From Anderson to Zeta

Abstract

For an irreducible crystallographic root system $\Phi$ and a positive integer $p$ relatively prime to the Coxeter number $h$ of $\Phi$, we give a natural bijection $\mathcal{A}$ from the set $\widetilde{W}^p$ of affine Weyl group elements with no inversions of height $p$ to the finite torus $\check{Q}/p\check{Q}$. Here $\check{Q}$ is the coroot lattice of $\Phi$. This bijection is defined uniformly for all irreducible crystallographic root systems $\Phi$ and is equivalent to the Anderson map $\mathcal{A}_{GMV}$ defined by Gorsky, Mazin and Vazirani when $\Phi$ is of type $A_{n-1}$. Specialising to $p=mh+1$, we use $\mathcal{A}$ to define a uniform $W$-set isomorphism $\zeta$ from the finite torus $\check{Q}/(mh+1)\check{Q}$ to the set of $m$-nonnesting parking functions $\mathsf{Park}_{\Phi}^{(m)}$ of $\Phi$. The map $\zeta$ is equivalent to the zeta map $\zeta_{HL}$ of Haglund and Loehr when $m=1$ and $\Phi$ is of type $A_{n-1}$.