The Affineq-Schur Algebra

We introduce an analogue of theq-Schur algebra associated to Coxeter systems of typeÁn − 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 typeÁr − 1, wheren ≥ r. This generalizes the originalq-Schur algebra as defined by Dipper and James, and the new algebra contains the ordinaryq-Schur algebra and the affine Hecke algebra as subalgebras. Using this approach we can prove a double centralizer property. The second construction realizes the affineq-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 ordinaryq-Schur algebra as a quotient ofU( n).