Transfer operators and Hankel transforms between relative trace formulas, II: Rankin–Selberg theory

Sakellaridis, Yiannis

doi:10.1016/j.aim.2021.108039

Cited by 2 publications

(2 citation statements)

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“…1.3. In this setting, [78] gave explicit formulas for transfer operators (4.1) between = the Kuznetsov stack for the group with dual group ˇ (see §5.2.2 below), and = ( × )/ diag . These operators transfer spaces of test measures to each other, but in a number of cases, studied in [31,70,71], properties such as the transfer of characters or the appropriate fundamental lemma are also known; thus, there is enough evidence to believe that these are the "correct" operators of functoriality for these comparisons.…”

Section: Rank 1 Spherical Varietiesmentioning

confidence: 99%

“…Such Hankel transforms have been described by Jacquet [42] for = the standard representation of GL (the paper [37] is closely related), and by me [71] for = the symmetric square representation of GL 2 . It would be interesting to examine if these formulas admit an interpretation in terms of quantization, like the transfer operators in this paper.…”

Section: S * ( (A))mentioning

confidence: 99%

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Spherical varieties, functoriality, and quantization

Sakellaridis¹

2021

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YIANNIS SAKELLARIDIS A. We discuss generalizations of the Langlands program, from reductive groups to the local and automorphic spectra of spherical varieties,and to more general representations arising as "quantizations" of suitable Hamiltonian spaces. To a spherical -variety , one associates a dual group and an -value (encoded in a representation of ), which conjecturally describe the local and automorphic spectra of the variety. This sets up a problem of functoriality, for any morphism → of dual groups. We review, and generalize, Langlands' "beyond endoscopy" approach to this problem. Then, we describe the cotangent bundles of quotient stacks of the relative trace formula, and show that transfer operators of functoriality between relative trace formulas in rank 1 can be interpreted as a change of "geometric quantization" for these cotangent stacks.

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