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Journal of Lie Theory 33 (2023), No. 2, 453--476
Copyright Heldermann Verlag 2023

An Explicit Plancherel Formula for Line Bundles over the One-Sheeted Hyperboloid

Frederik Bang-Jensen
Department of Mathematics, Aarhus University, Denmark

Jonathan Ditlevsen
Department of Mathematics, Aarhus University, Denmark


\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 one-sheeted 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, Fourier-Jacobi transform, direct integral.

MSC: 22E45

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