Descents and flag major index on conjugacy classes of colored permutation groups without short cycles

We consider the descent and flag major index statistics on the colored permutation groups, which are wreath products of the form Sn,r=Zᵣ Sₙ. We show that the k-th moments of these statistics on Sn,r will coincide with the corresponding moments on all conjugacy classes without cycles of lengths 1,2,…,2k. Using this, we establish the asymptotic normality of the descent and flag major index statistics on conjugacy classes of Sn,r with sufficiently long cycles. Our results generalize prior work of Fulman involving the descent and major index statistics on the symmetric group Sₙ. Our methods involve an intricate extension of Fulman's work on Sₙ combined with the theory of the degree for a colored permutation statistic, as introduced by Campion Loth, Levet, Liu, Sundaram, and Yin.