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Asymptotic formulas for coefficients of Kac–Wakimoto Characters

Published online by Cambridge University Press:  22 February 2013

KATHRIN BRINGMANN  and
KARL MAHLBURG
KATHRIN BRINGMANN
Affiliation:
Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany. e-mail: kbringma@math.uni-koeln.de
KARL MAHLBURG
Affiliation:
Department of Mathematics, Princeton University, NJ 08544, U.S.A. e-mail: mahlburg@math.princeton.edu Department of Mathematics, Louisiana State University, LA 70803, U.S.A. e-mail: mahlburg@math.lsu.edu
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Abstract

We study the coefficients of Kac and Wakimoto's character formulas for the affine Lie superalgebras sℓ(n+1|1)∧. The coefficients of these characters are the weight multiplicities of irreducible modules of the Lie superalgebras, and their asymptotic study begins with Kac and Peterson's earlier use of modular forms to understand the coefficients of characters for affine Lie algebras. In the affine Lie superalgebra setting, the characters are products of weakly holomorphic modular forms and Appell-type sums, which have recently been studied using developments in the theory of mock modular forms and harmonic Maass forms. Using our previously developed extension of the Circle Method for products of mock modular forms along with the Saddle Point Method, we find asymptotic series expansions for the coefficients of the characters with polynomial error.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 1 , July 2013 , pp. 51 - 72
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Andrews, G.On the theorems of Watson and Dragonette for Ramanujan's mock theta functions. Amer. J. Math. 88 (1966), 454–490.CrossRefGoogle Scholar
[2]Andrews, G.Partitions with short sequences and mock theta functions. Proc. Natl. Acad. Sci. (USA) 102 (2005), 4666–4671.CrossRefGoogle ScholarPubMed
[3]Borcherds, R.Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109 (1992), 405–444.CrossRefGoogle Scholar
[4]Bringmann, K. and Folsom, A. On the asymptotic behavior of Kac–Wakimoto characters. Proc. Amer. Math. Soc., to appear.Google Scholar
[5]Bringmann, K. and Mahlburg, K.An extension of the Hardy-Ramanujan Circle Method and applications to partitions without sequences. Amer. J. Math. 133 (2011), 1151–1178.CrossRefGoogle Scholar
[6]Bringmann, K. and Ono, K.The f(q) mock theta function conjecture and partition ranks. Invent. Math. 165 (2006), 243–266.CrossRefGoogle Scholar
[7]Bringmann, K. and Ono, K.Some characters of Kac and Wakimoto and nonholomorphic modular functions. Math. Ann. 345 (2009), 547–558.CrossRefGoogle Scholar
[8]Bringmann, K. and Ono, K. Coefficients of harmonic Maass forms. Proceedings of the 2008 University of Florida Conference on Partitions, q-series, and modular forms, accepted for publication.Google Scholar
[9]Bruinier, J. and Funke, J.On two geometric theta lifts. Duke Math. J. 125 (2004), 45–90.CrossRefGoogle Scholar
[10]Conway, J. and Norton, S.Monstrous Moonshine. Bull. London Math. Soc. 11 (1979), 308–339.CrossRefGoogle Scholar
[11]Dragonette, L.Some asymptotic formulae for the mock theta series of Ramanujan. Trans. Amer. Math. Soc. 72 (1952), 474–500.CrossRefGoogle Scholar
[12]Dyson, F.Some guesses in the theory of partitions. Eureka (Cambridge) 8 (1944), pages 10–15.Google Scholar
[13]Grosswald, E. and Rademacher, H.Dedekind Sums. Carus Math. Monographs, Math. Assoc. of 461 America (1972).CrossRefGoogle Scholar
[14]Hardy, G. and Ramanujan, S.Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. 17 (1918), 75–115.CrossRefGoogle Scholar
[15]Ingham, A.Some Tauberian theorems connected with the prime number theorem. J. London Math. Soc. 20 (1945), 171–180.CrossRefGoogle Scholar
[16]Kac, V.Infinite-dimensional Lie algebras and Dedekind's eta function. Funct. Anal. Appl. 8 (1974), 68–70.Google Scholar
[17]Kac, V.Lie superalgebras. Adv. Math. 26 (1977), 8–96.CrossRefGoogle Scholar
[18]Kac, V.Infinite dimensional Lie algebras, 3 edition (Cambridge University Press, 1990).CrossRefGoogle Scholar
[19]Kac, V. and Peterson, D.Infinite-dimensional Lie algebras, theta functions and modular forms. Adv. Math. 53 (1984), 125–264.CrossRefGoogle Scholar
[20]Kac, V. and Wakimoto, M.Integrable highest weight modules over affine superalgebras and number theory. Prog. Math. 123 (1994), 415–456.Google Scholar
[21]Kac, V. and Wakimoto, M.Integrable highest weight modules over affine superalgebras and Appell's function. Comm. Math. Phys. 215 (2001), 631–682.CrossRefGoogle Scholar
[22]Mumford, D. Tata Lectures on theta I. Prog. Math. no. 28 (1983).CrossRefGoogle Scholar
[23]Rademacher, H.Topics in analytic number theory. Die Grundlehren der mathematischen Wissenschaften, Band 169 (Springer Verlag, New York- Heidelberg, 1973).Google Scholar
[24]Rademacher, H. and Zuckerman, H.On the Fourier coefficients of certain modular forms of positive dimension. Ann. of Math. 39 (1938), 433–462.CrossRefGoogle Scholar
[25]Ramanujan, S.The Lost Notebook and Other Unpublished Papers (Narosa Publishing House, New Delhi, 1987).Google Scholar
[26]Shimura, G.On modular forms of half integral weight. Ann. of Math. 97 (1973), 440–481.CrossRefGoogle Scholar
[27]Zuckerman, H.Certain functions with singularities on the unit circle. Duke Math. J. 10 (1943), 381–395.CrossRefGoogle Scholar
[28]Zwegers, S. Mock theta functions. Ph.D. thesis. Universiteit Utrecht (2002).Google Scholar