Double Affine Hecke Algebras

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This project sits at the interface of the mathematical subfields of harmonic analysis, representation theory, and combinatorics. The project is expected to have applications to the classification of knots in three-dimensional space. It is also expected to have connections to theoretical physics. At the center of the project is a family of algebras, introduced by the PI in the middle 1990's, that have two actions by a symmetry group. These algebras are known as double affine Hecke algebras and they have found uses in several areas of mathematics, starting with their use in the proof of a conjecture of Ian Macdonald about the properties of certain orthogonal polynomials that arise from the study of symmetry.The major themes of the proposed work are expected to be the following: (1) the theory of invariants of torus knots using the structure of double affine Hecke algebras (abbreviated DAHA), (2) a new theory of Rogers-Ramanujan identities based on nil-DAHA, (3) a surprising formula for the minimal number of creation operators in terms of non-symmetric Macdonald polynomials, and (4) the action of the absolute Galois group on the ramified covers of elliptic curves associated with perfect DAHA modules.