Geodesic
An Explicit Plancherel Formula for Line Bundles over the One-Sheeted Hyperboloid
Journal of Lie theory, Tome 33 (2023) no. 2, pp. 453-476
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\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.
Classification : 22E45
Mots-clés : Plancherel formula, SL(2,R), intertwining operator, Fourier-Jacobi transform, direct integral 
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     author = {F. Bang-Jensen and J. Ditlevsen},
     title = {An {Explicit} {Plancherel} {Formula} for {Line} {Bundles} over the {One-Sheeted} {Hyperboloid}},
     journal = {Journal of Lie theory},
     pages = {453--476},
     year = {2023},
     volume = {33},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JLT_2023_33_2_JLT_2023_33_2_a0/}
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F. Bang-Jensen; J. Ditlevsen. An Explicit Plancherel Formula for Line Bundles over the One-Sheeted Hyperboloid. Journal of Lie theory, Tome 33 (2023) no. 2, pp. 453-476. http://geodesic.mathdoc.fr/item/JLT_2023_33_2_JLT_2023_33_2_a0/