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

Abstract

In this paper we consider $G=\mathrm{SL}(2,\mathbb{R})$ 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 ${\mathrm{Ind}}_H^G(\chi )$ into irreducible unitary representations, known as a Plancherel formula. This is done by studying explicit intertwining operators between ${\mathrm{Ind}}_H^G(\chi )$ and principal series representations of $G$. These operators depends holomorphically on the induction parameters.