Restriction of Unitary Representations — discrete and continuous spectrum

Toshiyuki Kobayashi

RIMS, Kyoto University toshi@kurims.kyoto-u.ac.jp

2000MSC: Primary 22E46; Secondary 43A85, 11F67, 53C50, 53D20

Let π be an irreducible unitary representation of a group G. A branching law is the irreducible decomposition of π with regard to its subgroup G′:

π|_{G′} ≃ ∫ ⊕ ; _{G′} m_{π}(τ)τ dμ(τ) (a direct integral).

Such a decomposition is unique, for example, if G′ is a real reductive group.

Special cases of branching problems include and/or reduce to the followings: Littlewood-Richardson rules, the decomposition of tensor product representations, character formulas, Blattner formulas, Plancherel theorems for homogeneous spaces, description of breaking symmetries in quantum mechanics, the theta-lifting in the theory of automorphic forms, etc.

Our concern is with non-compact subgroups G′, and we shall explain the algebraic and analytic theory of branching laws without continuous spectrum. Then, we shall discuss its recent applications which include:

1. (Representation theory) Understanding of “singular” representations.

2. (Discontinuous groups) The topology of modular varieties.

3. (L^{p}-analysis) Construction of new discrete series for homogeneous spaces.

Pan-African Congress of Mathematicians 2004, Tunisia Section: Plenary Lecture