Apollonian Packings and Kac-Moody Root Systems

We study Apollonian circle packings in relation to a certain rank 4 indefinite Kac-Moody root system $\Phi$. We introduce the generating function $Z(\mathbf{s})$ of a packing, an exponential series in four variables with an Apollonian symmetry group, which relates to Weyl-Kac characters of $\Phi$. By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of $\Phi$, with automorphic Weyl denominators, we express $Z(\mathbf{s})$ in terms of Jacobi theta functions and the Siegel modular form $\Delta_5$. We also show that the domain of convergence of $Z(\mathbf{s})$ is the Tits cone of $\Phi$, and discover that this domain inherits the intricate geometric structure of Apollonian packings.