*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)τ*d*μ(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.
- (
*L*^{p}-analysis) Construction of new discrete series for homogeneous spaces.

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© Toshiyuki Kobayashi