  We construct a family of solvable lattice models whose partition functions
include $p$-adic Whittaker functions for general linear groups from two very
different sources: from Iwahori-fixed vectors and from metaplectic covers.
Interpolating between them by Drinfeld twisting, we uncover unexpected
relationships between Iwahori and metaplectic Whittaker functions. This leads
to new Demazure operator recurrence relations for spherical metaplectic
Whittaker functions. In prior work of the authors it was shown that the row
transfer matrices of certain lattice models for spherical metaplectic Whittaker
functions could be represented as "half-vertex operators" operating on the
$q$-Fock space of Kashiwara, Miwa and Stern. In this paper the same is shown
for all the members of this more general family of lattice models including the
one representing Iwahori Whittaker functions.
