Affine Lie algebras and Hecke modular forms. (English) Zbl 0457.17007

Kac, V. G.; Peterson, D. H.

Bull. Am. Math. Soc., New Ser. 3, 1057-1061 (1980).

The character of a highest weight representation of an affine Lie algebra involves certain modular functions, called string functions. The transformation law for these string functions is found. Then the string functions can be computed in many interesting cases as the authors demonstrate. A special case yields a relationship with Hecke modular forms.

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MSC:

17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B65	Infinite-dimensional Lie (super)algebras
11F11	Holomorphic modular forms of integral weight

Keywords:

explicit formula for partition function; character; highest weight representation; affine Lie algebra; transformation law for string functions; Hecke modular forms

References:

[1]	Martin Eichler, Introduction to the theory of algebraic numbers and functions, Translated from the German by George Striker. Pure and Applied Mathematics, Vol. 23, Academic Press, New York-London, 1966. · Zbl 0152.19502
[2]	Alex J. Feingold and James Lepowsky, The Weyl-Kac character formula and power series identities, Adv. in Math. 29 (1978), no. 3, 271 – 309. · Zbl 0391.17009 · doi:10.1016/0001-8708(78)90020-8
[3]	I. B. Frenkel and V. G. Kac, Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980/81), no. 1, 23 – 66. · Zbl 0493.17010 · doi:10.1007/BF01391662
[4]	E. Hecke, Über einen Zusammenhang zwischen elliptischen Modulfunktionen und indefiniten quadratischen Formen, Mathematische Werke, Vandenhoeck and Ruprecht, Göttingen, 1959, pp. 418-427.
[5]	V. G. Kac, Simple irreducible graded Lie algebras of finite growth, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1323 – 1367 (Russian).
[6]	V. G. Kac, Infinite-dimensional Lie algebras, and the Dedekind \?-function, Funkcional. Anal. i Priložen. 8 (1974), no. 1, 77 – 78 (Russian).
[7]	V. G. Kac, Infinite-dimensional algebras, Dedekind’s \?-function, classical Möbius function and the very strange formula, Adv. in Math. 30 (1978), no. 2, 85 – 136. , https://doi.org/10.1016/0001-8708(78)90033-6 V. G. Kac, An elucidation of: ”Infinite-dimensional algebras, Dedekind’s \?-function, classical Möbius function and the very strange formula”. \?\(_{8}\)\?\textonesuperior \? and the cube root of the modular invariant \?, Adv. in Math. 35 (1980), no. 3, 264 – 273. · Zbl 0431.17009 · doi:10.1016/0001-8708(80)90052-3
[8]	V. G. Kac, Infinite-dimensional algebras, Dedekind’s \?-function, classical Möbius function and the very strange formula, Adv. in Math. 30 (1978), no. 2, 85 – 136. , https://doi.org/10.1016/0001-8708(78)90033-6 V. G. Kac, An elucidation of: ”Infinite-dimensional algebras, Dedekind’s \?-function, classical Möbius function and the very strange formula”. \?\(_{8}\)\?\textonesuperior \? and the cube root of the modular invariant \?, Adv. in Math. 35 (1980), no. 3, 264 – 273. · Zbl 0431.17009 · doi:10.1016/0001-8708(80)90052-3
[9]	Eduard Looijenga, Root systems and elliptic curves, Invent. Math. 38 (1976/77), no. 1, 17 – 32. · Zbl 0358.17016 · doi:10.1007/BF01390167
[10]	Robert V. Moody, A new class of Lie algebras, J. Algebra 10 (1968), 211 – 230. · Zbl 0191.03005 · doi:10.1016/0021-8693(68)90096-3
[11]	D. H. Peterson, Kostant-type partition functions (to appear).
[12]	V. G. Kac and D. H. Peterson (manuscript in preparation).