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On the direct integral decomposition in branching laws for real reductive groups

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The restriction of an irreducible unitary representation π of a real reductive group G to a reductive subgroup H decomposes into a direct integral of irreducible unitary representations τ of H with multiplicities m(π,τ) ∈ N ∪ {∞}. We show that on the smooth vectors of π, the direct integral is pointwise defined. This implies that m(π,τ) is bounded above by the dimension of the space Hom H| H) of intertwining operators between the smooth vectors, also called symmetry breaking operators, and provides a precise relation between these two concepts of multiplicity.

Original languageEnglish
JournalJournal of Lie Theory
Pages (from-to)191-196
Number of pages5
Publication statusPublished - 2022

    Research areas

  • Real reductive groups, branching laws, direct integral, pointwise defined, smooth vectors, symmetry breaking operators, unitary representations

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