Physics and Geometry Seminar
About 13 years ago, a new definition of the difference hypergeometric function was suggested; it is conceptually different from the definition Heine gave in 1846, which remained unchanged and unchallenged since then. Algebraically, the new functions, called "global" due to their analyticity everywhere, are closer to Bessel functions than to the classical hypergeometric and spherical functions. The connection to the basic hypergeometric functions is deep; it will be explained in the rank one case.
The construction is based on DAHA, which are deformations of the classical Heisenberg-Weyl algebras. The global functions are defined as the reproducing kernels of Fourier-DAHA transforms. The construction is based on nonsymmetric Macdonald polynomials (eigenfunctions of difference Dunkl operators). The nonsymmetric theory is a new powerful tool in the representations theory and the theory of special functions, generally, beyond the Lie theory.
The case of sl(2) will be mainly considered; no special knowledge of representation theory is assumed. The purpose is to reach (if time allows) the global q-Whittaker functions, which connect the Givental-Lee theory (Gromov-Witten invariants of flag varieties) with the algebraic theory of affine flag varieties, an equivalence of a physics A-model with the corresponding B-model in this setting.