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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
##### Keywords:
gamma factor; Rankin-Selberg integral
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##### References:
 [1] Aizenbud, A.; Gourevitch, D.; Rallis, S.; Schiffmann, G., Multiplicity one theorems, Ann. Math., 172, 1407-1434, (2010) · Zbl 1202.22012 [2] Banks, W., A corollary to bernstein’s theorem and Whittaker functionals on the metaplectic group, Math. Res. Lett., 5, 781-790, (1998) · Zbl 0944.22007 [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, 1-68, (1976) · Zbl 0348.43007 [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 SO_{$$n$$} × GL_{$$k$$}, 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 SO_{2$$n$$+1}: functorial lift to GL_{2}, endoscopy, and base change, Int. Math. Res. Notices, 729, 729-764, (2001) · Zbl 1060.11031 [11] Jacquet, H.; Piatetski-Shapiro, I.; Shalika, J.A., Rankin-Selberg convolutions, Am. J. Math., 105, 367-464, (1983) · Zbl 0525.22018 [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(2$$n$$ + 1) and applications, Ann. Math., 157, 743-806, (2003) · Zbl 1049.11055 [14] Kaplan, E., An invariant theory approach for the unramified computation of rankinv-Selberg integrals for quasi-split SO_{2$$n$$} × GL_{$$n$$}, J. Number Theory, 130, 1801-1817, (2010) · Zbl 1200.11035 [15] Kaplan, E., The unramified computation of rankinv-Selberg integrals for SO_{2$$l$$} × GL_{$$n$$}, Israel J. Math., 191, 137-184, (2012) · Zbl 1273.11086 [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. [17] Muić, G., A geometric construction of intertwining operators for reductive $$p$$-vadic groups, Manuscripta Math., 125, 241-272, (2008) · Zbl 1145.22010 [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, 297-355, (1981) · Zbl 0467.12013 [20] Shahidi, F., A proof of langlands’ conjecture on Plancherel measures; complementary series of $${\mathfrak{p}}$$ -adic groups, Ann. Math., 132, 273-330, (1990) · Zbl 0780.22005 [21] Soudry, D.: Rankin-Selberg convolutions for SO_{2$$l$$+1} × GL_{$$n$$}: local theory. Mem. Am. Math. Soc., 105(500) (1993) · Zbl 0805.22007 [22] Soudry, D., On the Archimedean theory of Rankin-Selberg convolutions for SO_{2$$l$$+1} × GL_{$$n$$}, Ann. Sci. Éc. Norm. Sup., 28, 161-224, (1995) · Zbl 0824.11034 [23] Soudry, D., Full multiplicativity of gamma factors for SO_{2$$l$$+1} × GL_{$$n$$}, Israel J. Math., 120, 511-561, (2000) · Zbl 1005.11027 [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, 235-333, (2003) · Zbl 1029.22016
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