Restriction of Unitary Representations — Discrete and continuous spectrum,
(plenary lecture), Sixth Pan-African Congress of Mathematicians (PACOM2004), Institut National des Sciences Appliquées et de la Technologie (INSAT), Tunis, Tunisia, September 2004.

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^⊕ m_π(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:

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.

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