  We study representations of simply-laced Weyl groups which are equipped with
canonical bases. Our main result is that for a large class of representations,
the separable elements of the Weyl group $W$ act on these canonical bases by
bijections up to lower-order terms. Examples of this phenomenon include the
action of separable permutations on the Kazhdan--Lusztig basis of irreducible
representations for the symmetric group, and the action of separable elements
of $W$ on dual canonical bases of weight zero in tensor product representations
of a Lie algebra. Our methods arise from categorical representation theory, and
in particular the study of the perversity of Rickard complexes acting on
triangulated categories.
