Tableaux on Periodic Skew Diagrams and Irreducible Representations of the Double Affine Hecke Algebra of Type A

As is well known, Young diagrams consisting of n boxes parameterize isomorphism classes of finite-dimensional irreducible representations of the symmetric group Sn of degree n, and moreover the structure of each irreducible representation is described in terms of tableaux on the corresponding Young diagram; namely, a basis of the representation is labeled by standard tableaux, with which the action of Sn generators is explicitly described. This combinatorial description due to A. Young has played an essential role in the study of the representation theory of the symmetric group (or the affine Hecke algebra), and its generalization to the (degenerate) affine Hecke algebra Hn(q) of GLn has been given in [3, 8, 9], where skew Young diagrams appear on combinatorial side. The purpose of this paper is to introduce an “affine analogue” of skew Young diagrams and tableaux, which give a parameterization and a combinatorial description of a family of irreducible representations of the double affine Hecke algebra Ḧn(q) of GLn over a field F, where q ∈ F is a parameter of the algebra. The double affine Hecke algebra was introduced by Cherednik [2, 4] and has since been used by him and by several authors to obtain important results about diagonal coinvariants, Macdonald polynomials, and certain Macdonald identities. In this paper, we focus on the case where q is not a root of 1, and we consider representations of Ḧn(q) that are X-semisimple; namely, we consider representations which