Tower equivalence and Lusztig’s truncated Fourier transform

If f f denotes the truncated Lusztig Fourier transform, we show that the image by f f of the normalized characteristic function of a Coxeter element is the alternate sum of the exterior powers of the reflection representation, and that any class function is tower equivalent to its image by f f . In particular this gives a proof of the results of Chapuy and Douvropoulos on “Coxeter factorizations with generalized Jucys-Murphy weights and matrix tree theorems for reflection groups” for irreducible spetsial reflection groups, based on Deligne-Lusztig combinatorics.