For G(r,1,n), category $\mathcal{O}$ is Koszul.  This is a consequence of a comparison theorem with a version of parabolic category $\mathcal{O}$ of affine type, recently proven by [Rouquier-Shan-Varagnolo-Vasserot](https://arxiv.org/abs/1305.4456) and by [Losev](https://arxiv.org/abs/1305.4894) when combined with slightly older work of [Shan-Varagnolo-Vasserot](https://arxiv.org/abs/1107.0146).  This technique doesn't have much hope of generalizing to other types (except maybe G(r,p,n)).