  Let $F$ be a non-archimedean local field with residue field $\mathbb{F}_q$
and let $G = GL_{2/F}$. Let $\mathbf{q}$ be an indeterminate and let
$H^{(1)}(\mathbf{q})$ be the generic pro-p Iwahori-Hecke algebra of the group
$G(F)$. Let $V_{\widehat{G}}$ be the Vinberg monoid of the dual group
$\widehat{G}$. We establish a generic version for $H^{(1)}(\mathbf{q})$ of the
Kazhdan-Lusztig-Ginzburg spherical representation, the Bernstein map and the
Satake isomorphism. We define the flag variety for the monoid $V_{\widehat{G}}$
and establish the characteristic map in its equivariant K-theory. These generic
constructions recover the classical ones after the specialization $\mathbf{q} =
q \in \mathbb{C}$. At $\mathbf{q} = q = 0 \in\overline{\mathbb{F}}_q$, the
spherical map provides a dual parametrization of all the irreducible
$H^{(1)}_{\overline{\mathbb{F}}_q}(0)$-modules.
