Graded $q$-Schur algebras

Generalizing recent work of Brundan and Kleshchev, we introduce grading on Dipper-James’ $q$-Schur algebra, and prove a graded analogue of the Leclerc and Thibon’s conjecture on the decomposition numbers of the $q$-Schur algebra when $q^2\mathrm{\ne }1$ and $q^3\mathrm{\ne }1$.