Calculus of archimedean Rankin–Selberg integrals with recurrence relations

Ishii, Taku; Miyazaki, Tadashi

doi:10.1090/ert/618

Calculus of archimedean Rankin–Selberg integrals with recurrence relations
HTML articles powered by AMS MathViewer

by Taku Ishii and Tadashi Miyazaki

Represent. Theory 26 (2022), 714-763

DOI: https://doi.org/10.1090/ert/618

Published electronically: July 6, 2022

PDF | Request permission

Abstract:

Let $n$ and $n’$ be positive integers such that $n-n’\in \{0,1\}$. Let $F$ be either $\mathbb {R}$ or $\mathbb {C}$. Let $K_n$ and $K_{n’}$ be maximal compact subgroups of $\mathrm {GL}(n,F)$ and $\mathrm {GL}(n’,F)$, respectively. We give the explicit descriptions of archimedean Rankin–Selberg integrals at the minimal $K_n$- and $K_{n’}$-types for pairs of principal series representations of $\mathrm {GL}(n,F)$ and $\mathrm {GL}(n’,F)$, using their recurrence relations. Our results for $F=\mathbb {C}$ can be applied to the arithmetic study of critical values of automorphic $L$-functions.

References

S. J. Ališauskas, A.-A. A. Jucys, and A. P. Jucys, On the symmetric tensor operators of the unitary groups, J. Mathematical Phys. 13 (1972), 1329–1333. MR 314374, DOI 10.1063/1.1666142
James W. Cogdell, Lectures on $L$-functions, converse theorems, and functoriality for $\textrm {GL}_n$, Lectures on automorphic $L$-functions, Fields Inst. Monogr., vol. 20, Amer. Math. Soc., Providence, RI, 2004, pp. 1–96. MR 2071506
Chao-Ping Dong and Huajian Xue, On the nonvanishing hypothesis for Rankin-Selberg convolutions for $\textrm {GL}_n(\Bbb C)\times \textrm {GL}_n(\Bbb C)$, Represent. Theory 21 (2017), 151–171. MR 3687651, DOI 10.1090/ert/502
I. M. Gel′fand and M. L. Cetlin, Finite-dimensional representations of the group of unimodular matrices, Doklady Akad. Nauk SSSR (N.S.) 71 (1950), 825–828 (Russian). MR 0035774
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
Harald Grobner and Michael Harris, Whittaker periods, motivic periods, and special values of tensor product $L$-functions, J. Inst. Math. Jussieu 15 (2016), no. 4, 711–769. MR 3569075, DOI 10.1017/S1474748014000462
Erich Hecke, Mathematische Werke, Vandenhoeck & Ruprecht, Göttingen, 1959 (German). Herausgegeben im Auftrage der Akademie der Wissenschaften zu Göttingen. MR 0104550
Miki Hirano, Taku Ishii, and Tadashi Miyazaki, Archimedean zeta integrals for $GL(3) \times GL(2)$, Mem. Amer. Math. Soc. 278 (2022), no. 1366, viii+122. MR 4426708, DOI 10.1090/memo/1366
Taku Ishii and Eric Stade, New formulas for Whittaker functions on $\textrm {GL}(n,\Bbb R)$, J. Funct. Anal. 244 (2007), no. 1, 289–314. MR 2294485, DOI 10.1016/j.jfa.2006.12.004
Taku Ishii and Eric Stade, Archimedean zeta integrals on $\mathrm {GL}_n\times \mathrm {GL}_m$ and $\mathrm {SO}_{2n+1}\times \mathrm {GL}_m$, Manuscripta Math. 141 (2013), no. 3-4, 485–536. MR 3062596, DOI 10.1007/s00229-012-0581-y
H. Jacquet and R. P. Langlands, Automorphic forms on $\textrm {GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654, DOI 10.1007/BFb0058988
H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367–464. MR 701565, DOI 10.2307/2374264
Hervé Jacquet, Automorphic forms on $\textrm {GL}(2)$. Part II, Lecture Notes in Mathematics, Vol. 278, Springer-Verlag, Berlin-New York, 1972. MR 0562503, DOI 10.1007/BFb0058503
