Functorial transfer between relative trace formulas in rank 1

According to the Langlands functoriality conjecture, broadened setting of spherical varieties (of which reductive groups are special cases), a map between L -groups should give rise functorial transfer their local and automorphic spectra. The “beyond endoscopy” proposal predicts that this will be realized as comparison limiting forms (relative) trace formulas these spaces. In paper, we establish for identity -groups, affine homogeneous spaces X = H \ G whose dual group is SL 2 or PGL (with split). More precisely, construct operator orbital integrals ( × ) / -relative formula, Kuznetsov formula . Besides -group, another invariant attached certain -value, space test measures enlarged accommodate given -value. explicitly in terms Fourier convolutions, making it suitable global by Poisson summation hence uniform proof, rank 1 , relations periods values -functions.