The aim of the project is to study double affine Hecke algebras, new and very useful objects in representation theory with many applications in mathematics and mathematical physics. The main objectives are as follows: 1) semisimple and unitary representations of double affine Hecke algebras and the harmonic analysis direction, 2) the counterpart of the Jantzen filtration and its connection to the Plancherel formula for affine Hecke algebras, 3) the non-semisimple theory and its applications to the decomposition of the polynomial representation, 4) the polynomial representation as |q|=1 and at roots of unity with possible relations to Lusztig's quantum groups.
The theory of double affine Hecke algebras attracts now quite a few specialists in representation theory and neighboring fields. It has deep relations to physics (the quantum many-body problem, Knizhnik-Zamolodchikov equation, Verlinde algebras and more). It also provides deep formalization of the concept of the Fourier transform, demonstrates the power of p-adic methods and tremendous potential of the q-functions. The project includes algebraic and analytic aspects of this quickly growing theory and applications, for instance, possible applications to the Langlands duality.
Major Findings: 1) New theory of nil-DAHA and Whittaker function, including the construction of Dunkl-type operators in the q-Toda theory, with applications in quantum and affine theory of flag varieties. 2) The analytic theory of difference spherical and Whittaker functions, generalizing the Harish-Chandra theory of the asymtotic decompositions of the classical spherical functions. 3) A breakthrough theory of Jones and other polynomials of torus knots via DAHA, with applications in physics (the theory of BPS states, the refined Chern-Simons theory) and in the categorization theory. 4) New theory of affine Hall functions and the affine Satake isomorphism based on DAHA, a generalization of the classical theory of p-adic spherical functions with applications to Kac-Moody algebras. Major Awards: 1) Fondation Sciences Mathematiques de Paris Fellowship (2010). 2) Fulbright-Israel Distinguished Chair Fellowship in the Natural Sciences and Engineering (2011-12). Broader Impacts: Organizing (with others) Physics-Mathematics Summer Institute (PhyMSI) in France (Luminy, Cargese; June 19- July 17, 2011) devoted to Double Affine Hecke Algebras, the Langlands Program, Affine Flag Varieties, Conformal Field Theory and Super Yang-Mills Theory. It was one of the major events in these and related fields with more than 140 participants from 13 countries. Among 61 participants from the USA and Canada, 25 were junior investigators; 20 speakers from 78 were junior investigators.
