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 and by Losev when combined with slightly older work of Shan-Varagnolo-Vasserot. This technique doesn't have much hope of generalizing to other types (except maybe G(r,p,n)).

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 and by Losev when combined with slightly older work of Shan-Varagnolo-Vasserot. This technique doesn't have much hope of generalizing to other types (except maybe G(r,p,n)).

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 and by Losev when combined with slightly older work of Shan-Varagnolo-Vasserot. This technique doesn't have much hope of generalizing to other types (except maybe G(r,p,n)).

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 and by Losev when combined with slightly older work of Shan-Varagnolo-Vasserot. This technique doesn't have much hope of generalizing to other types (except maybe G(r,p,n)).