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

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 $m(\pi ,\tau )\in \mathbb{N}\cup \{\mathrm{\infty }\}$. 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 ${\mathrm{Hom}}_H({\pi }^{\mathrm{\infty }}{\mathrm{\mid }}_H,{\tau }^{\mathrm{\infty }})$ of intertwining operators between the smooth vectors, also called symmetry breaking operators, and provides a precise relation between these two concepts of multiplicity.