Combinatorial actions and Gelfand property in affine Weyl groups

The affine Weyl group G of type C has many natural actions on finite
combinatorial objects (for example on the so called "arc permutations").
Their common feature is an action of the finite Weyl group of type C and
a "rotation", a one dimensional linear action of the translation
subgroup. The action of the finite Weyl group is multiplicity-free, so a
natural question arose, whether the original action is also
multiplicity-free? It turns out that it is not, but it is "almost," that
there exists a G equivariant pairing of the underlying objects that the
action on the pairs is multiplicity-free.
The whole structure of the question and the proof suggests a natural
generalisation to arbitrary types. Let G be any affine Weyl group,
acting by its finite Weyl group and a one dimensional "rotation." When
is this action multiplicity-free? And if not, is there a G-equivariant
pairing such that the action on pairs is multiplicity-free?
Part of the work was joint with Ron Adin and Yuval Roichman.