Eulerianity of Fourier coefficients of automorphic forms

Gourevitch, Dmitry; Gustafsson, Henrik; Kleinschmidt, Axel; Persson, Daniel; Sahi, Siddhartha

doi:10.1090/ert/565

Eulerianity of Fourier coefficients of automorphic forms
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by Dmitry Gourevitch, Henrik P. A. Gustafsson, Axel Kleinschmidt, Daniel Persson and Siddhartha Sahi

Represent. Theory 25 (2021), 481-507

DOI: https://doi.org/10.1090/ert/565

Published electronically: June 7, 2021

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Abstract:

We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another coefficient. We also establish a ‘hidden’ invariance property of Fourier coefficients. We apply these results to minimal and next-to-minimal automorphic representations, and deduce Eulerianity for a large class of Fourier and Fourier–Jacobi coefficients. In particular, we prove Eulerianity for parabolic Fourier coefficients with characters of maximal rank for a class of Eisenstein series in minimal and next-to-minimal representations of groups of ADE-type that are of interest in string theory.

References

O. Ahlén, H. P. A. Gustafsson, A. Kleinschmidt, B. Liu, and D. Persson, Fourier coefficients attached to small automorphic representations of ${\mathrm {SL}}_n(\mathbb {A})$, J. Number Theory 192 (2018) 80–142, arXiv:1707.08937.
Guillaume Bossard and Axel Kleinschmidt, Supergravity divergences, supersymmetry and automorphic forms, J. High Energy Phys. 8 (2015), 102, front matter + 45. MR 3400728, DOI 10.1007/JHEP08(2015)102
N. Bourbaki, Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1364, Hermann, Paris, 1975 (French). MR 0453824
Daniel Bump, Automorphic forms on $\textrm {GL}(3,\textbf {R})$, Lecture Notes in Mathematics, vol. 1083, Springer-Verlag, Berlin, 1984. MR 765698, DOI 10.1007/BFb0100147
David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060
W. Casselman and J. Shalika, The unramified principal series of $p$-adic groups. II. The Whittaker function, Compositio Math. 41 (1980), no. 2, 207–231. MR 581582
Dragomir Ž. Đoković, Note on rational points in nilpotent orbits of semisimple groups, Indag. Math. (N.S.) 9 (1998), no. 1, 31–34. MR 1618239, DOI 10.1016/S0019-3577(97)87565-9
Alexander Dvorsky and Siddhartha Sahi, Explicit Hilbert spaces for certain unipotent representations. II, Invent. Math. 138 (1999), no. 1, 203–224. MR 1714342, DOI 10.1007/s002220050347
Philipp Fleig, Henrik P. A. Gustafsson, Axel Kleinschmidt, and Daniel Persson, Eisenstein series and automorphic representations, Cambridge Studies in Advanced Mathematics, vol. 176, Cambridge University Press, Cambridge, 2018. With applications in string theory. MR 3793195, DOI 10.1017/9781316995860
Philipp Fleig, Axel Kleinschmidt, and Daniel Persson, Fourier expansions of Kac-Moody Eisenstein series and degenerate Whittaker vectors, Commun. Number Theory Phys. 8 (2014), no. 1, 41–100. MR 3260286, DOI 10.4310/CNTP.2014.v8.n1.a2
D. Gourevitch, H. P. A. Gustafsson, A. Kleinschmidt, D. Persson, and S. Sahi, A reduction principle for Fourier coefficients of automorphic forms, arXiv:1811.05966v5.
D. Gourevitch, H. P. A. Gustafsson, A. Kleinschmidt, D. Persson, and S. Sahi, Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups, Can. J. Math. online (2020) 1–48, arXiv:1908.08296.
Wee Teck Gan, Benedict H. Gross, and Dipendra Prasad, Symplectic local root numbers, central critical $L$ values, and restriction problems in the representation theory of classical groups, Astérisque 346 (2012), 1–109 (English, with English and French summaries). Sur les conjectures de Gross et Prasad. I. MR 3202556
R. Gomez, D. Gourevitch, and S. Sahi, Whittaker supports for representations of reductive groups, Annales de l’Institut Fourier, arXiv:1610.00284. To appear.
