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Infinite-dimensional Lie algebras, theta functions and modular forms. (English) Zbl 0584.17007
The authors give a full account of important results announced some years earlier [Bull. Am. Math. Soc., New Ser. 3, 1057–1061 (1980; Zbl 0457.17007)]. Several appendices discuss further implications of the theory presented and announce further results. The paper is written with recognition of the broad audience its content will interest. It includes background on affine Kac-Moody Lie algebras, irreducible highest weight representations, and classical theta functions and modular forms, as well as an account of recently discovered connections among those areas. In broad terms, the paper’s principal results fall into two main categories.
First, a theta-function interpretation of the Macdonald identities due to E. Looijenga [Invent. Math. 38, 17–32 (1976; Zbl 0358.17016)] is exploited through an observation of the first author [Adv. Math. 35, 264–273 (1980; Zbl 0431.17009)] that most generating functions for multiplicities appearing in the representation theory of affine Lie algebras become q-series of modular forms when multiplied by a suitable power of q. The character of a highest weight representation of an affine Lie algebra is rewritten in terms of theta functions of the modular forms. Then the authors use classical functional equations for theta functions to deduce transformation properties of the modular forms. The “very strange” formula [see the first author’s paper in Adv. Math. 30, 85–136 (1978; Zbl 0391.17010)] is then used to estimate the order of the poles at the cusps.
In short, the modular form theory makes it possible to compute with the forms, and a Tauberian theorem of Ingham is used to obtain the asymptotics of the multiplicities in question. If \(L\) is the affine algebra and \(\Lambda\) is the highest weight, then the key step in this part of the paper is establishing that \(mult_{\Lambda}(\lambda -n\delta)\) is an increasing function of \(n\). Here \(\lambda\) is in the dual space of the Cartan subalgebra \(H\) of the affine algebra \(L\) and \(\delta\) is the unique element of that space that annihilates \(\bar H \) (the Cartan subalgebra of the underlying classical simple finite dimensional algebra \(L)\) and \(c\) (where \(H=H+Cc+Cd\), \(d\) the derivation that acts on \(C[t,t^{- 1}]\otimes_ C \bar L\) as \(t(d/dt)\) and annihilates \(c)\) and maps \(d\) to 1. The approach taken by the authors involves use of a Heisenberg algebra. The results in this direction make it possible to explicitly determine the string functions in many cases. The multiplicities do not appear to be given by any simple combinatorial functions such as the classical partition function, but rather to depend on the fact that \(q^{1/24}(q,q)\) is a modular form.
The second main theme of the paper is the use of the second author’s explicit formulas for Kostant’s partition function. These make it possible to derive explicit formulas for generalized Kostant partition functions for certain affine algebras. That in turn affords a way of computing multiplicities directly for the algebra of type \(A_ 1^{(1)}\). The corresponding generating series are closely related to Hecke modular forms associated to real quadratic fields [see Math. Werke E. Hecke, 418–427 (1959; Zbl 0092.001)].
While the complexity of the formulas and identities obtained in the paper makes it impractical to be more specific here, it is to be noted that at the end of the paper, a collection of new (and old) identities for modular forms and elliptic theta functions is given. These formulas, which are natural consequences of the representation theory and its connections to modular forms in the simplest case \((A_ 1^{(1)})\), can be read independently of the rest of the paper.

MSC:
17B65 Infinite-dimensional Lie (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
11F11 Holomorphic modular forms of integral weight
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
14K25 Theta functions and abelian varieties
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