Double Hecke Algebras and Applications

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The purpose of this project is to study double affine Hecke algebras (DAHA), as previously introduced by the principal investigator. They have proved to be very useful in representation theory, the theory of special and spherical functions, conformal field theory, and combinatorics. The main objectives include: (1) A general classification of the semisimple, spherical, unitary, and Fourier-univariant representations of DAHA; (2) the classification of finite dimensional representations of DAHA for generic q, with applications to Macdonald's eta-type identities, Kac-Moody characters, and degenerate Bessel functions; (3) the theory of representations of DAHA at roots of unity, connections with the monodromy of the double affine KZ equations and elliptic braid groups; (4) classificaton of the Fourier-invariant uitary representations of DAHA with applications to Gauss-Selberg sums and Verlinde algebras; (5) harmonic analysis on semisimple representations of DAHA in the compact and noncompact cases, at roots of unity, and in the elliptic case.This is a project in the area of mathematics known as Representation Theory. The nature of representation theory, and what makes it so useful in so many areas of mathematics and science, is that it is a way of studying and encoding symmetries, whereever they may occur, in nature or abstractly. The purpose of this project is to study and develop the properties of a new representation-theoretic tool, "double affine Hecke algebras," that can be used in the application of representation theory to physics (conformal field theory) and combinatorics. Combinatorics is the science of counting the possible arrangements and ways of organizing collections of anything (e.g., atoms, pebbles, star clusters).