On the convergence of the Kac-Moody Correction Factor

TL;DR

This paper investigates the convergence and holomorphy of the inverse Kac-Moody correction factor, a function arising in affine and Kac-Moody algebra contexts, extending understanding of its mathematical properties.

Contribution

It proves the convergence and holomorphy of the inverse correction factor, providing new insights into its analytical behavior in Kac-Moody algebra theory.

Findings

Proves the inverse correction factor converges in a specific domain

Establishes the holomorphic nature of the inverse correction factor

Extends the understanding of correction factors in Kac-Moody algebra analysis

Abstract

The Kac-Moody correction factor, first studied by Macdonald in the affine case, corrects the failure of an identity found by Macdonald in finite-dimensional root systems in 1972. Subsequntly this factor appeared in several formulas in the affine or Kac-Moody analogue of $p$-adic spherical theory for reductive groups. In this article we view the inverse of this correction factor as a function, prove the convergency and holomorphy of this function on a certain domain.