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We formulate a $q$-Schur algebra associated to an arbitrary $W$-invariantfinite set $X_{\texttt f}$ of integral weights for a complex simple Lie algebrawith Weyl group $W$. We establish a $q$-Schur duality between the $q$-Schuralgebra and Hecke algebra associated to $W$. 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 $X_{\texttt f}$ inclassical types, we recover the $q$-Schur algebras in the literature. Our$q$-Schur algebras are closely related to the category $\mathcal O$, where thetype $G_2$ is studied in detail.

The $q$-Schur algebras and $q$-Schur dualities of finite type