Bessel Identities in the Waldspurger Correspondence over the Complex Numbers

TL;DR

This paper establishes identities between Bessel functions related to representations of PGL₂(ℂ) and SL₂(ℂ), reflecting the Waldspurger correspondence over complex numbers, and develops the local spectral theory of Jacquet's relative trace formula.

Contribution

It proves new identities between Bessel functions associated with different groups over ℂ, advancing the understanding of the Waldspurger correspondence in the complex setting.

Findings

Proved identities between relative Bessel functions for PGL₂(ℂ) and SL₂(ℂ).

Established regularity theorems for Bessel and relative Bessel distributions.

Developed the local spectral theory of Jacquet's relative trace formula over ℂ.

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 $\mathbb{C}$.