Affine quantum Schur--Weyl duality

Let $\text{mathpzc}K$ be an arbitrary commutative ring containing an invertible element $\epsilon$. Let ${\mathcal{H}}_{\text{\hspace{0.17em}⁣}△\text{\hspace{0.17em}⁣}}(r{)}_{\text{mathpzc}K}$ be the extended affine Hecke algebra of type $A$ with Hecke parameter $\epsilon$, let ${\mathrm{\Omega }}_{\text{mathpzc}K}^{\otimes r}$ be the affine tensor space, and let ${\mathcal{S}}_{\text{\hspace{0.17em}⁣}△\text{\hspace{0.17em}⁣}}(n,r{)}_{\text{mathpzc}K}$ be the corresponding affine quantum Schur algebra. We first prove that the natural right action of ${\mathcal{H}}_{\text{\hspace{0.17em}⁣}△\text{\hspace{0.17em}⁣}}(r{)}_{\text{mathpzc}K}$ on ${\mathrm{\Omega }}_{\text{mathpzc}K}^{\otimes r}$ is always faithful. Assume further that $\text{mathpzc}K$ is a field of characteristic $0$ and that $\epsilon$ is not a root of unity. We prove that, for any $n\ge 2$, the natural algebra homomorphism ${\xi }_r:{\mathcal{H}}_{\text{\hspace{0.17em}⁣}△\text{\hspace{0.17em}⁣}}(r{)}_{\text{mathpzc}K}\to {\mathrm{End}}_{{\mathcal{S}}_{\text{\hspace{0.17em}⁣}△\text{\hspace{0.17em}⁣}}(n,r{)}_{\text{mathpzc}K}}({\mathrm{\Omega }}_{\text{mathpzc}K}^{\otimes r}{)}^{\mathrm{o}\mathrm{p}}$ is an isomorphism. This proves Conjecture~3.8.8 of \cite{DDF}. As an application, we prove the conjecture formulated in \cite[5.2.4]{DDF} concerning the center of the affine quantum Schur algebra. We also prove that ${\mathcal{S}}_{\text{\hspace{0.17em}⁣}△\text{\hspace{0.17em}⁣}}(n,r{)}_{\text{mathpzc}K}$ is left and right Noetherian whenever $\text{mathpzc}K$ is a Noetherian commutative ring, which verify a conjecture in \cite[Rem. 1.7]{DY}.