Affine quantum Schur algebras and affine Hecke algebras

Let $\mathsf{F}$ be the Schur functor from the category of finite dimensional ${\mathcal{H}}_{△}(r{)}_{\mathbb{C}}$-modules to the category of finite dimensional ${\mathcal{S}}_{△}(n,r{)}_{\mathbb{C}}$-modules, where ${\mathcal{H}}_{△}(r{)}_{\mathbb{C}}$ is the extended affine Hecke algebra of type $A$ over $\mathbb{C}$ and ${\mathcal S}{\vartriangle}(n,r){$\mathbb{C}$}$istheaffinequantumSchuralgebrasover$$\mathbb{C}$$\mathrm{.}TheDrinfeldpolynomialsassociatedwith${\mathsf F}(V)$ were determined in \cite[7.6]{CP96} and \cite[4.4.2]{DDF} in the case of $n>r$, where $V$ is an irreducible ${\mathcal H}{{\vartriangle}}®\mathbb C$-module\mathrm{.}Wewillgeneralizetheresultin[loc\mathrm{.}cit\mathrm{.}]tothecaseof$n\leq r$\mathrm{.}Asanapplication,wewillclassifyfinitedimensionalirreducible${\mathcal S}{\vartriangle}(n,r){$\mathbb{C}$}$-modules, which has been proved in \cite[4.6.8]{DDF} using a different method. Furthermore we will use it to generalize \cite[(6.5f)]{Gr80} to the affine case.