Raul Gomez, Dmitry Gourevitch, and Siddhartha Sahi, Generalized and degenerate Whittaker models, Compos. Math. 153 (2017), no. 2, 223–256. MR 3705224, DOI 10.1112/S0010437X16007788
David Ginzburg and Joseph Hundley, Constructions of global integrals in the exceptional group $F_4$, Kyushu J. Math. 67 (2013), no. 2, 389–417. MR 3115211, DOI 10.2206/kyushujm.67.389
David Ginzburg, Certain conjectures relating unipotent orbits to automorphic representations, Israel J. Math. 151 (2006), 323–355. MR 2214128, DOI 10.1007/BF02777366
David Ginzburg, Eulerian integrals for $\textrm {GL}_n$, Multiple Dirichlet series, automorphic forms, and analytic number theory, Proc. Sympos. Pure Math., vol. 75, Amer. Math. Soc., Providence, RI, 2006, pp. 203–223. MR 2279938, DOI 10.1090/pspum/075/2279938
David Ginzburg, Towards a classification of global integral constructions and functorial liftings using the small representations method, Adv. Math. 254 (2014), 157–186. MR 3161096, DOI 10.1016/j.aim.2013.12.007
I. M. Gel′fand and D. A. Kajdan, Representations of the group $\textrm {GL}(n,K)$ where $K$ is a local field, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 95–118. MR 0404534
Henrik P. A. Gustafsson, Axel Kleinschmidt, and Daniel Persson, Small automorphic representations and degenerate Whittaker vectors, J. Number Theory 166 (2016), 344–399. MR 3486281, DOI 10.1016/j.jnt.2016.02.002
M. B. Green, S. D. Miller, and P. Vanhove, Small representations, string instantons, and Fourier modes of Eisenstein series, J. Number Theor. 146 (2015) 187–309, arXiv:1111.2983.
David Ginzburg, Stephen Rallis, and David Soudry, On the automorphic theta representation for simply laced groups, Israel J. Math. 100 (1997), 61–116. MR 1469105, DOI 10.1007/BF02773635
D. Ginzburg, S. Rallis, and D. Soudry, On Fourier coefficients of automorphic forms of symplectic groups, Manuscripta Math. 111 (2003), no. 1, 1–16. MR 1981592, DOI 10.1007/s00229-003-0355-7
David Ginzburg, Stephen Rallis, and David Soudry, The descent map from automorphic representations of $\textrm {GL}(n)$ to classical groups, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. MR 2848523, DOI 10.1142/9789814304993
Wee Teck Gan and Gordan Savin, On minimal representations definitions and properties, Represent. Theory 9 (2005), 46–93. MR 2123125, DOI 10.1090/S1088-4165-05-00191-3
Joachim Hilgert, Toshiyuki Kobayashi, and Jan Möllers, Minimal representations via Bessel operators, J. Math. Soc. Japan 66 (2014), no. 2, 349–414. MR 3201818, DOI 10.2969/jmsj/06620349
Joseph Hundley and Eitan Sayag, Descent construction for GSpin groups, Mem. Amer. Math. Soc. 243 (2016), no. 1148, v+124. MR 3517154, DOI 10.1090/memo/1148
Tamotsu Ikeda, On the theory of Jacobi forms and Fourier-Jacobi coefficients of Eisenstein series, J. Math. Kyoto Univ. 34 (1994), no. 3, 615–636. MR 1295945, DOI 10.1215/kjm/1250518935
Dihua Jiang and Baiying Liu, On Fourier coefficients of automorphic forms of $\textrm {GL}(n)$, Int. Math. Res. Not. IMRN 17 (2013), 4029–4071. MR 3096918, DOI 10.1093/imrn/rns153
Dihua Jiang, Baiying Liu, and Gordan Savin, Raising nilpotent orbits in wave-front sets, Represent. Theory 20 (2016), 419–450. MR 3564676, DOI 10.1090/ert/490
Dihua Jiang and Stephen Rallis, Fourier coefficients of Eisenstein series of the exceptional group of type $G_2$, Pacific J. Math. 181 (1997), no. 2, 281–314. MR 1486533, DOI 10.2140/pjm.1997.181.281
Dihua Jiang, Binyong Sun, and Chen-Bo Zhu, Uniqueness of Bessel models: the Archimedean case, Geom. Funct. Anal. 20 (2010), no. 3, 690–709. MR 2720228, DOI 10.1007/s00039-010-0077-4
Bertram Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), no. 2, 101–184. MR 507800, DOI 10.1007/BF01390249
D. Kazhdan and A. Polishchuk, Minimal representations: spherical vectors and automorphic functionals, Algebraic groups and arithmetic, Tata Inst. Fund. Res., Mumbai, 2004, pp. 127–198. MR 2094111
