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1 February 2021 Functorial transfer between relative trace formulas in rank 1
Yiannis Sakellaridis
Duke Math. J. 170(2): 279-364 (1 February 2021). DOI: 10.1215/00127094-2020-0046


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 \ 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 ) / 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.


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Yiannis Sakellaridis. "Functorial transfer between relative trace formulas in rank 1 ." Duke Math. J. 170 (2) 279 - 364, 1 February 2021.


Received: 11 September 2018; Revised: 28 March 2020; Published: 1 February 2021
First available in Project Euclid: 8 December 2020

Digital Object Identifier: 10.1215/00127094-2020-0046

Primary: 11F70
Secondary: 22E50

Rights: Copyright © 2021 Duke University Press


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Vol.170 • No. 2 • 1 February 2021
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