On the nonvanishing hypothesis for Rankin-Selberg convolutions for 𝐺𝐿_{𝑛}(ℂ)×𝔾𝕃_{𝕟}(ℂ)

Dong, Chao-Ping; Xue, Huajian

doi:10.1090/ert/502

On the nonvanishing hypothesis for Rankin-Selberg convolutions for $\mathrm {GL}_n(\mathbb {C})\times \mathrm {GL}_n(\mathbb {C})$
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by Chao-Ping Dong and Huajian Xue

Represent. Theory 21 (2017), 151-171

DOI: https://doi.org/10.1090/ert/502

Published electronically: August 21, 2017

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Abstract:

Inspired by Sun’s breakthrough in establishing the nonvanishing hypothesis for Rankin-Selberg convolutions for the groups $\mathrm {GL}_n (\mathbb {R})\times \mathrm {GL}_{n-1} (\mathbb {R})$ and $\mathrm {GL}_n (\mathbb {C})\times \mathrm {GL}_{n-1} (\mathbb {C})$, we confirm it for $\mathrm {GL}_{n} (\mathbb {C})\times \mathrm {GL}_n (\mathbb {C})$ at the central critical point.

References

A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, 2nd ed., Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000. MR 1721403, DOI 10.1090/surv/067
W. Casselman, Canonical extensions of Harish-Chandra modules to representations of $G$, Canad. J. Math. 41 (1989), no. 3, 385–438. MR 1013462, DOI 10.4153/CJM-1989-019-5
Laurent Clozel, Motifs et formes automorphes: applications du principe de fonctorialité, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988) Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 77–159 (French). MR 1044819
Roe Goodman and Nolan R. Wallach, Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol. 255, Springer, Dordrecht, 2009. MR 2522486, DOI 10.1007/978-0-387-79852-3
Loïc Grenié, Critical values of automorphic $L$-functions for $\textrm {GL}(r)\times \textrm {GL}(r)$, Manuscripta Math. 110 (2003), no. 3, 283–311. MR 1969002, DOI 10.1007/s00229-002-0333-5
Hervé Jacquet, Archimedean Rankin-Selberg integrals, Automorphic forms and $L$-functions II. Local aspects, Contemp. Math., vol. 489, Amer. Math. Soc., Providence, RI, 2009, pp. 57–172. MR 2533003, DOI 10.1090/conm/489/09547
Hendrik Kasten and Claus-Günther Schmidt, The critical values of Rankin-Selberg convolutions, Int. J. Number Theory 9 (2013), no. 1, 205–256. MR 2997500, DOI 10.1142/S1793042112501345
A. W. Knapp, Local Langlands correspondence: the Archimedean case, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 393–410. MR 1265560, DOI 10.1090/pspum/055.2/1265560
—, Lie Groups Beyond an Introduction, Birkhäuser, 2nd Edition, 2002.
Anthony W. Knapp and David A. Vogan Jr., Cohomological induction and unitary representations, Princeton Mathematical Series, vol. 45, Princeton University Press, Princeton, NJ, 1995. MR 1330919, DOI 10.1515/9781400883936
Joachim Mahnkopf, Cohomology of arithmetic groups, parabolic subgroups and the special values of $L$-functions on $\textrm {GL}_n$, J. Inst. Math. Jussieu 4 (2005), no. 4, 553–637. MR 2171731, DOI 10.1017/S1474748005000186
K. R. Parthasarathy, R. Ranga Rao, and V. S. Varadarajan, Representations of complex semi-simple Lie groups and Lie algebras, Ann. of Math. (2) 85 (1967), 383–429. MR 225936, DOI 10.2307/1970351
A. Raghuram, Critical values of Rankin-Selberg $L$-functions for $\text {GL}_n\times \text {GL}_{n-1}$ and the symmetric cube $L$-functions for $\text {GL}_2$, Forum Math. 28 (2016), no. 3, 457–489. MR 3510825, DOI 10.1515/forum-2014-0043
Binyong Sun, Positivity of matrix coefficients of representations with real infinitesimal characters, Israel J. Math. 170 (2009), 395–410. MR 2506332, DOI 10.1007/s11856-009-0034-9
Binyong Sun, Bounding matrix coefficients for smooth vectors of tempered representations, Proc. Amer. Math. Soc. 137 (2009), no. 1, 353–357. MR 2439460, DOI 10.1090/S0002-9939-08-09598-1
Binyong Sun, The nonvanishing hypothesis at infinity for Rankin-Selberg convolutions, J. Amer. Math. Soc. 30 (2017), no. 1, 1–25. MR 3556287, DOI 10.1090/jams/855
Binyong Sun and Chen-Bo Zhu, Multiplicity one theorems: the Archimedean case, Ann. of Math. (2) 175 (2012), no. 1, 23–44. MR 2874638, DOI 10.4007/annals.2012.175.1.2
Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR 929683
Nolan R. Wallach, Real reductive groups. II, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1992. MR 1170566
Huajian Xue, Homogeneous distributions on finite dimensional vector spaces, J. Lie Theory, 28 (2018), 33–41.
Oded Yacobi, An analysis of the multiplicity spaces in branching of symplectic groups, Selecta Math. (N.S.) 16 (2010), no. 4, 819–855. MR 2734332, DOI 10.1007/s00029-010-0033-z
D. P. Zhelobenko, Garmonicheskiĭ analiz na poluprostykh kompleksnykh gruppakh Li, Sovremennye Problemy Matematiki. [Current Problems in Mathematics], Izdat. “Nauka”, Moscow, 1974 (Russian). MR 0579170