Generic and mod p Kazhdan-Lusztig Theory for GL_2.

Let F be a non-archimedean local field with residue field \mathbb {F}_q and let \mathbf {G}=GL_{2/F}. Let \mathbf {q} be an indeterminate and let \mathcal {H}^{(1)}(\mathbf {q}) be the generic pro-p Iwahori-Hecke algebra of the p-adic group \mathbf {G}(F). Let V_{\mathbf {\widehat {G}}} be the Vinberg monoid of the dual group \mathbf {\widehat {G}}. We establish a generic version for \mathcal {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_{\mathbf {\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 \mathcal {H}^{(1)}_{\overline {\mathbb {F}}_q}(0)-modules. [ABSTRACT FROM AUTHOR]