Affine quantum Schur algebras and affine Hecke algebras

Abstract

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 ${\mathcal{S}}_{△}(n,r{)}_{\mathbb{C}}$ is the affine quantum Schur algebras over $\mathbb{C}$. The Drinfeld polynomials associated with $\mathsf{F}(V)$ were determined in \cite[7.6]{CP96} and \cite[4.4.2]{DDF} in the case of $n>r$, where $V$ is an irreducible ${\mathcal{H}}_{△}(r{)}_{\mathbb{C}}$-module. We will generalize the result in [loc. cit.] to the case of $n\le r$. As an application, we will classify finite dimensional irreducible ${\mathcal{S}}_{△}(n,r{)}_{\mathbb{C}}$-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.

References in corpus (1)

• Braid Group Action and Quantum Affine Algebras