The nonvanishing hypothesis at infinity for Rankin-Selberg convolutions

Sun, Binyong¹

¹ Institute of Mathematics and Hua Loo-Keng Key Laboratory of Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190, China

Journal of the American Mathematical Society, Tome 30 (2017) no. 1, pp. 1-25

Résumé

We prove the nonvanishing hypothesis at infinity for Rankin-Selberg convolutions for ${\mathrm {GL}}(n)\times {\mathrm {GL}}(n-1)$.

Détail

DOI : 10.1090/jams/855

Bibliographie

[1] Aizenbud, Avraham, Gourevitch, Dmitry Multiplicity one theorem for (𝐺𝐿_{𝑛+1}(ℝ),𝔾𝕃_{𝕟}(ℝ)) Selecta Math. (N.S.) 2009 271 294

[2] Borel, A., Wallach, N. Continuous cohomology, discrete subgroups, and representations of reductive groups 2000

[3] Bernstein, Joseph, Krã¶Tz, Bernhard Smooth Fréchet globalizations of Harish-Chandra modules Israel J. Math. 2014 45 111

[4] Casselman, W. Canonical extensions of Harish-Chandra modules to representations of 𝐺 Canad. J. Math. 1989 385 438

[5] Clozel, Laurent Motifs et formes automorphes: applications du principe de fonctorialité 1990 77 159

[6] Dixmier, Jacques Enveloping algebras 1996

[7] Flensted-Jensen, Mogens Discrete series for semisimple symmetric spaces Ann. of Math. (2) 1980 253 311

[8] Grobner, H., Harris, M. Whittaker periods, motivic periods, and special values of tensor product L-functions J. Inst. Math. Jussieu 2015

[9] Hecke, E. Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung Math. Ann. 1936 664 699

[10] Hecke, E. Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I Math. Ann. 1937 1 28

[11] Hecke, E. Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. II Math. Ann. 1937 316 351

[12] Jacquet, Hervã© Archimedean Rankin-Selberg integrals 2009 57 172

[13] Januszewski, Fabian Modular symbols for reductive groups and 𝑝-adic Rankin-Selberg convolutions over number fields J. Reine Angew. Math. 2011 1 45

[14] Januszewski, Fabian On 𝑝-adic 𝐿-functions for 𝐺𝐿(𝑛)×𝐺𝐿(𝑛-1) over totally real fields Int. Math. Res. Not. IMRN 2015 7884 7949

[15] Januszewski, F. p-adic L-functions for Rankin-Selberg convolutions over number fields

[16] Januszewski, F. On period relations for automorphic L-functions

[17] Kasten, Hendrik, Schmidt, Claus-Gã¼Nther The critical values of Rankin-Selberg convolutions Int. J. Number Theory 2013 205 256

[18] Kazhdan, D., Mazur, B., Schmidt, C.-G. Relative modular symbols and Rankin-Selberg convolutions J. Reine Angew. Math. 2000 97 141

[19] Knapp, Anthony W. Representation theory of semisimple groups 1986

[20] Knapp, Anthony W., Vogan, David A., Jr. Cohomological induction and unitary representations 1995

[21] Mahnkopf, Joachim Cohomology of arithmetic groups, parabolic subgroups and the special values of 𝐿-functions on 𝐺𝐿_{𝑛} J. Inst. Math. Jussieu 2005 553 637

[22] Parthasarathy, K. R., Ranga Rao, R., Varadarajan, V. S. Representations of complex semi-simple Lie groups and Lie algebras Ann. of Math. (2) 1967 383 429

[23] Raghuram, A. On the special values of certain Rankin-Selberg 𝐿-functions and applications to odd symmetric power 𝐿-functions of modular forms Int. Math. Res. Not. IMRN 2010 334 372

[24] Raghuram, A. Critical values of Rankin-Selberg L-functions for 𝐺𝐿(𝑛)×𝐺𝐿(𝑛-1) and the symmetric cube L-functions for 𝐺𝐿(2) Forum Math.

[25] Raghuram, A., Shahidi, Freydoon Functoriality and special values of 𝐿-functions 2008 271 293

[26] Raghuram, A., Shahidi, F. On certain period relations for cusp forms on 𝐺𝐿_{𝑛} Int. Math. Res. Not. IMRN 2008

[27] Schmidt, Claus-G. Relative modular symbols and 𝑝-adic Rankin-Selberg convolutions Invent. Math. 1993 31 76

[28] Schmidt, C.-G. Period relations and 𝑝-adic measures Manuscripta Math. 2001 177 201

[29] Sun, Binyong Positivity of matrix coefficients of representations with real infinitesimal characters Israel J. Math. 2009 395 410

[30] Sun, Binyong Bounding matrix coefficients for smooth vectors of tempered representations Proc. Amer. Math. Soc. 2009 353 357

[31] Sun, Binyong, Zhu, Chen-Bo Multiplicity one theorems: the Archimedean case Ann. of Math. (2) 2012 23 44

[32] Vogan, David A., Jr. Unitarizability of certain series of representations Ann. of Math. (2) 1984 141 187

[33] Vogan, David A., Jr. The unitary dual of 𝐺𝐿(𝑛) over an Archimedean field Invent. Math. 1986 449 505

[34] Vogan, David A., Jr., Zuckerman, Gregg J. Unitary representations with nonzero cohomology Compositio Math. 1984 51 90

[35] Wallach, Nolan R. Real reductive groups. I 1988

[36] Wallach, Nolan R. Real reductive groups. II 1992

[37] Wallach, Nolan R. On the unitarizability of derived functor modules Invent. Math. 1984 131 141

[38] Yacobi, Oded An analysis of the multiplicity spaces in branching of symplectic groups Selecta Math. (N.S.) 2010 819 855