Va. Gritsenko et Vv. Nikulin, SIEGEL AUTOMORPHIC FORM CORRECTIONS OF SOME LORENTZIAN KAC-MOODY LIE-ALGEBRAS, American journal of mathematics, 119(1), 1997, pp. 181-224
We find automorphic form corrections which are generalized Lorentzian
Kac-Moody superalgebras without odd real simple roots for two elliptic
Lorentzian Kac-Moody algebras of rank 3 with a lattice Weyl vector, a
nd calculate multiplicities of their simple and arbitrary roots. These
Kac-Moody algebras are defined by hyperbolic symmetrized generalized
Cartan matrices [GRAPHICS] of rank 3. Both these algebras have ellipti
c type (i.e., their Weyl groups have fundamental polyhedra of finite v
olume in corresponding hyperbolic spaces) and have a lattice Weyl vect
or. The correcting automorphic forms are Siegel modular forms. The for
m corresponding to G(1) is the classical Siegel cusp form of weight 5
which is the product of ten even theta-constants. In particular we fin
d an infinite product formula for this modular form.