When I talk about FI-modules and their relation with representations of the symmetric group, people often ask me, what about the other classical Weyl groups? The answer is that Jennifer Wilson, a student of Benson Farb’s at Chicago, has worked it all out, in her paper “FI_W–modules and stability criteria for representations of the classical Weyl groups,” available at her webpage. Lots of nice stuff in here — she revisits a theorem she already proved about stabilization of the cohomology of the string motion group, providing a much simpler proof, she gives a version of Murnaghan’s theorem for the hyperoctahedral group, she gets results on the cohomology of the complement of the hyperplane arrangement corresponding to the relevant Weyl group, etc.
It seems that type D is the hardest, and direct analogues of the approach that Benson, Tom and I used don’t work — in order to get there, she has to develop a notion of induction between these categories; just as one induces from representations of a smaller group to representations of a bigger one, she needs to induce from her category of FI_D-modules up to the more restrictive category of FI_B-modules. To accomplish this uses the (exotic, to me) theory of Kan extensions, and this ends up allowing her to use the theorems she proves in type B, which is closer to classical representation theory, to prove the desired theorems in type D. Cool!
I had sort of feared that there would be terrifying complications for types B and D in characteristic 2, but apparently not! That’s handsome.