The rectangular representation of the double affine Hecke algebra via elliptic Schur-Weyl duality

TL;DR

This paper constructs a combinatorial basis for a module of the double affine Hecke algebra using walks and tableaux, revealing its structure as a rectangular representation.

Contribution

It introduces a new combinatorial description of a module for the double affine Hecke algebra via walks and tableaux, connecting it to rectangular representations.

Findings

Established a weight basis for the module using walks and tableaux.

Identified the module with a rectangular irreducible representation of the double affine Hecke algebra.

Provided a combinatorial framework linking quantum differential operators and affine Hecke algebra representations.

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, i.e. 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.