Functorial transfer between relative trace formulas in rank 1

Abstract

According to the Langlands functoriality conjecture, broadened to the setting of spherical varieties (of which reductive groups are special cases), a map between $L$-groups of spherical varieties should give rise to a functorial transfer of their local and automorphic spectra. The “beyond endoscopy” proposal predicts that this transfer will be realized as a comparison between limiting forms of the (relative) trace formulas of these spaces. In this paper, we establish the local transfer for the identity map between $L$-groups, for spherical affine homogeneous spaces $X=H\backslash G$ whose dual group is ${SL}_2$ or ${PGL}_2$ (with $G$ and $H$ split). More precisely, we construct a transfer operator between orbital integrals for the $(X\times X)/G$-relative trace formula, and orbital integrals for the Kuznetsov formula of ${PGL}_2$ or ${SL}_2$. Besides the $L$-group, another invariant attached to $X$ is a certain $L$-value, and the space of test measures for the Kuznetsov formula is enlarged to accommodate the given $L$-value. The transfer operator is given explicitly in terms of Fourier convolutions, making it suitable for a global comparison of trace formulas by the Poisson summation formula, hence for a uniform proof, in rank $1$, of the relations between periods of automorphic forms and special values of $L$-functions.