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' \simeq ∫Ĝ⊕ mπ(τ)τdμ(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:
- (Representation theory) Understanding of "singular" representations.
- (Discontinuous groups) The topology of modular varieties.
- (Lp-analysis) Construction of new discrete series for homogeneous spaces.
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© Toshiyuki Kobayashi