This award provides funds for a project to study affine and double affine Hecke algebras. The latter algebras (recently introduced by the principal investigator) lead to a new approach to various problems in representation theory, special functions, harmonic analysis, conformal field theory, combinatorics, topology, and number theory. The main objectives are (1) the theory of differential and difference operators acting on polynomials, theta functions and various generalizations, (2) representations of double affine Hecke algebras at roots of unity, connections with the monodromy of the double affine KZ equations and elliptic braid groups, (3) the action of the modular group on the Macdonal polynomials, (4) relations to quantum groups of elliptic type, Kac-Moody algebras, matrix models, and W-algebras, and (5) applications to harmonic analysis. This research is in the general area of Combinatorics. Combinatorics attempts to find efficient methods to study how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research. This research also deals with orthogonal polynomials, which are useful in designing optical transmission lines.
