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.