  This work, to be published in Transformation Groups in two parts, is devoted
to the theory of nil-DAHA for the root system A_1 and its applications to
symmetric and nonsymmetric (spinor) global q-Whittaker functions. These
functions integrate the q-Toda eigenvalue problem and its Dunkl-type
nonsymmetric version.
  The global symmetric function can be interpreted as the generating function
of the Demazure characters for dominant weights, which describe the
algebraic-geometric properties of the corresponding affine Schubert varieties.
Its Harish-Chandra-type asymptotic expansion appeared directly related to the
solution of the q-Toda eigenvalue problem obtained by Givental and Lee in the
quantum K-theory of flag varieties. It provides an exact mathematical relation
between the corresponding physics A-type and B-type models.
  The spinor global functions extend the symmetric ones to the case of all
Demazure characters (not only those for the dominant weights); the
corresponding Gromov-Witten theory is not known. The main result of the paper
is a complete algebraic theory of these functions in terms of the induced
modules of the core subalgebra of nil-DAHA. It is the first instance of the
DAHA theory of canonical-crystal bases, quite non-trivial even for A_1.
