THEORY OF DISCRETELY DECOMPOSABLE RESTRICTIONS OF UNITARY REPRESENTATIONS OF SEMISIMPLE LIE GROUPS AND SOME APPLICATIONS

摘要

Branching problems ask how an irreducible representation of a group decomposes when restricted to a subgroup. This exposition surveys new aspects of branching problems for unitary representations of reductive Lie groups. The first half is written from the representation theoretic viewpoint. After an observation on the wild features of branching problems for non-compact subgroups in a general setting, we introduce the notion of admissible restrictions as a good framework that enjoys two properties: finiteness of multiplicities and discreteness of spectrum. A criterion for admissible restrictions is presented, of which the idea of proof stems from microlocal analysis and algebraic geometry. In this framework, we present a finite multiplicity theorem. Furthermore, an exclusive law of discrete spectrum is formulated for inductions and restrictions. The second half deals with applications. Once we know the non-existence of continuous spectrum in the restrictions, we could expect an algebraic approach to branching problems. In this framework, new branching formulas have been recently obtained in various settings, among which we present an example, namely, a generalization of the Kostant-Schmid formula to non-compact subgroups. Finally, we mention some applications of discretely decomposable branching laws to other fields of mathematics. The topics include (1) topologicat properties of modular varieties in locally symmetric spaces, (2) a construction of new discrete series representations for non-Riemannian non-symmetric homogeneous spaces. We end the exposition by a brief discussion on the mystery between tessellation of non-Riemannian homogeneous spaces and branching problems of unitary representations.