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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 the correction factor can be encoded as a collection of polynomials $m_lambda in mathbb{Z}[t]$ indexed by positive imaginary roots $lambda$. At $t=0$ these polynomials evaluate to the root multiplicities, so we consider $m_lambda$ to be a $t$-deformation of $mathrm{mult} (lambda)$. We generalize the Peterson algorithm and the Berman-Moody formula for root multiplicities to compute $m_lambda$. As a consequence we deduce fundamental properties of $m_lambda$.
In this paper, we calculate the dimension of root spaces $mathfrak{g}_{lambda}$ of a special type rank $3$ Kac-Moody algebras $mathfrak{g}$. We first introduce a special type of elements in $mathfrak{g}$, which we call elements in standard form. Then
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 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, w
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
In this note, we provide a complete description of the closed sets of real roots in a Kac-Moody root system.