Tensor product decomposition for rank-one spin groups I : unitary principal series representations

We provide an explicit direct integral decomposition for the tensor product representation ${\pi }_1\widehat{\otimes }{\pi }_2$ of the rank one spin group $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n,1)$ whenever ${\pi }_1$ is a unitary principal series representation and ${\pi }_2$ is an arbitrary irreducible unitary representation of $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n,1)$.