographs of James and Kerber [38] and Huppert [35]. Howewer, our approach is more analytical and, in particular, we interpret the exponentiation and the composition actions in terms of actions on suitable rooted trees. This is done in Section 2.1.2. In Section 2.3 we describe the conjugacy classes of wreath products F ≀ G, with particular emphasis on groups of the form C2 ≀ G (Section 2.3.2), and F ≀ Sn (Section 2.3.3), and then in Section 2.4 we use the little group method (Theorem 1.3.11) to determine a complete list of irreducible representations of wreath products. Finally in Sections 2.5 and 2.6 we analyze the representation theory of groups of the form C2 ≀ G and F ≀ Sn, respectively. This yields, in particular, a clear description of the representations of finite lamplighter groups (Sections 2.5.1 and 2.5.2) as well as of the groups Sm ≀ Sn (Section 2.6.1).

Basic properties of wreath products of finite groups

2.1.1 Definitions

Let G and F be two finite groups and suppose that G acts on a finite set X. Denote by FX the set of all maps f : X → F. The set FX is a group under pointwise multiplication: (f • f′)(x) = f(x)f′(x) for all f, f′ ∈ FX and x ∈ X.

The three classical Hilbert spaces of holomorphic functions in the unit disk are the Hardy, Bergman and Dirichlet spaces. There are several excellent texts covering the Hardy space and the Bergman space. However, to the best of our knowledge, up to now there has been no book devoted to the Dirichlet space. When we began our respective researches into the Dirichlet space, we found ourselves handicapped by the fact that the necessary background information was scattered around the literature, sometimes contained in articles that were difficult to follow. For this reason we began to think about writing an introduction that would be suitable for researchers and graduate students seeking a solid background in the subject. The more we learned about this topic, the more we became convinced that it contains many beautiful ideas that deserve a systematic exposition.

The name Dirichlet space derives from its definition in terms of the so-called Dirichlet integral, arising in Dirichlet's method for solving Laplace's equation (sometimes called the Dirichlet principle). As far as we can determine, the first appearance of the Dirichlet space under that name dates back to two articles of Beurling and Deny in 1958 and 1959, but in fact the notion existed and had been studied at least since Beurling's thesis, which was published in 1933 and written even a little earlier. In the years that followed, Beurling and Carleson laid the foundations of the theory and, after their pioneering work, many other distinguished mathematicians made important contributions.