The aim of the project is to study double affine Hecke algebras introduced by PI. They proved to be very useful in the representation theory and found many applications in mathematics and physics. The area of the algebraic analysis and its applications is the major theme of the project, including the harmonic analysis on symmetric spaces, the theory of hypergeometric, spherical and Whittaker functions. The theory of the difference counterparts of these functions, called ``global functions" due to their universality and excellent analytic properties, is one of the greatest applications of double affine Hecke algebras obtained by PI, Stokman and other researches. The global q-Whittaker functions have and expected to have multiple applications, including the Givental-Lee theory and the quantum Langlands program; the corresponding theory of nil-DAHA is very fruitful.
The theory of DAHA is a breakthrough development in the geometric, analytic and physically-inspired representation theory, which demonstrates the power of p-adic methods and tremendous potential of the q-functions. The directions of the project are mainly grouped around PI's theory of global difference spherical and Whittaker functions. There are important relations (known and expected) of these functions to the theory of homogeneous spaces of loop groups and (hopefully) to the geometric quantum Langlands program. The first results concerning the behavior of these functions at some special cases are an indication of their significance for the Number Theory. Also, these functions are expected to serve recent physics theories employing the integrability of the various quantum many-body problems.
