Nonsymmetric difference Whittaker functions

Starting with nonsymmetric global difference spherical functions, we define and calculate spinor (nonsymmetric) global q-Whittaker functions for arbitrary reduced root systems, which are reproducing kernels of the DAHA-Fourier transforms of Nil-DAHA and solutions of the q-Toda-Dunkl eigenvalue problem. We introduce the spinor q-Toda-Dunkl operators as limits of the difference Dunkl operators in DAHA theory under the spinor variant of the Ruijsenaars procedure. Their general algebraic theory (any reduced root systems) is the key part of this paper, based on the new technique of W-spinors and corresponding developments in combinatorics of affine root systems.