On Relative Trace Formulae: the Case of Jacquet-Rallis
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On Relative Trace Formulae: the Case of Jacquet-Rallis Pierre-Henri Chaudouard1 Received: 15 November 2017 / Revised: 26 November 2018 / Accepted: 5 December 2018 / Published online: 7 March 2019 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019
Abstract We give an account of recent works on Jacquet-Rallis’ approach to the Gan-Gross-Prasad conjecture for unitary groups. We report on the present state of the Jacquet-Rallis relative trace formulae and on some current applications of it. We give also a precise computation of the constant that appears in the statement “Fourier transform and transfer commute up to a constant”. Keywords Langlands program · Relative trace formula · Orbital integrals · Gan-Gross-Prasad conjectures Mathematics Subject Classification (2010) 11F70 · 22E50 · 22E55 · 11F66 · 11R39
1 Introduction 1.1 The Philosophy of the Jacquet-Rallis Trace Formula In the emerging relative Langlands program, the main concern is the study of periods of automorphic forms (namely some integrals over subgroups). These objects should have deep relations with special values of L-functions and Langlands functorialities (see [32]). In these questions, a relative trace formula should play a central role connecting periods to (relative) orbital integrals as advocated by Jacquet (see [19]). Gan, Gross, and Prasad made a series of precise conjectures about the link between the non-vanishing of certain periods and the non-vanishing of special values of L-functions (see [15]). Refining the Gan-Gross-Prasad conjecture, Ichino and Ikeda (cf. [18]) were able to give a conjectural Eulerian factorization of the square modulus of periods of orthogonal groups. Their conjecture extends to the case of unitary groups ([16]). In this survey, we will Lecture at the Annual Meeting 2017 of the Vietnam Institute for Advanced Study in Mathematics Pierre-Henri Chaudouard
[email protected] 1
Institut de Math´ematiques de Jussieu-Paris Rive Gauche, Universit´e Paris Diderot (Paris 7) et Institut Universitaire de France, UMR 7586, Bˆatiment Sophie Germain, Case 7012, F-75205 Paris Cedex 13, France
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P.-H. Chaudouard
focus on this case. In a seminal paper [21], Jacquet-Rallis suggested that the Gan-GrossPrasad conjecture for unitary groups should follow from three ingredients: • • •
the classical work of Jacquet-Piatetskii-Shapiro-Shalika on integral representations of L-functions of pairs for GL(n) (cf. [20]); two relative trace formulae (one for unitary groups and the other for linear groups) that express periods in terms of relative orbital integrals; a comparison of relating orbital integrals on unitary groups and linear groups.
The idea is roughly that periods and orbital integrals should be dual objects. The comparison of orbital integrals should be dual to a comparison of periods and so it should be possible to transfer part of Jacquet-Piatetskii-Shapiro-Shalika’s results to unitary groups. In particular, one has to explore harmonic analysi
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