Apollonian packings and Kac-Moody root systems

We study Apollonian circle packings using the properties of a certain rank 4 indefinite Kac-Moody root system Φ<!-- Φ --> Φ . We introduce the generating function Z ( s ) Z(s) of a packing, an exponential series in four variables with an Apollonian symmetry group, which is a symmetric function for Φ<!-- Φ --> Φ . By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of Φ<!-- Φ --> Φ , with automorphic Weyl denominators, we express Z ( s ) Z(s) in terms of Jacobi theta functions and the Siegel modular form Δ<!-- Δ --> 5 Δ _5 . We also show that the domain of convergence of Z ( s ) Z(s) is the Tits cone of Φ<!-- Φ --> Φ , and discover that this domain inherits the intricate geometric structure of Apollonian packings.