## Branching problems of unitary representations.(English)Zbl 1008.43009

Li, Ta Tsien (ed.) et al., Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20-28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press. 615-627 (2002).
Basic theory and new perspectives of admissible restrictions are presented from both the analytic and the algebraic point of view, such as:
1. Admissible restrictions to subgroups, including a criterion for admissible restrictions and its converse statement;
2. infinitesimal discrete decomposability, including a discrete branching law, a criterion for discrete decomposability and the size of irreducible summands;
3. some applications to representation theory;
4. new discrete series for homogeneous spaces, nonsymmetric spaces, symmetric spaces and an exclusive law of discrete spectrum for restriction and induction;
5. modular varieties and vanishing theorems, including vanishing theorems for modular symbols.
For the entire collection see [Zbl 0993.00022].
Reviewer: Su Weiyi (Nanjing)

### MSC:

 43A85 Harmonic analysis on homogeneous spaces 22E46 Semisimple Lie groups and their representations 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics 53D20 Momentum maps; symplectic reduction
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