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

Abstract

We formulate a $q$-Schur algebra associated to 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 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 $X_{\text{f}}$ 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.

v3, 33 pages, mild corrections, J. Inst. Math. Jussieu (to appear)

References in corpus (1)

• Affine flag varieties and quantum symmetric pairs

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