Double Affine Hecke Algebras

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

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.