# Mathematics Colloquia and Seminars

The irreducible representations (irreps) of the symmetric group $S_n$ are parameterized by combinatorial objects called Young diagrams'' (or partitions) $\lambda$. A given irrep has a basis indexed by Young tableaux'' of that shape $\lambda$. In fact, this basis consists of weight vectors (simultaneous eigenvectors) for a commutative subalgebra $X$ of the group algebra ${\mathbb C} S_n$.
The double affine Hecke algebra (DAHA) is a deformation of the group algebra of the {\it affine \/} symmetric group ${\widehat{S_n}}$, and also contains a commutative subalgebra $X$. (The DAHA has received attention recently for its connections to special functions, conformal field theory, harmonic analysis of symmetric spaces, the classical theories of hypergeometric functions and $q$-hypergeometric functions, etc.)
Not every irrep of the DAHA has a basis of weight vectors (and in fact it is quite difficult to parameterize all of its irreps), but if we restrict our attention to those that do, these are parameterized by affine shapes'' ${\widehat{ \lambda / \mu} }$ and have a basis (of $X$-weight vectors) indexed by the affine tableaux'' of that shape. In this talk, we will construct these irreps.