The Rectangular Representation of the Double Affine Hecke Algebra via Elliptic Schur–Weyl Duality

Abstract

Abstract Given a module $M$ for the algebra ${\mathcal{D}}_{\mathtt{q}}(G)$ of quantum differential operators on $G$, and a positive integer $n$, we may equip the space $F_n^G(M)$ of invariant tensors in $V^{\otimes n}\otimes M$, with an action of the double affine Hecke algebra of type $A_{n-1}$. Here $G= SL_N$ or $GL_N$, and $V$ is the $N$-dimensional defining representation of $G$. In this paper, we take $M$ to be the basic ${\mathcal{D}}_{\mathtt{q}}(G)$-module, that is, the quantized coordinate algebra $M= {\mathcal{O}}_{\mathtt{q}}(G)$. We describe a weight basis for $F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$ combinatorially in terms of walks in the type $A$ weight lattice, and standard periodic tableaux, and subsequently identify $F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$ with the irreducible “rectangular representation” of height $N$ of the double affine Hecke algebra.