The flag major index on conjugacy classes of wreath products

A classical result, independently proved by Garsia and by Wachs, gives a simple and elegant closed expression for the evaluation, at a primitive root of unity, of the major index generating function over a conjugacy class in the symmetric group. In this paper, we generalize this result to the flag major index generating function over conjugacy classes in any wreath product $\mathbb{Z}_{r}\wr S_{n}$. Our approach is new, and applies conjugation by a Coxeter element as a cyclic group action on the conjugacy class. This leads to a simple formula in the general case.