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 $\mathbb{C}$.
Local integrability of Bessel functions on split groups
Realization of irreducible unitary representations of $ \mathfrak{osp}(M/N; \mathbb{R}) $ on Fock spaces
Characters of irreducible unitary representations of $U(n, n+1)$ via double lifting from $U(1)$
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