Regularized spectral expansion of a Rankin--Selberg period and moments

We establish a regularized spectral decomposition of the squared magnitude of a maximal degenerate Eisenstein series as a Schwartz distribution on ${\mathrm{G}\mathrm{L}}_n$. Although the original problem is on ${\mathrm{G}\mathrm{L}}_n$, its spectral components are described entirely by the automorphic spectrum of ${\mathrm{G}\mathrm{L}}_2$. As an application, we prove a partial reciprocity formula relating the second moment of ${\mathrm{G}\mathrm{L}}_n\times {\mathrm{G}\mathrm{L}}_n$ Rankin–Selberg central $L$-values and a mixed moment of $L$-values of ${\mathrm{G}\mathrm{L}}_2$. We also prove, assuming the generalized Ramanujan conjecture, an asymptotic formula for the second moment of the above Rankin–Selberg $L$-functions over a conductor-aspect family with an error term of square-root-cancellation strength.