According to the Langlands functoriality conjecture, broadened to the setting of spherical varieties (of which reductive groups are special cases), a map between -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 -groups, for spherical affine homogeneous spaces whose dual group is or (with and split). More precisely, we construct a transfer operator between orbital integrals for the -relative trace formula, and orbital integrals for the Kuznetsov formula of or . Besides the -group, another invariant attached to is a certain -value, and the space of test measures for the Kuznetsov formula is enlarged to accommodate the given -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 , of the relations between periods of automorphic forms and special values of -functions.
"Functorial transfer between relative trace formulas in rank ." Duke Math. J. 170 (2) 279 - 364, 1 February 2021. https://doi.org/10.1215/00127094-2020-0046