Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T16:14:38.920Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A CERTAIN LOCAL IDENTITY FOR LAPID–MAO’S CONJECTURE AND FORMAL DEGREE CONJECTURE : EVEN UNITARY GROUP CASE

Part of: Discontinuous groups and automorphic forms Lie groups

Published online by Cambridge University Press:  08 January 2021

Kazuki Morimoto Open the ORCID record for Kazuki Morimoto [Opens in a new window]
Kazuki Morimoto*
Affiliation:
Department of Mathematics, Graduate School of Science, Kobe University, 1-1, Rokkodai-cho, Nada-ku, Kobe, 657-8501, Japan (morimoto@math.kobe-u.ac.jp)
Get access Rights & Permissions [Opens in a new window]

Abstract

Lapid and Mao formulated a conjecture on an explicit formula of Whittaker–Fourier coefficients of automorphic forms on quasi-split reductive groups and metaplectic groups as an analogue of the Ichino–Ikeda conjecture. They also showed that this conjecture is reduced to a certain local identity in the case of unitary groups. In this article, we study the even unitary-group case. Indeed, we prove this local identity over p-adic fields. Further, we prove an equivalence between this local identity and a refined formal degree conjecture over any local field of characteristic zero. As a consequence, we prove a refined formal degree conjecture over p-adic fields and get an explicit formula of Whittaker–Fourier coefficients under certain assumptions.

Keywords

automorphic representationsspecial values of automorphic L-functionsrepresentation theory of p-adic reductive groups

