Igusa modular forms and 'the~simplest' Lorentzian Kac--Moody algebras

V. A. Gritsenko ; V. V. Nikulin

Sbornik. Mathematics, Tome 187 (1996) no. 11, pp. 1601-1641

Résumé

Automorphic corrections for the Lorentzian Kac–Moody algebras with the simplest generalized Cartan matrices of rank 3, $$ A_{1,0}=\begin{pmatrix} \hphantom{-}{2}\hphantom{-}{0}{-1} \\ \hphantom{-}{0}\hphantom {-}{2}{-2} \\ {-1}{-2}\hphantom {-}{2} \end{pmatrix} \quad\text{and}\quad A_{1,\mathrm {I}}=\begin {pmatrix} \hphantom {-}{2}{-2}{-1} \\ {-2}\hphantom {-}{2}{-1} \\ {-1}{-1}\hphantom {-}{2} \end{pmatrix} $$ are found. For $A_1,0$ this correction, which is a generalized Kac–Moody Lie super algebra, is delivered by $\chi_{35}(Z)$, the Igusa $\operatorname{Sp}_4(\mathbb Z)$-modular form of weight $35$, while for $A_{1,\mathrm{I}}$ it is given by some Siegel modular form $\widetilde \Delta_{30}(Z)$ of weight 30 with respect to a 2-congruence subgroup of $\operatorname{Sp}_4(\mathbb Z)$. Expansions of $\chi_{35}(Z)$ and $\widetilde\Delta_{30}(Z)$ in infinite products are obtained and the multiplicities of all the roots of the corresponding generalized Lorentzian Kac–Moody superalgebras are calculated. These multiplicities are determined by the Fourier coefficients of certain Jacobi forms of weight 0 and index 1. The method adopted for constructing $\chi_{35}(Z)$ and $\widetilde\Delta_{30}(Z)$ leads in a natural way to an explicit construction (as infinite products or sums) of Siegel modular forms whose divisors are Humbert surfaces with fixed discriminants. A geometric construction of these forms was proposed by van der Geer in 1982. To show the prospects for further studies, the list of all hyperbolic symmetric generalized Cartan matrices $A$ with the following properties is presented: $A$ is a matrix of rank 3 and of elliptic or parabolic type, has a lattice Weyl vector, and contains a parabolic submatrix $\widetilde{\mathbb A}_1$.