Abstract
Following Jacquet, Lapid and Rogawski, we define regularized periods of automorphic forms on
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 26287003
Award Identifier / Grant number: 26800017
Funding source: European Research Council
Award Identifier / Grant number: 290766 (AAMOT)
Funding statement: A. Ichino is partially supported by JSPS Grant-in-Aid for Scientific Research (B) 26287003. S. Yamana is partially supported by JSPS Grant-in-Aid for Young Scientists (B) 26800017. The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 290766 (AAMOT).
Acknowledgements
We thank the referee for suggesting significant improvements to this paper.
References
[1] A. Aizenbud, D. Gourevitch, S. Rallis and G. Schiffmann, Multiplicity one theorems, Ann. of Math. (2) 172 (2010), 1407–1434. 10.4007/annals.2010.172.1407Search in Google Scholar
[2] J. Arthur, A trace formula for reductive groups. II: Applications of a truncation operator, Compos. Math. 40 (1980), 87–121. Search in Google Scholar
[3] J. Arthur, The trace formula in invariant form, Ann. of Math. (2) 114 (1981), 1–74. 10.2307/1971376Search in Google Scholar
[4] J. Arthur, On the inner product of truncated Eisenstein series, Duke Math. J. 49 (1982), 35–70. 10.1215/S0012-7094-82-04904-3Search in Google Scholar
[5] Y. Benoist and H. Oh, Polar decomposition for p-adic symmetric spaces, Int. Math. Res. Not. IMRN 2007 (2007), no. 24, Article ID rnm121. 10.1093/imrn/rnm121Search in Google Scholar
[6] W. Casselman, Canonical extensions of Harish-Chandra modules to representations of G, Canad. J. Math. 41 (1989), 385–438. 10.4153/CJM-1989-019-5Search in Google Scholar
[7] P. Delorme and V. Sécherre, An analogue of the Cartan decomposition for p-adic reductive symmetric spaces of split p-adic reductive groups, Pacific J. Math. 251 (2011), 1–21. 10.2140/pjm.2011.251.1Search in Google Scholar
[8]
J. Franke,
Harmonic analysis in weighted
[9] W. T. Gan, B. H. Gross and D. Prasad, Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups, Astérisque 346 (2012), 1–109. Search in Google Scholar
[10] D. Ginzburg, D. Jiang and S. Rallis, Nonvanishing of the central critical value of the third symmetric power L-functions, Forum Math. 13 (2001), no. 1, 109–132. 10.1515/form.2001.001Search in Google Scholar
[11] D. Ginzburg, D. Jiang and S. Rallis, On the nonvanishing of the central value of the Rankin–Selberg L-functions, J. Amer. Math. Soc. 17 (2004), no. 3, 679–722. 10.1515/9783110892703.157Search in Google Scholar
[12] D. Ginzburg, D. Jiang and S. Rallis, On the nonvanishing of the central value of the Rankin–Selberg L-functions. II, Automorphic representations, L-functions and applications. Progress and prospects, Ohio State Univ. Math. Res. Inst. Publ. 11, De Gruyter, Berlin (2005), 157–191. 10.1515/9783110892703.157Search in Google Scholar
[13] D. Ginzburg, D. Jiang and S. Rallis, Models for certain residual Representations, Automorphic forms and L-functions. I: Global aspects, Contemp. Math. 488, American Mathematical Society, Providence (2009), 125–146. 10.1090/conm/488/09567Search in Google Scholar
[14] D. Ginzburg, S. Rallis and D. Soudry, The descent map from automorphic representations of GL(n) to classical groups, World Scientific, Hackensack 2011. 10.1142/7742Search in Google Scholar
[15] N. Harris, The refined Gross–Prasad conjecture for unitary groups, Int. Math. Res. Not. IMRN 2014 (2014), no. 2, 303–389. 10.1093/imrn/rns219Search in Google Scholar
[16] A. Ichino and T. Ikeda, On the periods of automorphic forms on special orthogonal groups and the Gross–Prasad conjecture, Geom. Funct. Anal. 19 (2010), no. 5, 1378–1425. 10.1007/s00039-009-0040-4Search in Google Scholar
[17]
A. Ichino and S. Yamana,
Periods of automorphic forms: The case of
[18] H. Jacquet, Integral representation of Whittaker functions, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins University, Baltimore (2004), 373–419. Search in Google Scholar
[19] H. Jacquet, Archimedean Rankin–Selberg integrals, Automorphic forms and L-functions II. Local aspects, Contemp. Math. 489, American Mathematical Society, Providence (2009), 57–172. 10.1090/conm/489/09547Search in Google Scholar
