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Higher level affine Schur and Hecke algebras

We define a higher level version of the affine Hecke algebra and prove that, after completion, this algebra is isomorphic to a completion of Webster's tensor product algebra of type A. We then introduce a higher level version of the affine Schur algebra and establish, again after completion, an isomorphism with the quiver Schur algebra. An important observation is that the higher level affine Schur algebra surjects to the Dipper-James-Mathas cyclotomic q-Schur algebra. Moreover, we give nice diagrammatic presentations for all the algebras introduced in this paper.