Homology representations arising from the half cube, II

In a previous work, we defined a family of subcomplexes of the n-dimensional half cube by removing the interiors of all half cube shaped faces of dimension at least k, and we proved that the homology of such a subcomplex is concentrated in degree k − 1. This homology group supports a natural action of the Coxeter group W (D n) of type D. In this paper, we explicitly determine the characters (over C) of these homology representations, which turn out to be multiplicity free. Regarded as representations of the symmetric group S n by restriction, the ho-mology representations turn out to be direct sums of certain representations induced from parabolic subgroups. We conjecture that the latter representations of S n agree with the representations of S n on the (k − 2)-nd homology of the complement of the k-equal hyperplane arrangement.