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

TL;DR

This paper develops Kazhdan-Lusztig bases and explores asymptotic forms for affine q-Schur algebras, confirming Lusztig's conjectures in this context and analyzing their homological properties.

Contribution

It introduces Kazhdan-Lusztig bases for affine q-Schur algebras and verifies Lusztig's conjectures for these structures, extending known results.

Findings

Lusztig's conjectures hold for affine q-Schur algebras.

Affine q-Schur algebra alS_{q,k}^{ riangle}(2,2) has finite global dimension when char k=0 and 1+q.

Established the structure of asymptotic forms for these 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 $\mathcal{S}_{q,k}^{\vartriangle}(2,2)$ over a field $k$ has finite global dimension when char $k=0$ and $1+q\neq 0.$