I start with the interaction of representation theory of Lie groups with various branches of mathematics, and explain the fundamental role of reductive groups. I exploited Benoist-Kobayashi's criterion for tempered homogeneous spaces by using geometric group theory, and also a state-of-the-art of the classification theory of irreducible (infinite-dimensional) representations of real reductive Lie groups such as Langland's classification, Vogan's classification, and the one based on D-modules. A short discussion of the orbit method as a geometric quantization of coadjoint orbits which are typical examples of homogeneous symplectic manifolds are also discussed. For broad audience, structure theory of real reductive Lie groups such as maximal compact subgroups and the Cartan decomposition, and reductive pairs such as symmetric pairs are included followed by some applications such as the criterion for the Calabi-Markus phenomenon for discontinuous groups. (数理大学院・4年生共通講義)
© Toshiyuki Kobayashi