Material discussed in the lectures

Exercises

Location and time:

The aim of this advanced course is to develop the basic theory of harmonic analysis on semisimple Lie groups and symmetric spaces, and to give an overview of important results in this area.

Description:
This course will be an advanced course on harmonic analysis on groups and homogeneous spaces. By this we mean a generalization of classical Fourier theory where an arbitrary function is decomposed in terms of basic "wave functions." This decomposition is usually called the Plancherel decomposition.

For locally compact abelian groups the Plancherel decomposition amounts to Pontrjagin duality, and for compact non-abelian groups, it amounts to the Peter-Weyl theorem.

In the present course we will start with a quick recall of the Peter-Weyl theorem, and show that the building blocks of the decomposition can also be defined as eigenfunctions for the commutative algebra of bi-invariant differential operators on the group. The eigenvalues are described through the so-called Harish-Chandra isomorphism.

After this introduction, we will concentrate on the harmonic analysis on real semisimple Lie groups of the non-compact type and the associated Riemannian symmetric spaces. In this setting a far-reaching generalization of the Hermann Weyl's theory for the compact group has been developped in the years 1955 - 1980 by Harish-Chandra. The structure theory of real semisimple groups has particular features that will be discussed in detail: maximal compact subgroup, Cartan decomposition, restricted root system, and Iwasawa decomposition.

As we know from the Peter-Weyl theorem for compact Lie groups, the building blocks for Fourier decomposition are defined in terms of irreducible unitary representations of the group. For real semisimple Lie groups of the non-compact type such representations, when not trivial, must be infinite dimensional. This presents technical difficulties, which can be overcome by the introduction of algebraically defined representations, the so-called Harish-Chandra modules. These will be introduded and serve as the basis for a further development of the theory.

At the end of the course we will be able to give a precise description of the Plancherel theorems for real semisimple Lie groups and Riemannian symmetric spaces. The representations appearing in these Plancherel theorems are the representations of the so-called discrete series, and representations obtained through induction from parabolic subgroups.

Subjects that will be discussed along the way are: the notion of standard intertwining operators, the subrepresentation theorem, the Langlands classification of admissible representations. The course will be a mixture of rigorous development of the basic theory and a survey of important results, with references to the literature.

Lecture Notes. These will be made available during the course.

Literature:
(for background reading)