The Affine q-Schur algebra

Abstract

We introduce an analogue of the $q$-Schur algebra associated to Coxeter systems of type ${\hat{A}}_{n-1}$. We give two constructions of this algebra. The first construction realizes the algebra as a certain endomorphism algebra arising from an affine Hecke algebra of type ${\hat{A}}_{r-1}$, where $n\ge r$. This generalizes the original $q$-Schur algebra as defined by Dipper and James, and the new algebra contains the ordinary $q$-Schur algebra and the affine Hecke algebra as subalgebras. Using this approach we can prove a double centralizer property. The second construction realizes the affine $q$-Schur algebra as the faithful quotient of the action of a quantum group on the tensor power of a certain module, analogous to the construction of the ordinary $q$-Schur algebra as a quotient of $U(\mathfrak{g}{\mathfrak{l}}_n)$.

29 pages AMSTeX; 1 eps figure

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