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

TL;DR

This paper derives an explicit Plancherel formula for the space of functions on the one-sheeted hyperboloid, using intertwining operators between induced and principal series representations of SL(2,R).

Contribution

It provides a new explicit Plancherel formula for line bundles over the hyperboloid by analyzing intertwining operators and their holomorphic dependence.

Findings

Explicit decomposition of induced representations into irreducibles.

Construction of holomorphic intertwining operators.

Holomorphic dependence of operators on induction parameters.

Abstract

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