Bessel Identities in the Waldspurger Correspondence over the Complex Numbers

Abstract

We prove certain identities between relative Bessel functions attached to irreducible unitary representations of $\mathrm{PGL}_2(\mathbb{C})$ and Bessel functions attached to irreducible unitary representations of $\mathrm{SL}_2 (\mathbb{C})$. These identities reflect the Waldspurger correspondence over $\mathbb{C}$. We also prove several regularity theorems for Bessel and relative Bessel distributions which appear in the relative trace formula. This paper constitutes the local spectral theory of Jacquet's relative trace formula over $\mathbb{C}$.