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

TY - JOUR

T1 - On the direct integral decomposition in branching laws for real reductive groups

AU - Frahm, Jan

PY - 2022

Y1 - 2022

N2 - 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(π ∞| H,τ ∞) of intertwining operators between the smooth vectors, also called symmetry breaking operators, and provides a precise relation between these two concepts of multiplicity.

AB - 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(π ∞| H,τ ∞) of intertwining operators between the smooth vectors, also called symmetry breaking operators, and provides a precise relation between these two concepts of multiplicity.

KW - Real reductive groups

KW - branching laws

KW - direct integral

KW - pointwise defined

KW - smooth vectors

KW - symmetry breaking operators

KW - unitary representations

UR - http://www.scopus.com/inward/record.url?scp=85132992660&partnerID=8YFLogxK

M3 - Journal article

SN - 0949-5932

VL - 32

SP - 191

EP - 196

JO - Journal of Lie Theory

JF - Journal of Lie Theory

IS - 1

ER -