On A Generalization of Motohashi's Formula

We study a spectral reciprocity formula relating ${\mathrm{G}\mathrm{L}}_3\times {\mathrm{G}\mathrm{L}}_2$ with ${\mathrm{G}\mathrm{L}}_3\times {\mathrm{G}\mathrm{L}}_1$ and ${\mathrm{G}\mathrm{L}}_1$ moments of $L$-functions discovered by Kwan. Globally we give an adelic and distributional treatment. Our test automorphic function is of general type. To achieve this generality we develop an extension of the generalized Godement sections. Locally we give the weight function transforms in both directions for the fixed tempered representation $\mathrm{\Pi }$ of $\mathrm{G}{\mathrm{L}}_3(\mathbf{F})$. We obtain the transform by a theory of the Voronoi–Hankel transforms, which extends Miller–Schmid’s local theory of the Voronoi formula for ${\mathrm{G}\mathrm{L}}_n$.