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Branching laws for square integrable representations. (English) Zbl 1196.22009

The authors study square integrable representations of a real reductive Lie group \(G\). They show that for a square integrable representation, its restriction to a closed connected reductive subgroup \(H\) is admissible if and only if the restriction to the intersection \(L\) of \(H\) with a maximal compact subgroup is admissible. This gives a strategy for the computation of the multiplicities in the restriction to \(H\) for the admissible case via the multiplicities in the restriction to \(L\). The authors give nice formulae (in terms of partition functions) for the multiplicities in the restriction to \(L\) under some natural additional assumptions. Finally, the authors also consider the semi-classical analogue of these results for coadjoint orbits.

MSC:

22E46 Semisimple Lie groups and their representations
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
05E15 Combinatorial aspects of groups and algebras (MSC2010)
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