We formulate a $q$-Schur algebra associated with an arbitrary $W$-invariant finite set $X_{\text{f}}$ of integral weights for a complex simple Lie algebra with Weyl group $W$. We establish a $q$-Schur duality between the $q$-Schur algebra and Hecke algebra associated with $W$. We then realize geometrically the $q$-Schur algebra and duality and construct a canonical basis for the $q$-Schur algebra with positivity. With suitable choices of $X_{\text{f}}$ in classical types, we recover the $q$-Schur algebras in the literature. Our $q$-Schur algebras are closely related to the category ${\mathcal{O}}$, where the type $G_{2}$ is studied in detail.

中文翻译：

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有限型的-SCHUR代数和-SCHUR对偶

我们制定一个$q$-Schur代数与任意关联$W$-不变的有限集$X_{\text{f}}$具有 Weyl 群的复杂简单李代数的积分权重$W$. 我们建立一个$q$-舒尔之间的对偶$q$-Schur 代数和 Hecke 代数与$W$. 然后我们几何地实现$q$-Schur 代数和对偶，并为$q$-Schur 代数与正数。有合适的选择$X_{\text{f}}$在经典类型中，我们恢复$q$——文献中的舒尔代数。我们的$q$-Schur代数与范畴密切相关${\mathcal{O}}$, 其中类型$G_{2}$进行了详细研究。