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
Hervé Jacquet and Joseph Shalika, Rankin-Selberg convolutions: Archimedean theory, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 2, Weizmann, Jerusalem, 1990, pp. 125–207. MR 1159102
A. A. Jucis, The isoscalar factors of the Clebsch-Gordan coefficients of unitary groups, Litovsk. Fiz. Sb. 10 (1970), 5–12 (Russian, with English and Lithuanian summaries). MR 285204
Anthony W. Knapp, Representation theory of semisimple groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001. An overview based on examples; Reprint of the 1986 original. MR 1880691
Bertram Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), no. 2, 101–184. MR 507800, DOI 10.1007/BF01390249
Tadashi Miyazaki, The local zeta integrals for $GL(2,\mathbf {C})\times GL(2,\mathbf {C})$, Proc. Japan Acad. Ser. A Math. Sci. 94 (2018), no. 1, 1–6. MR 3743720, DOI 10.3792/pjaa.94.1
Alexandru A. Popa, Whittaker newforms for Archimedean representations, J. Number Theory 128 (2008), no. 6, 1637–1645. MR 2419183, DOI 10.1016/j.jnt.2007.06.005
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
R. A. Rankin, Contributions to the theory of Ramanujan’s function $\tau (n)$ and similar arithmetical functions. I. The zeros of the function $\sum ^\infty _{n=1}\tau (n)/n^s$ on the line ${\mathfrak {R}}s=13/2$. II. The order of the Fourier coefficients of integral modular forms, Proc. Cambridge Philos. Soc. 35 (1939), 351–372. MR 411, DOI 10.1017/S0305004100021095
Atle Selberg, Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. 43 (1940), 47–50 (German). MR 2626
Birgit Speh and David A. Vogan Jr., Reducibility of generalized principal series representations, Acta Math. 145 (1980), no. 3-4, 227–299. MR 590291, DOI 10.1007/BF02414191
Eric Stade, Mellin transforms of $\textrm {GL}(n,\Bbb R)$ Whittaker functions, Amer. J. Math. 123 (2001), no. 1, 121–161. MR 1827280, DOI 10.1353/ajm.2001.0004
Eric Stade, Archimedean $L$-factors on $\textrm {GL}(n)\times \textrm {GL}(n)$ and generalized Barnes integrals, Israel J. Math. 127 (2002), 201–219. MR 1900699, DOI 10.1007/BF02784531
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
N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie groups and special functions. Vol. 3, Mathematics and its Applications (Soviet Series), vol. 75, Kluwer Academic Publishers Group, Dordrecht, 1992. Classical and quantum groups and special functions; Translated from the Russian by V. A. Groza and A. A. Groza. MR 1206906, DOI 10.1007/978-94-017-2881-2
Nolan R. Wallach, Real reductive groups. II, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1992. MR 1170566
N. R. Wallach, $C^\infty$ vectors, Representations of Lie groups and quantum groups (Trento, 1993) Pitman Res. Notes Math. Ser., vol. 311, Longman Sci. Tech., Harlow, 1994, pp. 205–270. MR 1431308
Takao Watanabe, Global theta liftings of general linear groups, J. Math. Sci. Univ. Tokyo 3 (1996), no. 3, 699–711. MR 1432113
Shou-Wu Zhang, Gross-Zagier formula for $\textrm {GL}_2$, Asian J. Math. 5 (2001), no. 2, 183–290. MR 1868935, DOI 10.4310/AJM.2001.v5.n2.a1
D. P. Zhelobenko, On Gel′fand-Zetlin bases for classical Lie algebras, Representations of Lie groups and Lie algebras (Budapest, 1971) Akad. Kiadó, Budapest, 1985, pp. 79–106. MR 829046