Kac-Wakimoto characters and non-holomorphic Jacobi forms

Bringmann, Kathrin; Olivetto, René

doi:10.1090/tran/6709

Kac-Wakimoto characters and non-holomorphic Jacobi forms

Authors: Kathrin Bringmann and René Olivetto
Journal: Trans. Amer. Math. Soc. 369 (2017), 1163-1184
MSC (2010): Primary 11F03, 11F22, 11F37, 11F50
DOI: https://doi.org/10.1090/tran/6709
Published electronically: April 15, 2016
MathSciNet review: 3572269
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Abstract: In this paper, we investigate the automorphic properties of certain characters introduced by Kac and Wakimoto pertaining to $s\ell (m,n)^{\wedge }$ highest weight modules. Extending previous work of the first author and Ono, the first author and Folsom, and the second author, we investigate the general case, not specializing the Jacobi variables. We prove that the Kac-Wakimoto characters are essentially holomorphic parts of multivariable mixed H-harmonic Maass-Jacobi forms, which are certain non-holomorphic generalizations of classical holomorphic Jacobi forms. This also gives extra structure to the previous considered cases.

[1] Rolf Berndt and Ralf Schmidt, Elements of the representation theory of the Jacobi group, Progress in Mathematics, vol. 163, Birkhäuser Verlag, Basel, 1998. MR 1634977
[2] Richard E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), no. 2, 405–444. MR 1172696, https://doi.org/10.1007/BF01232032
[3] Kathrin Bringmann and Amanda Folsom, On the asymptotic behavior of Kac-Wakimoto characters, Proc. Amer. Math. Soc. 141 (2013), no. 5, 1567–1576. MR 3020844, https://doi.org/10.1090/S0002-9939-2012-11439-X
[4] Kathrin Bringmann and Amanda Folsom, Almost harmonic Maass forms and Kac-Wakimoto characters, J. Reine Angew. Math. 694 (2014), 179–202. MR 3259042, https://doi.org/10.1515/crelle-2012-0102
[5] Kathrin Bringmann, Amanda Folsom, and Karl Mahlburg, Quasimodular forms and 𝑠ℓ(𝑚|𝑚)^{∧} characters, Ramanujan J. 36 (2015), no. 1-2, 103–116. MR 3296713, https://doi.org/10.1007/s11139-014-9621-4
[6] Kathrin Bringmann and Ken Ono, Some characters of Kac and Wakimoto and nonholomorphic modular functions, Math. Ann. 345 (2009), no. 3, 547–558. MR 2534107, https://doi.org/10.1007/s00208-009-0364-2
[7] Kathrin Bringmann and Olav K. Richter, Zagier-type dualities and lifting maps for harmonic Maass-Jacobi forms, Adv. Math. 225 (2010), no. 4, 2298–2315. MR 2680205, https://doi.org/10.1016/j.aim.2010.03.033
[8] Kathrin Bringmann, Martin Raum, and Olav K. Richter, Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6647–6670. MR 3356950, https://doi.org/10.1090/S0002-9947-2015-06418-6
[9] Kathrin Bringmann and Sander Zwegers, Rank-Crank type PDE’s and non-holomorphic Jacobi forms, Math. Res. Lett. 17 (2010), no. 4, 589–600. MR 2661165, https://doi.org/10.4310/MRL.2010.v17.n4.a1
[10] Jan Hendrik Bruinier and Jens Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45–90. MR 2097357, https://doi.org/10.1215/S0012-7094-04-12513-8
[11] J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), no. 3, 308–339. MR 554399, https://doi.org/10.1112/blms/11.3.308
[12] A. Dabholkar, S. Murthy, and D. Zagier, Quantum black holes, wall crossing, and mock modular forms, to appear in Cambridge Monographs in Mathematical Physics.
[13] Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735
[14] Amanda Folsom, Kac-Wakimoto characters and universal mock theta functions, Trans. Amer. Math. Soc. 363 (2011), no. 1, 439–455. MR 2719689, https://doi.org/10.1090/S0002-9947-2010-05181-5
[15] I. B. Frenkel, J. Lepowsky, and A. Meurman, A natural representation of the Fischer-Griess Monster with the modular function 𝐽 as character, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 10, , Phys. Sci., 3256–3260. MR 747596, https://doi.org/10.1073/pnas.81.10.3256
[16] F. Hirzebruch and D. Zagier, Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Invent. Math. 36 (1976), 57–113. MR 453649, https://doi.org/10.1007/BF01390005
[17] V. G. Kac, Infinite-dimensional Lie algebras, and the Dedekind 𝜂-function, Funkcional. Anal. i Priložen. 8 (1974), no. 1, 77–78 (Russian). MR 0374210
[18] Victor G. Kac and Minoru Wakimoto, Integrable highest weight modules over affine superalgebras and Appell’s function, Comm. Math. Phys. 215 (2001), no. 3, 631–682. MR 1810948, https://doi.org/10.1007/s002200000315
[19] Masanobu Kaneko and Don Zagier, A generalized Jacobi theta function and quasimodular forms, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 165–172. MR 1363056, https://doi.org/10.1007/978-1-4612-4264-2_6
[20] Anatoly Libgober, Elliptic genera, real algebraic varieties and quasi-Jacobi forms, Topology of stratified spaces, Math. Sci. Res. Inst. Publ., vol. 58, Cambridge Univ. Press, Cambridge, 2011, pp. 95–120. MR 2796409
[21] Hans Maass, Lectures on modular functions of one complex variable, Notes by Sunder Lal. Tata Institute of Fundamental Research Lectures on Mathematics, No. 29, Tata Institute of Fundamental Research, Bombay, 1964. MR 0218305
[22] R. Olivetto, Harmonic Maass forms, meromorphic Jacobi forms, and applications to Lie superalgebras, Ph.D. Thesis, Univerity of Cologne, in preparation.
[23] René Olivetto, On the Fourier coefficients of meromorphic Jacobi forms, Int. J. Number Theory 10 (2014), no. 6, 1519–1540. MR 3248169, https://doi.org/10.1142/S1793042114500419
[24] Ameya Pitale, Jacobi Maaßforms, Abh. Math. Semin. Univ. Hambg. 79 (2009), no. 1, 87–111. MR 2541345, https://doi.org/10.1007/s12188-008-0013-9
[25] Hans Rademacher, Topics in analytic number theory, Springer-Verlag, New York-Heidelberg, 1973. Edited by E. Grosswald, J. Lehner and M. Newman; Die Grundlehren der mathematischen Wissenschaften, Band 169. MR 0364103
[26] C. Ziegler, Jacobi forms of higher degree, Abh. Math. Sem. Univ. Hamburg 59 (1989), 191–224. MR 1049896, https://doi.org/10.1007/BF02942329
[27] S. Zwegers, Multivariable Appell functions, preprint (2010).
[28] S. Zwegers, Mock theta functions, Ph.D. Thesis, Universiteit Utrecht (2002).