
Journal of Lie Theory 33 (2023), No. 2, 453476 Copyright Heldermann Verlag 2023 An Explicit Plancherel Formula for Line Bundles over the OneSheeted Hyperboloid Frederik BangJensen Department of Mathematics, Aarhus University, Denmark bangjensen@math.au.dk Jonathan Ditlevsen Department of Mathematics, Aarhus University, Denmark ditlevsen@math.au.dk [Abstractpdf] \def\Ind{\rm Ind\,} \def\SL{\rm SL\,} We consider $G=\SL(2,\mathbb{R})$ and $H$ the subgroup of diagonal matrices. Then $X=G/H$ is a unimodular homogeneous space which can be identified with the onesheeted hyperboloid. For each unitary character $\chi$ of $H$ we decompose the induced representations $\Ind_H^G(\chi)$ into irreducible unitary representations, known as a Plancherel formula. This is done by studying explicit intertwining operators between $\Ind_H^G(\chi)$ and principal series representations of $G$. These operators depends holomorphically on the induction parameters. Keywords: Plancherel formula, SL(2,R), intertwining operator, FourierJacobi transform, direct integral. MSC: 22E45 [ Fulltextpdf (212 KB)] for subscribers only. 