Generic and mod $p$ Kazhdan-Lusztig Theory for $GL_2$

Abstract:

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.