Functorial transfer between relative trace formulas in rank one

TL;DR

This paper establishes a local transfer between trace formulas for spherical varieties with dual groups SL(2) or PGL(2), advancing the Langlands functoriality conjecture through explicit Fourier convolution operators.

Contribution

It constructs an explicit transfer operator for orbital integrals in rank one, linking relative trace formulas and Kuznetsov formulas, and incorporates L-values into the test measure space.

Findings

Constructed a transfer operator using Fourier convolutions.

Proved the fundamental lemma for the transfer operator.

Facilitated a global comparison of trace formulas via Poisson summation.

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 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\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 x 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 fundamental lemma for this transfer operator is proven in a forthcoming paper of Johnstone and Krishna. 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 one, of the relations between periods of automorphic forms and special values of L-functions.