Invariants in the cohomology of the complement of quaternionic reflection arrangements

Abstract

Let $\mathcal A$ be a hyperplane arrangement in a vector space $V$ and $G \leq GL(V)$ a group fixing $\mathcal A$. In case when $G$ is a complex reflection group and $\mathcal A=\mathcal A(G)$ is its reflection arrangement in $V$, Douglass, Pfeiffer, and R\"ohrle studied the invariants of the $Q G$-module $H^*(M(\mathcal A);Q)$, the rational, singular cohomology of the complement space $M(\mathcal A)$ in $V$. In this paper we generalize the work in Douglass, Pfeiffer, and R\"ohrle to the case of quaternionic reflection groups, first classified by Cohen in 1980. We obtain a straightforward generalization of the Hilbert--Poincar\'e series of the ring of invariants in the cohomology from the complex case when the quaternionic reflection group is complex-reducible according to Cohen's classification. Surprisingly, only one additional family of new types of Poincar\'e polynomials occurs in the quaternionic setting which is not realised in the complex case, namely those of a particular class of imprimitive irreducible quaternionic reflection groups. We utilize a new description of the lattices of intersections of imprimitive groups as Dowling lattices. Finally, we discuss bases of the space of $G$-invariants in $H^*(M(\mathcal A);Q)$.