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

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 ${\mathrm{SL}}_2(\mathbb{C})$.

17 pages. Analysis greatly simplified thanks to an observation of the referee. To appear in J. Anal. Math