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

Abstract: The restriction of an irreducible unitary representation $\pi$ of a real reductive group $G$ to a reductive subgroup $H$ decomposes into a direct integral of irreducible unitary representations $\tau$ of $H$ with multiplicities $m(\pi,\tau)\in\mathbb{N}\cup{\infty}$. We show that on the smooth vectors of $\pi$, the direct integral is pointwise defined. This implies that $m(\pi,\tau)$ is bounded above by the dimension of the space $\operatorname{Hom}_H(\pi^\infty|_H,\tau^\infty)$ of intertwining operators between the smooth vectors, also called symmetry breaking operators, and provides a precise relation between these two concepts of multiplicity.