Rank 2 symmetric hyperbolic Kac–Moody algebras and Hilbert modular forms

Abstract

In this paper we study rank 2 symmetric hyperbolic Kac–Moody algebras H(a) with the Cartan matrices (2−a−a2), a⩾3, and their automorphic correction in terms of Hilbert modular forms. We associate a family of H(a)'s to the quadratic field Q(p) for each odd prime p and show that there exists a chain of embeddings in each family. When p=5,13,17, we show that the first H(a) in each family, i.e. H(3), H(11), H(66), is contained in a generalized Kac–Moody superalgebra whose denominator function is a Hilbert modular form given by a Borcherds product. Hence, our results provide automorphic correction for those H(a)'s. We also compute asymptotic formulas for the root multiplicities of the generalized Kac–Moody superalgebras using the fact that the exponents in the Borcherds products are Fourier coefficients of weakly holomorphic modular forms of weight 0.