MSC classification

Primary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 4 , July 2022 , pp. 1107 - 1161
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anandavardhanan, U. K., Root numbers of Asai L-functions, Int. Math. Res. Not. IMRN 2008 (2008), rnn125.CrossRefGoogle Scholar
Anandavardhanan, U. K., Kable, A. C. and and Tandon, R., Distinguished representations and poles of twisted tensor $L$ -functions, Proc. Amer. Math. Soc. 132(10) (2004), 2875–2883.CrossRefGoogle Scholar
Arthur, J., The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups , Amer. Math. Soc. Colloq. Publ. 61 (Providence, American Mathematical Society, 2013).Google Scholar
Atobe, H., On the uniqueness of generic representations in an $L$ -packet, Int. Math. Res. Not. IMRN 2017(23) (2017), 7051–7068.Google Scholar
Baruch, E. M., Bessel functions for $\mathrm{GL}(n)$ over a $p$ -adic field, in Automorphic Representations, L-functions and Applications: Progress and Prospects, Ohio State University Mathematical Research Institute Publications 11 (Berlin, de Gruyter, 2005), 1–40.Google Scholar
Ben-Artzi, A. and Soudry, D., On $L$ -functions for ${\mathrm{U}}_{2k}\times {\mathrm{R}}_{E/F}{\mathrm{GL}}_m$ , ( $k<m$ ), Contemp. Math. 664 (2016), 69–104.CrossRefGoogle Scholar
Beuzart-Plessis, R., ‘Plancherel formula for ${\mathrm{GL}}_n(F)\setminus {\mathrm{GL}}_n(E)$ and applications to the Ichino–Ikeda and formal degree conjectures for unitary groups’, Preprint, 2018, arXiv:1812.00047. Google Scholar
Bernstein, I.N. and Zelevinsky, A.V., Induced representations of reductive p-adic groups. I., Ann. Sci. Éc. Norm. Supér. (4) 10(4) (1977), 441–472.CrossRefGoogle Scholar
Chaudouard, P-H. and Laumon, G., Le lemme fondamental pondéré. II. Énoncés cohomologiques, Ann. of Math. (2) 176(3) (2012), 1647–1781.CrossRefGoogle Scholar
Delorme, P., Formule de Plancherel pour les fonctions de Whittaker sur un groupe réductif $p$ -adique, Ann. Inst. Fourier (Grenoble) 63(1) (2013), 155–217.CrossRefGoogle Scholar
Gan, W. T. and Ichino, A., Formal degrees and local theta correspondence, Invent. Math. 195(3) (2014), 509–672.CrossRefGoogle Scholar
Gelbart, S. and Rogawski, J. D., $L$ -functions and Fourier–Jacobi coefficients for unitary group $\mathrm{U}(3)$ , Invent. Math. 105(3) (1991), 445–472.CrossRefGoogle Scholar
Ginzburg, D., Rallis, S. and Soudry, D., The Descent Map from Automorphic Representations of $\mathrm{GL}(n)$ to Classical Groups (Hackensack, World Scientific Publishing Co., 2011).Google Scholar
Gross, B. and Prasad, D., On the decomposition of a representation of ${\mathrm{SO}}_n$ when restricted to ${\mathrm{SO}}_{n-1}$ , Canad. J. Math. 44 (1992), 974–1002.CrossRefGoogle Scholar
Gross, B. and Reeder, M., Arithmetic invariants of discrete Langlands parameters, Duke Math. J. 154(3) (2010), 431–508.CrossRefGoogle Scholar
Henniart, G., Caractérisation de la correspondance de Langlands locale par les facteurs $\varepsilon$ de paires, Invent. Math. 113 (1993), 339–350.CrossRefGoogle Scholar
Hiraga, K., Ichino, A. and Ikeda, T., Formal degrees and adjoint $\gamma$ -factors, J. Amer. Math. Soc. 21(1) (2008), 283–304.CrossRefGoogle Scholar
Hiraga, K., Ichino, A. and Ikeda, T., Correction to: ‘Formal degrees and adjoint $\gamma$ -factors’, J. Amer. Math. Soc. 21(4) (2008), 1211–1213.CrossRefGoogle Scholar
Ichino, A. and Ikeda, T., On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture, Geom. Funct. Anal. 19(5) (2010), 1378–1425.CrossRefGoogle Scholar
Ichino, A., Lapid, E. and Mao, Z., On the formal degrees of square-integrable representations of odd special orthogonal and metaplectic groups, Duke Math. J. 166(7) (2017), 1301–1348.CrossRefGoogle Scholar
Jacquet, H., Piatetski-Shapiro, I. I. and Shalika, J., Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), 367–464.CrossRefGoogle Scholar
Jiang, D. and Soudry, D., Generic representation and local Langlands reciprocity law for p-adic ${\mathrm{SO}}_{2n+1}$ , in Contributions to Automorphic Forms, Geometry, and Number Theory (Baltimore, Johns Hopkins University Press, 2004), 457–519.Google Scholar
Kable, A. C., Asai $L$ -functions and Jacquet’s conjecture, Amer. J. Math. 126(4) (2004), 789–820.CrossRefGoogle Scholar
Kaletha, T., Minguez, A., Shin, S.W. and White, P.-J., ‘Endoscopic classification of representations: inner forms of unitary groups’, Preprint, 2014, arXiv:1409.3731. Google Scholar
Kim, H. and Krishnamurthy, M., Stable base change lift from unitary groups to ${\mathrm{GL}}_n$ , Int. Math. Res. Papers 2005(1) (2005), 1–52.CrossRefGoogle Scholar
