Infinite-dimensional Lie algebras, theta functions and modular forms.

*(English)*Zbl 0584.17007The 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.

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.

Reviewer: James F. Hurley (Storrs)

##### 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 |

##### Keywords:

multiplicity; affine Kac-Moody Lie algebras; irreducible highest weight representations; theta functions; modular forms; Macdonald identities; string functions; generalized Kostant partition functions; identities for modular forms; elliptic theta functions
PDF
BibTeX
XML
Cite

\textit{V. G. Kac} and \textit{D. H. Peterson}, Adv. Math. 53, 125--264 (1984; Zbl 0584.17007)

Full Text:
DOI

##### References:

[1] | Andrews, G.E, The theory of partitions, Encyclopedia of mathematics, Vol. 2, (1976) · Zbl 0371.10001 |

[2] | Bernstein, I; Schwartzman, O, Chevalley theorem for complex crystallographic Coxeter groups, Functional anal. appl., 12, (1978) |

[3] | Bourbaki, N; Bourbaki, N, Groupes et algebres de Lie, (1968), Hermann Paris, Chaps. IV-VI · Zbl 0186.33001 |

[4] | Conway, J.H; Norton, S.P, Monstrous moonshine, Bull. London math. soc., 11, 308-339, (1979) · Zbl 0424.20010 |

[5] | Eichler, M, Introduction to the theory of algebraic numbers and functions, (1966), Academic Press New York · Zbl 0152.19502 |

[6] | Feingold, A; Lepowsky, J, The Weyl-kač character formula and power series identities, Adv. in math., 29, 271-309, (1978) · Zbl 0391.17009 |

[7] | Frenkel, I.B; Kač, V.G, Basic representations of affine Lie algebras and dual resonance models, Invent. math., 62, 23-66, (1980) · Zbl 0493.17010 |

[8] | Gabber, O; Kač, V.G, On defining relations of certain infinite-dimensional Lie algebras, Bull. amer. math. soc., 5, 185-189, (1981) · Zbl 0474.17007 |

[9] | Hecke, E, Über einen neuen zusammenhang zwischen elliptischen modulfunktionen und indefiniten quadratischen formen, (), 418-427 · JFM 51.0292.04 |

[10] | Helgason, S, Differential geometry, Lie groups and symmetric spaces, (1980), Academic Press New York |

[11] | Humphreys, J.E, Introduction to Lie algebras and representation theory, (1972), Springer-Verlag New York/Berlin · Zbl 0254.17004 |

[12] | Igusa, J, Theta functions, (1972), Springer-Verlag New York/Berlin · Zbl 0251.14016 |

[13] | Jacobi, C.G.J, Fundamenta nova theoriae functionum ellipticarum (1829), (), 49-239 |

[14] | Kač, V.G, Simple irreducible graded Lie algebras of finite growth, Math. USSR-izv., 2, 1271-1311, (1968) · Zbl 0222.17007 |

[15] | Kač, V.G, Automorphisms of finite order of semisimple Lie algebras, J. funct. anal. appl., 3, 252-254, (1969) · Zbl 0274.17002 |

[16] | Kač, V.G, Infinite-dimensional Lie algebras and Dedekind’s η-function, J. funct. anal. appl., 8, 68-70, (1974) · Zbl 0299.17005 |

[17] | Kač, V.G, Infinite-dimensional algebras, Dedekind’s η-function, classical Möbius function and the very strange formula, Adv. in math., 30, 85-136, (1978) · Zbl 0391.17010 |

[18] | Kač, V.G, An elucidation of “infinite-dimensional algebras and the very strange formula”, E(1)8 and the cube root of the modular invariant j, Adv. in math., 35, 264-273, (1980) · Zbl 0431.17009 |

[19] | Kač, V.G, Infinite root systems, representations of graphs, and invariant theory, Invent. math., 56, 57-92, (1980) · Zbl 0427.17001 |

[20] | Kač, V.G, On simplicity of certain infinite-dimensional Lie algebras, Bull. amer. math. soc., 2, 311-314, (1980) · Zbl 0427.17012 |

[21] | Kač, V.G, Simple Lie groups and the Legendre symbol, (), 110-123 · Zbl 0498.22013 |

