Branching problems ask how an irreducible representation of a group decomposes when restricted to a subgroup. In the Lectures, I plan to give a survey on new aspects on branching problems of unitary representations of reductive Lie groups.[ Math Colloquium | Special Lectures in Real & Complex Geometry | announcement | abstract (pdf) ]The first half will be based on representation theoretic viewpoints. Following an observation of wild feature of branching problems with respect to non-compact subgroups in a general setting, I will introduce the notion of admissible restrictions as a nice framework that enjoys two properties: finiteness of multiplicities and discreteness of spectrum. A criterion for finite multiplicity and discretely decomposable restrictions will be presented. The idea of our proof stems from microlocal analysis and algebraic geometry. Furthermore, an exclusive law of discrete spectrum is formalized for inductions and restrictions.
The second half deals with applications. Admissible restrictions may give tools for the study of representations for bigger groups via restrictions, and for the construction of irreducibles of smaller groups via branching laws. Once we know that restrictions are discretely decomposable, we may employ an algebraic approach to branching problems. In this direction, new branching formulas have been recently obtained in various settings, of which I plan to illustrate by examples. Finally, I will discuss some applications of discretely decomposable branching laws to other areas of mathematics. The topics include (1) topological properties of modular varieties in locally symmetric spaces, (2) a construction of new discrete series representations for non-Riemannian non-symmetric homogeneous spaces.
If time allows, I may mention also a mystery between tessellation of (and discontinuous groups for) non-Riemannian homogeneous spaces and branching problems of unitary representations.
© Toshiyuki Kobayashi