Kneser, M., Semi-simple algebraic groups, in Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) (Washington, D.C., Thompson, 1967), 250–265.Google Scholar
Konno, T., A note on Langlands’ classification and irreducibility of induced representations of $p$ -adic group, Kyushu J. Math. 57(2) (2003), 383–409.CrossRefGoogle Scholar
Lapid, E. and Mao, Z., Stability of certain oscillatory integrals, Int. Math. Res. Not. IMRN 2013(3) (2013), 525–547.CrossRefGoogle Scholar
Lapid, E. and Mao, Z., On a new functional equation for local integrals, Contemp. Math. 614 (2014), 261–294.CrossRefGoogle Scholar
Lapid, E. and Mao, Z., A conjecture on Whittaker-Fourier coefficients of cusp forms, J. Number Theory 146 (2015), 448–505.CrossRefGoogle Scholar
Lapid, E. and Mao, Z., Model transition for representations of metaplectic type. With an appendix by Marko Tadić, Int. Math. Res. Not. IMRN 2015(19) (2015), 9486–9568.CrossRefGoogle Scholar
Lapid, E. and Mao, Z., On Whittaker-Fourier coefficients of automorphic forms on unitary groups : reduction to a local conjecture, Contemp. Math. 664 (2016), 295–320.CrossRefGoogle Scholar
Lapid, E. and Mao, Z., On an analogue of the Ichino-Ikeda conjecture for Whittaker coefficients on the metaplectic group, Algebra Number Theory 11(3) (2017), 713–765.CrossRefGoogle Scholar
Lapid, E. and Mao, Z., Whittaker-Fourier coefficients of cusp forms on $\widetilde{\mathrm{Sp}}_n$ : reduction to a local statement, Amer. J. Math. 139(1) (2017), 1–55.CrossRefGoogle Scholar
Liu, B., Genericity of representations of $p$ -adic ${\mathrm{Sp}}_{2n}$ and local Langlands parameters, Canad. J. Math. 63(5) (2011), 1107–1136.CrossRefGoogle Scholar
Matringe, N., Conjectures about distinction and local Asai $L$ -functions, Int. Math. Res. Not. IMRN 2009(9) (2009), 1699–1741.Google Scholar
Matringe, N., Distinguished generic representations of $\mathrm{GL}(n)$ over $p$ -adic fields, Int. Math. Res. Not. IMRN 2011(1) (2011), 74–95.CrossRefGoogle Scholar
Moeglin, C., and Tadić, M., Construction of discrete series for classical $p$ -adic groups, J. Amer. Math. Soc. 15(3) (2002), 715–786.CrossRefGoogle Scholar
Moeglin, C., Vigneras, M. F. and Waldspurger, J.-L., Correspondances de Howe sur un corps $p$ -adique, Lecture Notes in Mathematics 1291 (Berlin, Springer-Verlag, 1987).Google Scholar
Moeglin, C. and Waldspurger, J.-L., Stabilisation de la Formule des Traces Tordue , Vol. 1, Progr. Math. 316 (Cham, Birkhäuser/Springer, 2016).Google Scholar
Moeglin, C. and Waldspurger, J.-L., Stabilisation de la Formule des Traces Tordue , Vol. 2, Progr. Math. 317 (Cham, Birkhäuser/Springer, 2016).Google Scholar
Mok, C. P., Endoscopic classification of representations of quasi-split unitary groups, Mem. Amer. Math. Soc. 235 (2015), 1108.Google Scholar
Morimoto, K., On the irreducibility of global descents for even unitary groups and its applications, Trans. Amer. Math. Soc. 370(9) (2018), 6245–6295.Google Scholar
Morimoto, K., Model transition for representations of unitary type, Int. Math. Res. Not. IMRN 2020(4) (2020), 1112–1203.Google Scholar
Morimoto, K., ‘On gamma factors of Rankin-Selberg integrals for ${\mathrm{U}}_{2\ell}\times {\mathrm{GL}}_n$ ’, Preprint, 2019.Google Scholar
Offen, O., On local root numbers and distinction, J. Reine Angew. Math. 652 (2011), 165–205.Google Scholar
Sakellaridis, Y. and Venkatesh, A., Periods and harmonic analysis on spherical varieties, Astérisque 396 (2017).Google Scholar
Shahidi, F., On multiplicativity of local factors, in Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of His Sixtieth Birthday, Part II (Ramat Aviv, 1989), Israel Math. Conf. Proc. 3 (Jerusalem, Weizmann, 1990), 279–289.Google Scholar
Shahidi, F., A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$ -adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330. CrossRefGoogle Scholar
Soudry, D. and Tanay, Y., On local descent for unitary groups, J. Number Theory 146 (2015), 557–626.CrossRefGoogle Scholar
Steinberg, R., Lectures on Chevalley Groups, Notes prepared by John Faulkner and Robert Wilson (New Haven, Yale University, 1968).Google Scholar
Wallach, N. R., Real Reductive Groups, I , Pure Appl. Math. 132 (Boston, Academic Press, 1988).Google Scholar
Wallach, N. R., Real Reductive Groups , II, Pure Appl. Math. 132 (Boston, Academic Press, 1992).Google Scholar
Zelevinsky, A. V., Induced representations of reductive $p$ -adic groups. II. On irreducible representations of $\mathrm{GL}(n)$ , Ann. Sci. Éc. Norm. Supér. (4) 13(2) (1980), 165–210.Google Scholar