From Anderson to Zeta

For an irreducible crystallographic root system $Φ$ and a positive integer $p$ relatively prime to the Coxeter number $h$ of $Φ$, 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 $Φ$. This bijection is defined uniformly for all irreducible crystallographic root systems $Φ$ and is equivalent to the Anderson map $\mathcal{A}_{GMV}$ defined by Gorsky, Mazin and Vazirani when $Φ$ is of type $A_{n-1}$. Specialising to $p=mh+1$, we use $\mathcal{A}$ to define a uniform $W$-set isomorphism $ζ$ from the finite torus $\check{Q}/(mh+1)\check{Q}$ to the set of $m$-nonnesting parking functions $\mathsf{Park}_Φ^{(m)}$ of $Φ$. The map $ζ$ is equivalent to the zeta map $ζ_{HL}$ of Haglund and Loehr when $m=1$ and $Φ$ is of type $A_{n-1}$.

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