[20] H. Jacquet, E. Lapid and J. Rogawski, Periods of automorphic forms, J. Amer. Math. Soc. 12 (1999), 173–240. 10.1090/S0894-0347-99-00279-9Search in Google Scholar
[21]
H. Jacquet, I. I. Piatetski-Shapiro and J. Shalika,
Automorphic forms on
[22] H. Jacquet and S. Rallis, On the Gross–Prasad conjecture for unitary groups, On certain L-functions, Clay Math. Proc. 13, American Mathematical Society, Providence (2011), 205–264. Search in Google Scholar
[23] H. Jacquet and J. Shalika, On Euler products and the classification of automorphic representations. I, Amer. J. Math. 103 (1981), 499–558. 10.2307/2374103Search in Google Scholar
[24] D. Jiang, B. Sun and C.-B. Zhu, Uniqueness of Bessel model: The archimedean case, Geom. Funct. Anal. 20 (2010), 690–709. 10.1007/s00039-010-0077-4Search in Google Scholar
[25] D. Jiang and L. Zhang, A product of tensor product L-functions of quasi-split classical groups of Hermitian type, Geom. Funct. Anal. 24 (2014), 552–609. 10.1007/s00039-014-0266-7Search in Google Scholar
[26] D. Jiang and L. Zhang, Arthur parameters and cuspidal automorphic modules of classical groups, preprint (2015), http://arxiv.org/abs/1508.03205. 10.4007/annals.2020.191.3.2Search in Google Scholar
[27] T. Kaletha, A. Minguez, S. W. Shin and P.-J. White, Endoscopic classification of representations: Inner forms of unitary groups, preprint (2014), http://arxiv.org/abs/1409.3731. Search in Google Scholar
[28] S. Kato and K. Takano, Square integrability of representations on p-adic symmetric spaces, J. Funct. Anal. 258 (2010), 1427–1451. 10.1016/j.jfa.2009.10.026Search in Google Scholar
[29] H. Kim and W. Kim, On local L-functions and normalized intertwining operators. II: Quasi-split groups, On certain L-functions, Clay Math. Proc. 13, American Mathematical Society, Providence (2011), 265–295. Search in Google Scholar
[30] H. Kim and M. Krishnamurthy, Base change lift for odd unitary groups, Functional analysis VIII, Various Publ. Ser. (Aarhus) 47, Aarhus University, Aarhus (2004), 116–125. Search in Google Scholar
[31]
H. Kim and M. Krishnamurthy,
Stable base change lift from unitary groups to
[32] E. Lapid and J. Rogawski, Periods of Eisenstein series: The Galois case, Duke Math. J. 120 (2003), no. 1, 153–226. 10.1215/S0012-7094-03-12016-5Search in Google Scholar
[33] W. Luo, Z. Rudnick and P. Sarnak, On the generalized Ramanujan conjecture for GL(n), Automorphic forms, automorphic representations, and arithmetic, Proc. Sympos. Pure Math. 66/2, American Mathematical Society, Providence (1999), 301–310. 10.1090/pspum/066.2/1703764Search in Google Scholar
[34] C. Mœglin and J.-L. Waldspurger, Spectral decomposition and Eisenstein series, Cambridge Tracts in Math. 113, Cambridge University Press, Cambridge 1995. 10.1017/CBO9780511470905Search in Google Scholar
[35] C. P. Mok, Endoscopic classification of representations of quasi-split unitary groups, Mem. Amer. Math. Soc. 235 (2015), no. 1108. 10.1090/memo/1108Search in Google Scholar
[36] G. Shimura, Arithmetic of hermitian forms, Doc. Math. 13 (2008), 739–774. 10.1007/978-3-319-32548-4_9Search in Google Scholar
[37] B. Sun and C.-B. Zhu, Multiplicity one theorems: The Archimedean case, Ann. of Math. (2) 175 (2012), 23–44. 10.4007/annals.2012.175.1.2Search in Google Scholar
[38] N. Wallach, Real reductive groups. II, Pure Appl. Math. 132, Academic Press, Boston 1992. Search in Google Scholar
[39] S. Yamana, Periods of automorphic forms: The trilinear case, J. Inst. Math. Jussieu (2015), 10.1017/S1474748015000377. 10.1017/S1474748015000377Search in Google Scholar
[40] S. Yamana, Periods of residual automorphic forms, J. Funct. Anal. 268 (2015), 1078–1104. 10.1016/j.jfa.2014.11.009Search in Google Scholar
[41]
A. Zelevinsky,
Induced representations of reductive
[42] W. Zhang, Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups, Ann. of Math. (2) 180 (2014), 971–1049. 10.4007/annals.2014.180.3.4Search in Google Scholar
[43] W. Zhang, Automorphic period and the central value of Rankin–Selberg L-function, J. Amer. Math. Soc. 27 (2014), 541–612. 10.1090/S0894-0347-2014-00784-0Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston