A Whittaker-Plancherel Inversion Formula for $\mathrm{SL}_2(\mathbb{C})$

TL;DR

This paper derives a Whittaker-Plancherel inversion formula for the complex special linear group, linking Whittaker-Fourier coefficients to Bessel coefficients via the Bessel transform, advancing harmonic analysis on this group.

Contribution

It introduces a new inversion formula for $ ext{SL}_2( ext{C})$ based on the Bessel transform, providing a deeper understanding of harmonic analysis on this group.

Findings

Decomposition of Whittaker-Fourier coefficients in terms of Bessel coefficients

Extension of harmonic analysis techniques to $ ext{SL}_2( ext{C})$

Analytic framework connecting Bessel transform and representation theory

Abstract

In this paper, we establish a Whittaker-Plancherel inversion formula for $\mathrm{SL}_2(\mathbb{C})$ from the analytic perspective of the Bessel transform of Bruggeman and Motohashi. The formula gives a decomposition of the Whittaker-Fourier coefficient of a compactly supported function on $\mathrm{SL}_2(\mathbb{C})$ in terms of its Bessel coefficients attached to irreducible unitary tempered representations of $\mathrm{SL}_2(\mathbb{C})$.