Higher Period Integrals and Derivatives of L-functions

We propose a geometric framework to produce a formula relating higher period integrals to higher central derivatives of $L$-functions over function fields, extending the framework of relative Langlands duality à la Ben-Zvi–Sakellaridis–Venkatesh to higher derivatives. For a strongly tempered affine smooth $G$-variety $X$, we give a geometric construction of the action of $L$-observables on the geometric period integral of a Hecke eigensheaf. By taking a suitable version of Frobenius trace of this action, we recover higher central derivatives of the $L$-function attached to the dual symplectic representation. As an application, in the Rankin–Selberg case $({\mathrm{G}\mathrm{L}}_n\times {\mathrm{G}\mathrm{L}}_{n-1},{\mathrm{G}\mathrm{L}}_{n-1})$, we obtain a formula for higher derivatives of the Rankin–Selberg $L$-function. This provides a conceptual generalization of Yun–Zhang’s higher Gross–Zagier formula to higher-dimensional spherical varieties.