Kazhdan-Lusztig bases and the asymptotic forms for affine $q$-Schur algebras

We define Kazhdan-Lusztig bases and study asymptotic forms for affine $q$-Schur algebras following Du and McGerty. We will show that the analogues of Lusztig’s conjectures for Hecke algebras with unequal parameters hold for affine $q$-Schur algebras. We will also show that the affine $q$-Schur algebra ${\mathcal{S}}_{q,k}^{△}(2,2)$ over a field $k$ has finite global dimension when char $k=0$ and $1+q\mathrm{\ne }0.$