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 \`a 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 $(\mathrm{GL}_n\times\mathrm{GL}_{n-1},\mathrm{GL}_{n-1})$, 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.