Arthur Parameters and Fourier coefficients for Automorphic Forms on Symplectic Groups

Jiang, Dihua; Liu, Baiying

Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 477-519.

Nous étudions la structures des coefficients de Fourier des formes automorphes sur des groupes symplectiques à partir de leurs structures locale et globale liée aux paramètres d’Arthur. Ceci est la première étape pour prouver une conjecture du premier auteur concernant le lien entre la structure des coefficients de Fourier et les paramètres d’Arthur pour les formes automorphes dans le spectre discret.

We study the structures of Fourier coefficients of automorphic forms on symplectic groups based on their local and global structures related to Arthur parameters. This is a first step towards the general conjecture on the relation between the structure of Fourier coefficients and Arthur parameters for automorphic forms occurring in the discrete spectrum, given by the first named author.

Classification : 11F70, 22E50, 11F85, 22E55
Keywords: Arthur Parameters, Fourier Coefficients, Unipotent Orbits, Automorphic Forms
Mot clés : Paramètres d’Arthur, Coefficients de Fourier, Orbites Unipotentes, Formes Automorphes

[1] Achar, Pramod N. An order-reversing duality map for conjugacy classes in Lusztig’s canonical quotient, Transform. Groups, Volume 8 (2003) no. 2, pp. 107-145 | DOI

[2] Arthur, James The endoscopic classification of representations, American Mathematical Society Colloquium Publications, 61, American Mathematical Society, Providence, RI, 2013, xviii+590 pages (Orthogonal and symplectic groups)

[3] Barbasch, Dan The unitary spherical spectrum for split classical groups, J. Inst. Math. Jussieu, Volume 9 (2010) no. 2, pp. 265-356 | DOI

[4] Barbasch, Dan; Vogan, David A. Jr. Unipotent representations of complex semisimple groups, Ann. of Math. (2), Volume 121 (1985) no. 1, pp. 41-110 | DOI

[5] Collingwood, David H.; McGovern, William M. Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993, xiv+186 pages

[6] Gan, Wee Teck; Gross, Benedict H.; Prasad, Dipendra Symplectic local root numbers, central critical $L$ values, and restriction problems in the representation theory of classical groups, Astérisque (2012) no. 346, pp. 1-109 (Sur les conjectures de Gross et Prasad. I)

[7] Ginzburg, David Constructing automorphic representations in split classical groups, Electron. Res. Announc. Math. Sci., Volume 19 (2012), pp. 18-32 | DOI

[8] Ginzburg, David; Jiang, Dihua; Rallis, Stephen On the nonvanishing of the central value of the Rankin-Selberg $L$-functions, J. Amer. Math. Soc., Volume 17 (2004) no. 3, p. 679-722 (electronic) | DOI

[9] Ginzburg, David; Jiang, Dihua; Rallis, Stephen; Soudry, David $L$-functions for symplectic groups using Fourier-Jacobi models, Arithmetic geometry and automorphic forms (Adv. Lect. Math. (ALM)), Volume 19, Int. Press, Somerville, MA, 2011, pp. 183-207

[10] Ginzburg, David; Rallis, Stephen; Soudry, David On Fourier coefficients of automorphic forms of symplectic groups, Manuscripta Math., Volume 111 (2003) no. 1, pp. 1-16 | DOI

[11] Ginzburg, David; Rallis, Stephen; Soudry, David Construction of CAP representations for symplectic groups using the descent method, Automorphic representations, $L$-functions and applications: progress and prospects (Ohio State Univ. Math. Res. Inst. Publ.), Volume 11, de Gruyter, Berlin, 2005, pp. 193-224 | DOI

[12] Ginzburg, David; Rallis, Stephen; Soudry, David The descent map from automorphic representations of $\mathrm{GL}\left(n\right)$ to classical groups, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011, x+339 pages | DOI

[13] Jiang, Dihua Automorphic integral transforms for classical groups I: Endoscopy correspondences, Automorphic forms and related geometry: assessing the legacy of I. I. Piatetski-Shapiro (Contemp. Math.), Volume 614, Amer. Math. Soc., Providence, RI, 2014, pp. 179-242 | DOI

[14] Jiang, Dihua; Liu, Baiying Fourier coefficients for automorphic forms on quasisplit classical groups (Accepted by a special volume in honor of J. Cogdell, Comtemp. Math., AMS, 2015.)

[15] Jiang, Dihua; Liu, Baiying On Fourier coefficients of automorphic forms of $\mathrm{GL}\left(n\right)$, Int. Math. Res. Not. IMRN (2013) no. 17, pp. 4029-4071

[16] Jiang, Dihua; Liu, Baiying On special unipotent orbits and Fourier coefficients for automorphic forms on symplectic groups, J. Number Theory, Volume 146 (2015), pp. 343-389 | DOI

[17] Jiang, Dihua; Liu, Baiying; Zhang, Lei Poles of certain residual Eisenstein series of classical groups, Pacific J. Math., Volume 264 (2013) no. 1, pp. 83-123 | DOI

[18] Jiang, Dihua; Zhang, Lei A product of tensor product $L$-functions of quasi-split classical groups of Hermitian type, Geom. Funct. Anal., Volume 24 (2014) no. 2, pp. 552-609 | DOI

[19] Kudla, S. Note on the local theta correspondence (Preprint, 1996)

[20] Lang, Serge Algebraic number theory, Graduate Texts in Mathematics, 110, Springer-Verlag, New York, 1994, xiv+357 pages | DOI

[21] Liu, Baiying Fourier Coefficients of Automorphic Forms and Arthur Classification, ProQuest LLC, Ann Arbor, MI, 2013, 127 pages Thesis (Ph.D.)–University of Minnesota

[22] Muić, Goran On the non-unitary unramified dual for classical $p$-adic groups, Trans. Amer. Math. Soc., Volume 358 (2006) no. 10, p. 4653-4687 (electronic) | DOI

[23] Muić, Goran On certain classes of unitary representations for split classical groups, Canad. J. Math., Volume 59 (2007) no. 1, pp. 148-185 | DOI

[24] Muić, Goran; Tadić, Marko Unramified unitary duals for split classical $p$-adic groups; the topology and isolated representations, On certain $L$-functions (Clay Math. Proc.), Volume 13, Amer. Math. Soc., Providence, RI, 2011, pp. 375-438

[25] Nevins, Monica On nilpotent orbits of ${\mathrm{SL}}_n$ and ${\mathrm{Sp}}_{2n}$ over a local non-Archimedean field, Algebr. Represent. Theory, Volume 14 (2011) no. 1, pp. 161-190 | DOI

[26] O’Meara, O. T. Introduction to quadratic forms, Springer-Verlag, New York-Heidelberg, 1971, xi+342 pages (Second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 117)

[27] Piatetski-Shapiro, I. I. Multiplicity one theorems, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1 (Proc. Sympos. Pure Math., XXXIII), Amer. Math. Soc., Providence, R.I., 1979, pp. 209-212

[28] Shalika, J. A. The multiplicity one theorem for ${\mathrm{GL}}_n$, Ann. of Math. (2), Volume 100 (1974), pp. 171-193 | DOI

[29] Waldspurger, Jean-Loup Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés, Astérisque (2001) no. 269, vi+449 pages