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 FS _n. 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.