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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.
In this note, we provide a complete description of the closed sets of real roots in a Kac-Moody root system.
Weyl group multiple Dirichlet series, introduced by Brubaker, Bump, Chinta, Friedberg and Hoffstein, are expected to be Whittaker coefficients of Eisenstein series on metaplectic groups. Chinta and Gunnells constructed these multiple Dirichlet series
Let R be a finitely generated commutative ring with 1, let A be an indecomposable 2-spherical generalized Cartan matrix of size at least 2 and M=M(A) the largest absolute value of a non-diagonal entry of A. We prove that there exists an integer n=n(A
We study a correction factor for Kac-Moody root systems which arises in the theory of $p$-adic Kac-Moody groups. In affine type, this factor is known, and its explicit computation is the content of the Macdonald constant term conjecture. The data of
We initiate an investigation of lattices in a new class of locally compact groups, so called locally pro-$p$-complete Kac-Moody groups. We discover that in rank 2 their cocompact lattices are particularly well-behaved: under mild assumptions, a cocom