Branching problems ask how an irreducible representation of a group decomposes when restricted to a subgroup. This exposition surveys new aspects on branching problems of unitary representations of reductive Lie groups.
The first half is written from the representation theoretic viewpoint. After an observation of wild feature of branching problems to 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
We end the exposition by a brief discussion on the mystery between tessellation of non-Riemannian homogeneous spaces and branching problems of unitary representations.
- topological properties of modular varieties in locally symmetric spaces,
- a construction of new discrete series representations for non-Riemannian non-symmetric homogeneous spaces.
* Apparently, an online version of this paper is not published by the AMS as of now. Please contact the author (Email: toshi (at) ms.u-tokyo.ac.jp) if you want an offprint. The RIMS preprint (links below) is virtually identical to the published version except for the typesetting and the update of references.[ Original | RIMS preprint(pdf) | RIMS preprint(ps.gz) | preprint version(dvi) | related papers ]
© Toshiyuki Kobayashi