Bessel identities in the Waldspurger correspondence over the complex numbers

摘要

We prove certain identities between relative Bessel functions attached to irreducible unitary representations of PGL(2)(DOUBLE-STRUCK CAPITAL C) and Bessel functions attached to irreducible unitary representations of SL2(DOUBLE-STRUCK CAPITAL C). These identities reflect the Waldspurger correspondence over DOUBLE-STRUCK CAPITAL 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 DOUBLE-STRUCK CAPITAL C.