The Rectangular Representation Of The Double Affine Hecke Algebra Via Elliptic Schur-Weyl Duality

Given a module M for the algebra D-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-circle times n circle times M, with an action of the double affine Hecke algebra of type A(n-1). Here G = SLN or GL(N), and V is the N-dimensional defining representation of G. In this paper, we take M to be the basic D-q (G)-module, that is, the quantized coordinate algebra M = 0, 1 (G). We describe a weight basis for F-n(G) (O-q(G)) combinatorially in terms of walks in the type A weight lattice, and standard periodic tableaux, and subsequently identify F-n(G) (O-q(G)) with the irreducible "rectangular representation" of height N of the double affine Hecke algebra.