Zelevinsky's involution at roots of unity

We give a combinatorial algorithm for computing Zelevinsky’s involution of the set of isomorphism classes of irreducible representations of the affine Hecke algebra \H_m(t) when $t$ is a primitive $n$th root of 1. We show that the same map can also be interpreted in terms of aperiodic nilpotent orbits of $\text{Zb}\mathrm{/}n\text{Zb}$-graded vector spaces.