Automorphic forms on \(O_{s+2,2}(\mathbb R)\) and infinite products.

*(English)*Zbl 0932.11028From the text: The denominator function of a generalized Kac-Moody algebra is often an automorphic form for a group \(O_{s+2,2} (\mathbb R)\) which can be written as an infinite product. The author studies such forms and constructs some infinite families of them. This has applications to the theory of generalized Kac-Moody algebras, unimodular lattices, and reflection groups. He also uses these forms to write several well-known modular forms, such as the elliptic modular function \(j\) and the Eisenstein series \(E_4\) and \(E_6\), as infinite products.

The wealth of information treated in this highly interesting paper can be seen by looking at the table of contents: Automorphic forms and Jacobi forms, Classical theory, Hecke operators for Jacobi groups, Analytic continuation, Vector systems and the Macdonald identities, The Weierstrass \(\wp\) function, Generators for \(O_M(\mathbb Z)^+\), The positive weight case, The zero weight case, The negative weight case, Invariant modular products, Heights of vectors, Product formulas for modular forms, Generalized Kac-Moody algebras, Hyperbolic reflection groups, Open problems.

The construction of automorphic forms as infinite products depends on three results. The first result (theorem 5.1) states that under mild conditions a modular product can be analytically continued as a meromorphic function to the whole of the Hermitian symmetric space \(H\) of \(O_{s+2,2} (\mathbb R)\), and its poles and zeros can only lie on certain special divisors, called quadratic divisors. (A modular product is, roughly speaking, an infinite product whose exponents are given by the coefficients of some nearly holomorphic modular forms.) The proof of this uses the Hardy-Ramanujan-Rademacher asymptotic series for the coefficients of a nearly holomorphic modular form. The second result (theorem 6.5) is a generalization of the Macdonald identities from affine root systems to “affine vector systems”. This generalization states (roughly) that an infinite product over the vectors of an affine vector system is a Jacobi form. (For affine root systems the usual Macdonald identities follow easily from this using the fact that any Jacobi form can be written as a finite sum of products of theta functions and modular forms.) The third result used is a description of Hecke operators \(V_l\) for certain parabolic subgroups (“Jacobi subgroups”) of discrete subgroups of \(O_{s+2,2} (\mathbb R)\).

Putting these three results together, one sometimes finds that an expression of the form \(\exp(\rho+ \sum_{l\geq 0}\varphi|V_l)\), where \(\varphi\) is a nearly holomorphic Jacobi form, is an automorphic form on \(O_{s+2,2} (\mathbb R)\). He proves this by showing that it transforms like an automorphic form under two parabolic subgroups \(J(\mathbb Z)^+\) and \(F(\mathbb Z)^+\), and then checking (in theorem 8.1) that these two subgroups generate a discrete subgroup of \(O_{s+2,2} (\mathbb R)^+\) of finite covolume. The invariance under the Jacobi group \(J(\mathbb Z)^+\) follows from the results on Hecke generators on Jacobi forms, and the invariance under the Fourier group \(F(\mathbb Z)^+\) follows by calculating the Fourier coefficients explicitly and checking that these are invariant under \(F(\mathbb Z)^+\).

When the Jacobi form \(\varphi\) is holomorphic, this is similar to a method for constructing automorphic forms on \(Sp_4(\mathbb R)\) found by Maass, and generalized to \(O_{s+2,2} (\mathbb R)\) by Gritsenko, who showed that \(\sum_{l\geq 0}\varphi|V_l\) was an automorphic form. The two main extra complications to be dealt with when \(\varphi\) is not holomorphic are firstly that this sum no longer converges everywhere and so has to be analytically continued, and secondly that the “Weyl vector” \(\rho\) has to be chosen correctly.

A large number of examples and a list of 12 open problems illustrate the results and challenge further research.

The wealth of information treated in this highly interesting paper can be seen by looking at the table of contents: Automorphic forms and Jacobi forms, Classical theory, Hecke operators for Jacobi groups, Analytic continuation, Vector systems and the Macdonald identities, The Weierstrass \(\wp\) function, Generators for \(O_M(\mathbb Z)^+\), The positive weight case, The zero weight case, The negative weight case, Invariant modular products, Heights of vectors, Product formulas for modular forms, Generalized Kac-Moody algebras, Hyperbolic reflection groups, Open problems.

The construction of automorphic forms as infinite products depends on three results. The first result (theorem 5.1) states that under mild conditions a modular product can be analytically continued as a meromorphic function to the whole of the Hermitian symmetric space \(H\) of \(O_{s+2,2} (\mathbb R)\), and its poles and zeros can only lie on certain special divisors, called quadratic divisors. (A modular product is, roughly speaking, an infinite product whose exponents are given by the coefficients of some nearly holomorphic modular forms.) The proof of this uses the Hardy-Ramanujan-Rademacher asymptotic series for the coefficients of a nearly holomorphic modular form. The second result (theorem 6.5) is a generalization of the Macdonald identities from affine root systems to “affine vector systems”. This generalization states (roughly) that an infinite product over the vectors of an affine vector system is a Jacobi form. (For affine root systems the usual Macdonald identities follow easily from this using the fact that any Jacobi form can be written as a finite sum of products of theta functions and modular forms.) The third result used is a description of Hecke operators \(V_l\) for certain parabolic subgroups (“Jacobi subgroups”) of discrete subgroups of \(O_{s+2,2} (\mathbb R)\).

Putting these three results together, one sometimes finds that an expression of the form \(\exp(\rho+ \sum_{l\geq 0}\varphi|V_l)\), where \(\varphi\) is a nearly holomorphic Jacobi form, is an automorphic form on \(O_{s+2,2} (\mathbb R)\). He proves this by showing that it transforms like an automorphic form under two parabolic subgroups \(J(\mathbb Z)^+\) and \(F(\mathbb Z)^+\), and then checking (in theorem 8.1) that these two subgroups generate a discrete subgroup of \(O_{s+2,2} (\mathbb R)^+\) of finite covolume. The invariance under the Jacobi group \(J(\mathbb Z)^+\) follows from the results on Hecke generators on Jacobi forms, and the invariance under the Fourier group \(F(\mathbb Z)^+\) follows by calculating the Fourier coefficients explicitly and checking that these are invariant under \(F(\mathbb Z)^+\).

When the Jacobi form \(\varphi\) is holomorphic, this is similar to a method for constructing automorphic forms on \(Sp_4(\mathbb R)\) found by Maass, and generalized to \(O_{s+2,2} (\mathbb R)\) by Gritsenko, who showed that \(\sum_{l\geq 0}\varphi|V_l\) was an automorphic form. The two main extra complications to be dealt with when \(\varphi\) is not holomorphic are firstly that this sum no longer converges everywhere and so has to be analytically continued, and secondly that the “Weyl vector” \(\rho\) has to be chosen correctly.

A large number of examples and a list of 12 open problems illustrate the results and challenge further research.

Reviewer: O.Ninnemann (Berlin)

##### MSC:

11F22 | Relationship to Lie algebras and finite simple groups |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

11F12 | Automorphic forms, one variable |

11F50 | Jacobi forms |

##### Keywords:

automorphic forms; denominator function; generalized Kac-Moody algebra; unimodular lattices; reflection groups; elliptic modular function; Eisenstein series; infinite products; Jacobi forms; Hecke operators; Jacobi groups; Macdonald identities; Weierstrass \(\wp\) function##### References:

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