Quiver Schur algebras and Koszul duality

Stroppel and Webster have introduced a grading on the cyclotomic q  -Schur algebra Sds. We prove that the obtained graded algebra is graded Morita equivalent to a Koszul algebra. The proof is based on a result of Rouquier, Shan, Varagnolo and Vasserot that identifies the category mod(Sds) with a subcategory of an affine parabolic category OO of type A. This subcategory admits a Koszul grading constructed by Shan, Varagnolo and Vasserot. We identify this Koszul grading with the grading on mod(Sds).