Correction factors for Kac-Moody groups and $t$-deformed root multiplicities

TL;DR

This paper introduces a $t$-deformation of root multiplicities in Kac-Moody root systems, generalizing existing algorithms to compute correction factors and revealing new properties of these polynomials.

Contribution

It generalizes the Peterson algorithm and Berman-Moody formula to compute $t$-deformed root multiplicities for Kac-Moody algebras.

Findings

Explicit computation of $m_$ polynomials for positive imaginary roots

Generalization of root multiplicity formulas to include $t$-deformations

Fundamental properties of the $t$-deformed multiplicity polynomials

Abstract

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 $m_\lambda$.