D. Kazhdan, B. Pioline, and A. Waldron, Minimal representations, spherical vectors, and exceptional theta series, Commun. Math. Phys. 226 (2002) 1–40, arXiv:hep-th/0107222.
D. Kazhdan and G. Savin, The smallest representation of simply laced groups, 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. 209–223. MR 1159103
Toshiyuki Kobayashi and Gordan Savin, Global uniqueness of small representations, Math. Z. 281 (2015), no. 1-2, 215–239. MR 3384868, DOI 10.1007/s00209-015-1481-0
Robert P. Langlands, Euler products, Yale Mathematical Monographs, vol. 1, Yale University Press, New Haven, Conn.-London, 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1967. MR 0419366
Hung Yean Loke and Gordan Savin, Rank and matrix coefficients for simply laced groups, J. Reine Angew. Math. 599 (2006), 201–216. MR 2279102, DOI 10.1515/CRELLE.2006.082
Stephen D. Miller and Siddhartha Sahi, Fourier coefficients of automorphic forms, character variety orbits, and small representations, J. Number Theory 132 (2012), no. 12, 3070–3108. MR 2965210, DOI 10.1016/j.jnt.2012.05.032
Jan Möllers and Benjamin Schwarz, Bessel operators on Jordan pairs and small representations of semisimple Lie groups, J. Funct. Anal. 272 (2017), no. 5, 1892–1955. MR 3596711, DOI 10.1016/j.jfa.2016.10.026
C. Mœglin and J.-L. Waldspurger, Modèles de Whittaker dégénérés pour des groupes $p$-adiques, Math. Z. 196 (1987), no. 3, 427–452 (French). MR 913667, DOI 10.1007/BF01200363
C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de $\textrm {GL}(n)$, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 605–674 (French). MR 1026752, DOI 10.24033/asens.1595
C. Mœglin and J.-L. Waldspurger, Spectral decomposition and Eisenstein series, Cambridge Tracts in Mathematics, vol. 113, Cambridge University Press, Cambridge, 1995. Une paraphrase de l’Écriture [A paraphrase of Scripture]. MR 1361168, DOI 10.1017/CBO9780511470905
Omer Offen and Eitan Sayag, Global mixed periods and local Klyachko models for the general linear group, Int. Math. Res. Not. IMRN 1 (2008), Art. ID rnm 136, 25. MR 2417789, DOI 10.1093/imrn/rnm136
Aaron Pollack, The Fourier expansion of modular forms on quaternionic exceptional groups, Duke Math. J. 169 (2020), no. 7, 1209–1280. MR 4094735, DOI 10.1215/00127094-2019-0063
Boris Pioline and Daniel Persson, The automorphic NS5-brane, Commun. Number Theory Phys. 3 (2009), no. 4, 697–754. MR 2610484, DOI 10.4310/CNTP.2009.v3.n4.a5
François Rodier, Whittaker models for admissible representations of reductive $p$-adic split groups, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 425–430. MR 0354942
J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 0344216, DOI 10.1007/978-1-4684-9884-4
J. A. Shalika, The multiplicity one theorem for $\textrm {GL}_{n}$, Ann. of Math. (2) 100 (1974), 171–193. MR 348047, DOI 10.2307/1971071
Freydoon Shahidi, Functional equation satisfied by certain $L$-functions, Compositio Math. 37 (1978), no. 2, 171–207. MR 498494
Freydoon Shahidi, On certain $L$-functions, Amer. J. Math. 103 (1981), no. 2, 297–355. MR 610479, DOI 10.2307/2374219
Nicolas Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, vol. 946, Springer-Verlag, Berlin-New York, 1982 (French). MR 672610, DOI 10.1007/BFb0096302
Birgit Speh, Unitary representations of $\textrm {Gl}(n,\,\textbf {R})$ with nontrivial $({\mathfrak {g}},\,K)$-cohomology, Invent. Math. 71 (1983), no. 3, 443–465. MR 695900, DOI 10.1007/BF02095987
Gordan Savin and Michael Woodbury, Structure of internal modules and a formula for the spherical vector of minimal representations, J. Algebra 312 (2007), no. 2, 755–772. MR 2333183, DOI 10.1016/j.jalgebra.2007.01.014
David A. Vogan Jr., Singular unitary representations, Noncommutative harmonic analysis and Lie groups (Marseille, 1980) Lecture Notes in Math., vol. 880, Springer, Berlin-New York, 1981, pp. 506–535. MR 644845