  We give a proof of the results of Chapuy and Douvropoulos [3] for irreducible
spetsial reflection groups based on Deligne-Lusztig combinatorics. In
particular, if f denotes the truncated Lusztig Fourier transform, we show that
the image by f of the normalized characteristic function of a Coxeter element
is the alternate sum of the exterior powers of the reflection representation,
and that a class function is tower equivalent to its image by f .
