A Type B analog of the Whitehouse representation

We give a Type $B$ analog of Whitehouse’s lifts of the Eulerian representations from $S_n$ to $S_{n+1}$ by introducing a family of $B_n$-representations that lift to $B_{n+1}$. As in Type $A$, we interpret these representations combinatorially via a family of orthogonal idempotents in the Mantaci-Reutenauer algebra, and topologically as the graded pieces of the cohomology of a certain ${\mathbb{Z}}_2$-orbit configuration space of ${\mathbb{R}}^3$. We show that the lifted $B_{n+1}$-representations also have a configuration space interpretation, and further parallel the Type $A$ story by giving analogs of many of its notable properties, such as connections to equivariant cohomology and the Varchenko-Gelfand ring.