Disintegrating tensor representations of nilpotent Lie groups

Let $G$ be a simply connected nilpotent Lie group and $H$ a closed connected subgroup of $G$. Given an irreducible unitary representation $\pi$ of $G$, we present an explicit disintegration of the restriction $\pi_{\vert H}$ of $\pi$ to $H$. Such a disintegration relies on the description of the double cosets space $H \diagdown G\diagup B$ for an arbitrary closed connected subgroup $B$ of $G$, and the well-known smooth disintegration of monomial representations of nilpotent Lie groups. As an application we get a concrete disintegration and a criterion of irreducibility for tensor products of a finite number of irreducible representations of $G$.