Refined Gelfand models for wreath products

In [F. Caselli, Involutory reflection groups and their models, J. Algebra 24 (2010), 370–393] there is constructed a uniform Gelfand model for all non-exceptional irreducible complex reflection groups which are involutory. This model can be naturally decomposed into the direct sum of submodules indexed by symmetric conjugacy classes, and in this paper we present a simple combinatorial description of the irreducible decomposition of these submodules if the group is the wreath product of a cyclic group with a symmetric group. This is attained by showing that such decomposition is compatible with the generalized Robinson–Schensted correspondence for these groups.