Vinberg is the relevant name
Is there a more traditional construction of this differential in general? I.e., that for any M in g-mod^B, there is a differential on the complex DualVerma(C^*(n,M)) with cohomology M.
Q: How well does this work if you remove the T(O)-equivariance? (maybe to be a little less ambitious, replace it with the first congruence subgroup)
What do you want to put on the left? (instead of KL(T,\kappa))?