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Rademacher-type formulas for the multiplicities of irreducible highest- weight representations of affine Lie algebras. (English) Zbl 0619.17013

The weight lattice of an integrable irreducible highest weight representation of an affine Lie algebra is a union of infinite strings. Kac and Peterson have shown that the multiplicities in the same string are the Fourier coefficients of a modular form of negative weight, called a string function. In this paper the authors use the results of Kac and Peterson to adapt Rademacher’s circle method to string functions and derive formulas for their coefficients and the formulas obtained are of the type proved by Rademacher for the partition function. This approach yields explicit determination of Fourier coefficients of affine Lie algebras of different types and in particular that of type \(C_ 4^{(1)}\).
Reviewer: M.Cheema

MSC:

17B65 Infinite-dimensional Lie (super)algebras
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
11F11 Holomorphic modular forms of integral weight
11P81 Elementary theory of partitions
05A19 Combinatorial identities, bijective combinatorics
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References:

[1] V. G. Kac and D. H. Peterson, Affine Lie algebras and Hecke modular forms, Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 3, 1057 – 1061. · Zbl 0457.17007
[2] Victor G. Kac and Dale H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. in Math. 53 (1984), no. 2, 125 – 264. · Zbl 0584.17007 · doi:10.1016/0001-8708(84)90032-X
[3] 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. · Zbl 0253.10002
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