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 AA from the set W˜p of affine Weyl group elements with no inversions of height p   to the finite torus Q∨/pQ∨Q∨/pQ∨. Here Q∨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  AGMVAGMV defined by Gorsky, Mazin and Vazirani when Φ is of type An−1An−1.Specialising to p=mh+1p=mh+1, we use AA to define a uniform W-set isomorphism ζ   from the finite torus Q∨/(mh+1)Q∨Q∨/(mh+1)Q∨ to the set of m  -nonnesting parking functions ParkΦ(m) of Φ. The map ζ is equivalent to the zeta map  ζHLζHL of Haglund and Loehr when m=1m=1 and Φ is of type An−1An−1.