At the foundation of harmonic analysis are the well-known principles that a geometric space may be studied though its space of functions, and that the analysis of these functions is simplified by taking into account the symmetries of the space.Over the past 60 years, considerable developments have occurred in the global analysis on spaces acted upon by real reductive groups as a generalization of Fourier analysis.

These lectures will focus on several fundamental aspects of this study in which irreducible tempered representations act as “building blocks” of Langlands’ classification theory. Topics will include: real spherical manifolds, the representation theory of real reductive groups, branching laws, tempered homogeneous spaces, examples and applications of tempered varieties.

Lecture 1. Is representation theory useful for global analysis on manifolds X? In the first lecture we highlight “multiplicities” of irreducible representations occurring in the space of functions. We discuss what kind of geometry guarantees a “good control” of the transformation group on the function space, and the answer brings us naturally to the notion of spherical varieties/real spherical manifolds. The main approach of Lecture 1 is PDEs.

Lecture 2. Tempered homogeneous spaces and tempered subgroups We discuss asymptotic behaviors of matrix coefficients on L^2(X) beyond (real) spherical spaces, and explain a criterion to detect for which homogeneous space X the regular representation L^2(X) is tempered. If time permits, we discuss the notion of tempered subgroups à la Margulis and its connection with discontinuous groups. The main approach of Lecture 2 is dynamical system and unitary representation theory.

Lecture 3. Classification theory of non-tempered homogeneous spaces We address a question which reductive homogeneous space is tempered. It turns out that this question is subtle even for semisimple symmetric spaces, however, quite surprisingly, one can give a complete description of (reductive) non- tempered homogeneous spaces. Lecture 3 focuses on some basic ideas in the classification theory of non-tempered homogeneous spaces. The main approach of Lecture 3 is combinatorics for convex polyhedra.

Lecture 4. Interaction of tempered homogeneous spaces with other disciplines. We discuss a bird's eye view on tempered homogeneous spaces in the variety of all Lie subalgebras in a fixed Lie algebra. We also discuss tempered homogeneous spaces from the viewpoint of “orbit philosophy”. These viewpoints enrich the concept of “tempered homogeneous spaces” not only from analysis but also from topology and geometry. If time permits, we explain some open problems. The main disciplines of Lecture 4 are topology (collapsing Lie algebras) and geometric quantization.

[ Program ]

© Toshiyuki Kobayashi