  From a root system, one may consider the arrangement of reflecting
hyperplanes, as well as its toric and elliptic analogues. The corresponding
Weyl group acts on the complement of the arrangement and hence on its
cohomology. We consider a sequence of linear, toric, or elliptic arrangements
which arise from a family of root systems of type A, B, C, or D, and we show
that the rational cohomology stabilizes as a sequence of Weyl group
representations. Our techniques combine a Leray spectral sequence argument
similar to that of Church in the type A case along with FI$_W$-module theory
which Wilson developed and used in the linear case. A key to the proof relies
on a combinatorial description, using labelled partitions, of the poset of
connected components of intersections of subvarieties in the arrangement.
