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

Li Luo, Weiqiang Wang

Published 2017-10-28Version 1

We formulate a $q$-Schur algebra associated to an arbitrary $W$-invariant finite set $X_{\texttt f}$ 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 $X_{\texttt f}$ in classical types, we recover the $q$-Schur algebras in the literature. We study the $q$-Schur algebra of type $G_2$ in detail, and establish its connection to the category $\mathcal O$.