Double Affine Hecke Algebras

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

Double Affine Hecke Algebras Ivan Cherednik The aim of the project is to study double affine Hecke algebras (DAHA) and their representations, especially those with applications in harmonic analysis. The focus is the structure of the polynomial representation of DAHA, its decomposition and special quotients for generic q, at roots of unity and for unimodular q. The main examples are finite dimensional quotients of the polynomial representation generalizing the Verlinde algebras from conformal field theory; infinite dimensional and non-semisimple quotients are a natural next step. Deep connections to the classical theory of affine Hecke algebras, the theory of Kac-Moody algebras, and modern mathematical physics are expected. Lie groups and Lie algebras formalize the concept of symmetry in the theory of special functions, physics, geometry, and combinatorics. In a similar way, DAHA are candidates for a formalization of the notion of the Fourier transform. In the simplest one-dimensional case, they are directly related to the celebrated sl(2). There are indications that they serve the multidimensional theory of Fourier transform better than Lie and Kac-Moody algebras. DAHA are also a source of new unitary infinite dimensional theories, that, presumably, will find fundamental applications in mathematics and physics, the theory of special functions and combinatorics. There are other important directions of the DAHA theory, mainly of algebraic nature, but unitary representaions and Fourier analysis remain of high priority.