(1) Department of Mathematics, Technion, 32000 Haifa, Israel;(2) Department of Mathematics, Rutgers University, 07102 Newark, NJ, USA
Abstract:
We prove certain identities between Bessel functions attached to irreducible unitary representations ofPGL2(R) and Bessel functions attached to irreducible unitary representations of the double cover ofSL2(R). These identities give a correspondence between such representations which turns out to be the Waldspurger correspondence.
In the process we prove several regularity theorems for Bessel distributions which appear in the relative trace formula. In
the heart of the proof lies a classical result of Weber and Hardy on a Fourier transform of classical Bessel functions. This
paper constitutes the local (real) spectral theory of the relative trace formula for the Waldspurger correspondence for which
the global part was developed by Jacquet.
Research of first author was partially supported by NSF grant DMS-0070762.
Research of second author was partially supported by NSF grant DMS-9729992 and DMS 9971003.