\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
Affiliations des auteurs :
Frederik Bang-Jensen
1
;
Jonathan Ditlevsen
1
1
Department of Mathematics, Aarhus University, Denmark
Frederik Bang-Jensen; Jonathan 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/JOLT_2023_33_2_a0/
@article{JOLT_2023_33_2_a0,
author = {Frederik Bang-Jensen and Jonathan 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/JOLT_2023_33_2_a0/}
}
TY - JOUR
AU - Frederik Bang-Jensen
AU - Jonathan Ditlevsen
TI - An Explicit Plancherel Formula for Line Bundles over the One-Sheeted Hyperboloid
JO - Journal of Lie Theory
PY - 2023
SP - 453
EP - 476
VL - 33
IS - 2
UR - http://geodesic.mathdoc.fr/item/JOLT_2023_33_2_a0/
ID - JOLT_2023_33_2_a0
ER -
%0 Journal Article
%A Frederik Bang-Jensen
%A Jonathan Ditlevsen
%T An Explicit Plancherel Formula for Line Bundles over the One-Sheeted Hyperboloid
%J Journal of Lie Theory
%D 2023
%P 453-476
%V 33
%N 2
%U http://geodesic.mathdoc.fr/item/JOLT_2023_33_2_a0/
%F JOLT_2023_33_2_a0
