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Multiplicativity of the gamma factors of Rankin-Selberg integrals for \(\mathrm{SO} _{2l} \times\mathrm{GL}_n\). (English) Zbl 1318.11158

Let \(F\) be a \(p\)-adic field. For positive integers \(l\) and \(n\), let \(H\) be a quasisplit form of the special orthogonal group \(\mathrm{SO}_{2l}\), and let \(G = \mathrm{GL}_n\). Let \(\pi\) and \(\tau\) be irreducible smooth generic representations of \(H(F)\) and \(G(F)\), respectively. The article under review studies the Rankin-Selberg integrals attached to the pair \(\pi,\tau\) by S. Gelbart et al. [Explicit constructions of automorphic \(L\)-functions. Berlin: Springer (1987; Zbl 0612.10022)] and D. Ginzburg [J. Reine Angew. Math. 405, 156–180 (1990; Zbl 0684.22009)]. These integrals can be used to define the \(\gamma\)-factor \(\Gamma (\pi\times\tau,\psi,s)\), where \(\psi\) is a fixed additive character of \(F\).
The author proves that the association \((\pi, \tau) \rightsquigarrow\Gamma (\pi\times\tau,\psi,s)\) behaves multiplicatively in the arguments \(\pi\) and \(\tau\) with respect to parabolic induction. Using archimedean results of D. Soudry [Ann. Sci. Éc. Norm. Supér. (4) 28, No. 2, 161–224 (1995; Zbl 0824.11034)] and global arguments, the author then shows that the function \(\Gamma (\pi\times\tau,\psi,s)\) coincides with the standard \(\gamma\)-factor defined by Shahidi. The results of this paper have been used to show that the local \(L\)-factor attached to \(\pi ,\tau\) (in the tempered case) defined via Rankin-Selberg integrals coincides with the \(L\)-function \(L(\pi\times\tau,s)\) defined by Shahidi.

MSC:

11S40 Zeta functions and \(L\)-functions
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
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[1] Aizenbud A., Gourevitch D., Rallis S., Schiffmann G.: Multiplicity one theorems. Ann. Math. 172(2), 1407-1434 (2010) · Zbl 1202.22012 · doi:10.4007/annals.2010.172.1413
[2] Banks W.: A corollary to Bernstein’s theorem and Whittaker functionals on the metaplectic group. Math. Res. Lett. 5(6), 781-790 (1998) · Zbl 0944.22007 · doi:10.4310/MRL.1998.v5.n6.a7
[3] Bernstein I.N., Zelevinsky A.V.: Representations of the group GL(n,F) where F is a local non-Archimedean field. Russ. Math. Surveys 31(3), 1-68 (1976) · Zbl 0348.43007 · doi:10.1070/RM1976v031n03ABEH001532
[4] Casselman W., Shalika J.A.: The unramified principal series of p-adic groups II: the Whittaker function. Compos. Math. 41, 207-231 (1980) · Zbl 0472.22005
[5] Cogdell J.W., Kim H.H., Piatetski-Shapiro I., Shahidi F.: Functoriality for the classical groups. Publ. Math. IHES 99, 163-233 (2004) · Zbl 1090.22010
[6] Gan, W.T., Gross, B.H., Prasad, D.: Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups, Asterisque 346 (2012). http://arxiv.org/abs/0909.2999v1 · Zbl 1280.22019
[7] Gelbart, S., Piatetski-Shapiro, I., Rallis, S.: L-functions for G × GL(n). Lecture Notes in Mathemetics, vol. 1254, Springer, New York (1987) · Zbl 0612.10022
[8] Ginzburg D.: L-functions for SOn × GLk. J. Reine Angew. Math. 405, 156-180 (1990) · Zbl 0684.22009
[9] Ginzburg, D., Piatetski-Shapiro, I., Rallis, S.: L-functions for the orthogonal group. Mem. Am. Math. Soc. 128(611), (1997) · Zbl 0884.11022
[10] Ginzburg D., Rallis S., Soudry D.: Generic automorphic forms of SO2n+1: functorial lift to GL2, endoscopy, and base change. Int. Math. Res. Notices 729(14), 729-764 (2001) · Zbl 1060.11031 · doi:10.1155/S1073792801000381
[11] Jacquet H., Piatetski-Shapiro I., Shalika J.A.: Rankin-Selberg convolutions. Am. J. Math. 105(2), 367-464 (1983) · Zbl 0525.22018 · doi:10.2307/2374264
[12] Jacquet, H., Shalika, J.A.: Rankin-Selberg convolutions: Archimedean theory, Festschrift in Honor of I. Piatetskiv-Shapiro, Part I, pp. 125-207. Weizmann Science Press, Jerusalem (1990) · Zbl 0472.22005
[13] Jiang D., Soudry D.: The local converse theorem for SO(2n + 1) and applications. Ann. Math. 157(3), 743-806 (2003) · Zbl 1049.11055 · doi:10.4007/annals.2003.157.743
[14] Kaplan E.: An invariant theory approach for the unramified computation of Rankinv-Selberg integrals for quasi-split SO2n × GLn. J. Number Theory 130, 1801-1817 (2010) · Zbl 1200.11035 · doi:10.1016/j.jnt.2010.02.004
[15] Kaplan E.: The unramified computation of Rankinv-Selberg integrals for SO2l × GLn. Israel J. Math. 191(1), 137-184 (2012) · Zbl 1273.11086 · doi:10.1007/s11856-011-0203-5
[16] Moeglin, C., Waldspurger, J.-L.: La conjecture locale de Gross-Prasad pour les groupes spéciaux orthogonaux: le cas général (2010). http://arxiv.org/abs/1001.0826v1. · Zbl 1276.22007
[17] Muić G.: A geometric construction of intertwining operators for reductive p-vadic groups. Manuscripta Math. 125, 241-272 (2008) · Zbl 1145.22010 · doi:10.1007/s00229-007-0146-7
[18] Shahidi F.: Functional equation satisfied by certain L-functions. Compos. Math. 37, 171-208 (1978) · Zbl 0393.12017
[19] Shahidi F.: On certain L-functions. Am. J. Math. 103(2), 297-355 (1981) · Zbl 0467.12013 · doi:10.2307/2374219
[20] Shahidi F.: A proof of Langlands’ conjecture on Plancherel measures; complementary series of \[{\mathfrak{p}} \] -adic groups. Ann. Math. 132(2), 273-330 (1990) · Zbl 0780.22005 · doi:10.2307/1971524
[21] Soudry, D.: Rankin-Selberg convolutions for SO2l+1 × GLn: local theory. Mem. Am. Math. Soc., 105(500) (1993) · Zbl 0805.22007
[22] Soudry D.: On the Archimedean theory of Rankin-Selberg convolutions for SO2l+1 × GLn. Ann. Sci. Éc. Norm. Sup. 28(2), 161-224 (1995) · Zbl 0824.11034
[23] Soudry D.: Full multiplicativity of gamma factors for SO2l+1 × GLn. Israel J. Math. 120(1), 511-561 (2000) · Zbl 1005.11027 · doi:10.1007/BF02834849
[24] Soudry, D.: Rankin-Selberg integrals, the descent method, and Langlands functoriality. In: Proceedings of the International Congress of Mathematicians, pp. 1311-1325. EMS, Madrid (2006) · Zbl 1130.11024
[25] Waldspurger J.-L.: La formule de Plancherel d’après Harish-Chandra. J. Inst. Math. Jussieu 2(2), 235-333 (2003) · Zbl 1029.22016 · doi:10.1017/S1474748003000082
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