[22] | Kač, V.G; Peterson, D.H, Affine Lie algebras and Hecke modular forms, Bull. amer. math. soc., 3, 1057-1061, (1980) · Zbl 0457.17007 |

[23] | Kač, V.G; Peterson, D.H, Spin and wedge representations of infinite-dimensional Lie algebras and groups, (), 3308-3312 · Zbl 0469.22016 |

[24] | Knopp, M, Modular functions in analytic number theory, (1970), Markham Chicago · Zbl 0259.10001 |

[25] | Kostant, B, On Macdonald’s η-function formula, the Laplacian and generalized exponents, Adv. in math., 20, 179-212, (1976) · Zbl 0339.10019 |

[26] | Lion, G; Vergne, M, The Weil representation, Maslov index and theta series, (1980), Birkhäuser Basel · Zbl 0444.22005 |

[27] | Looijenga, E, Root systems and elliptic curves, Invent. math., 38, 17-32, (1976) · Zbl 0358.17016 |

[28] | Looijenga, E, Invariant theory for generalized root systems, Invent. math., 61, 1-32, (1980) · Zbl 0436.17005 |

[29] | Macdonald, I, Affine root systems and Dedekind’s η-function, Invent. math., 15, 91-143, (1972) · Zbl 0244.17005 |

[30] | Milnor, J; Husemoller, D, Symmetric bilinear forms, (1973), Springer-Verlag New York/Berlin · Zbl 0292.10016 |

[31] | Moody, R.V, A new class of Lie algebras, J. algebra, 10, 211-230, (1968) · Zbl 0191.03005 |

[32] | Moreno, C, The higher reciprocity laws: an example, J. number theory, 12, 57-70, (1980) · Zbl 0426.10024 |

[33] | Mumford, D, Tata lectures on theta, (1982), Birkhäuser Boston · Zbl 0744.14033 |

[34] | \scD. H. Peterson, Kostant-type partition functions, to appear. |

[35] | \scD. H. Peterson, On independence of fundamental characters of certain infinite-dimensional groups, to appear. |

[36] | \scD. H. Peterson, An infinite class of identities connecting definite and indefinite quadratic forms, to appear. |

[37] | Riemann, B, Theorie der abelschen functionen, J. reine angew. math., 54, 115-155, (1857) |

[38] | Serre, J.-P, Modular forms of weight one and Galois representations, () |

[39] | Serre, J.-P, Cours d’arithmetique, (1970), Presses Universitaires de France Paris · Zbl 0225.12002 |

[40] | Tannery, J; Molk, J, Éléments de la théorie des fonctions elliptiques, (1898), Paris · JFM 29.0379.11 |

[41] | Vinberg, E.B, Discrete linear groups generated by reflections, Math. USSR-izv., 5, 1083-1119, (1971) · Zbl 0256.20067 |

[42] | Fegan, H.D, The heat equation and modular forms, J. differential geom., 13, 589-602, (1978) · Zbl 0437.22010 |

[43] | Ingham, A.E, A Tauberian theorem for partitions, Ann. of math., 42, 1075-1090, (1941) · Zbl 0063.02973 |

[44] | Lepowsky, J, Generalized Verma modules, loop space cohomology and Macdonald-type identities, Ann. sci. école norm. sup., 12, 169-234, (1979) · Zbl 0414.17007 |

[45] | Bernstein, I.N; Gelfand, I.M; Gelfand, S.I, Schubert cells and flag space cohomologies, Funct. anal. appl., 7, (1973) · Zbl 0282.20035 |

[46] | Verma, D.-N, The role of affine Weyl groups in the representation theory of algebraic Chevalley groupś and their Lie algebras, () · Zbl 0316.20030 |

[47] | Frenkel, I.B, Spinor representations of affine Lie algebras, (), 6303-6306 · Zbl 0451.17004 |

[48] | Frenkel, I.B, Representations of affine Lie algebras, Hecke modular forms and Korteweg-de Vries type equations, () · Zbl 0505.17008 |

[49] | \scD. H. Peterson, Level one modules of affine Lie algebras, to appear. |

[50] | Kac, V.G, Infinite dimensional Lie algebras, (1983), Birkhäuser Boston · Zbl 0425.17009 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.