The aim of this project is to study the double affine Hecke algebra introduced by Dr. Cherednik which promises important application to many other branches of mathematics. Specific objectives include the study of Macdonald polynomials of double affine Hecke algebras at roots of unity, as well as induced and spherical representations of the algebras. Dr. Cherednik also intends to explore the analytic theory of q-spherical functions, and the difference Fourier transformation including q-deformations of the Gauss integrals, Zeta functions, and L- functions. Finally applications of double affine Hecke algebra to Harish-Chandra theory and the harmonic analysis on Kac-Moody groups will be considered.
This is a project in the area of mathematics known as Algebra. In the 20-th century, algebra has become the language of all of mathematics. Mathematicians studying geometry, number theory, analysis, mathematical physics, statistics, or just about anything have noticed that there are underlying structures in their studies that transcend their origins. In Algebra these underlying structures are studied in their pure form. As the fundamental properties of these structures are uncovered, the implications reverberate back to all the original settings in which these structures occur. In earlier research Dr. Cherednik introduced a new structure to algebra known as a double affine Hecke algebra which appears in representation theory, special functions, harmonic analysis, combinatorics, and conformal field theory. In this project he will continue his study of this new algebraic object.
