An explicit Plancherel formula for line bundles over the one-sheeted hyperboloid

In this paper we consider $G=\operatorname{SL}(2,\mathbb{R})$ and $H$ the subgroup of diagonal matrices. Then $X=G/H$ is a unimodular homogeneous space which can be identified with the one-sheeted hyperboloid. For each unitary character $\chi$ of $H$ we decompose the induced representations $\operatorname{Ind}_H^G(\chi)$ into irreducible unitary representations, known as a Plancherel formula. This is done by studying explicit intertwining operators between $\operatorname{Ind}_H^G(\chi)$ and principal series representations of $G$. These operators depends holomorphically on the induction parameters.