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

Abstract

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 $1+q\ne 0.$

I will withdraw the claim "idempotence of the lowest two-sided ideal" in Section 6, and I am very sorry for my mistakes. Instead, I will study some relatively simple examples. arXiv admin note: text overlap with arXiv:0810.2335 by other authors