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

The irreducible representations of the symmetric group Sn are parameterized by combinatorial objects called Young diagrams, or shapes. A given irreducible representation has a basis indexed by Young tableaux of that shape. In fact, this basis consists of weight vectors (simultaneous eigenvectors) for a commutative subalgebra F[X] of the group algebra FSn. The double affine Hecke algebra (DAHA) is a deformation of the group algebra of the affine symmetric group and it also contains a commutative subalgebra F[X]. Not every irreducible representation of the DAHA has a basis of weight vectors (and in fact it is quite difficult to parameterize all of its irreducible representations), but if we restrict our attention to those that do, these irreducible representations are parameterized by “affine shapes” and have a basis (of X-weight vectors) indexed by the “affine tableaux” of that shape. In this talk, we will construct these irreducible representations. Introduction. We introduce and study an affine analogue of skew Young diagrams and tableaux on them. The double affine Hecke algebra of type A acts on the space spanned by standard tableaux on each diagram. We show that the modules obtained this way are irreducible, and they exhaust all irreducible modules of a certain class over the double affine Hecke algebra. In particular, the classification of irreducible modules of this class, announced by Cherednik, is recovered. As is well-known, Young diagrams consisting of n boxes parameterize isomorphism classes of finite dimensional irreducible representations of the symmetric group Sn, 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, on 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 for the (degenerate) affine Hecke algebra Hn(q) of GLn has been given in [Ch1, Ra1, Ra2], 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 I. Cherednik [Ch2, Ch3] 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