A Whittaker-Plancherel inversion formula for \(\mathrm{SL}_2(\mathbb{C} )\) (Q2073031)

In this article, the authors prove a Whittaker-Plancherel inversion formula which 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 \(\mathrm{SL}_2(\mathbb{C})\). Note that the similar Whittaker-Plancherel formula for \(\mathrm{SL}_2(\mathbb{R})\) was obtained by \textit{E. M. Baruch} and \textit{Z. Mao} [J. Funct. Anal. 238, No. 1, 221--244 (2006; Zbl 1104.44001)].