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 $G=G{L}_{2/F}$. 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 $V_{\hat{G}}$ be the Vinberg monoid of the dual group $\hat{G}$. 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 $V_{\hat{G}}$ 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 $H_{{\overline{\mathbb{F}}}_q}^{(1)}(0)$-modules.

46 pages, some typos/minor inaccuracies fixed, added a subsection on central characters

Cited by in corpus (2)

• Central elements in affine mod $p$ Hecke algebras via perverse ${\mathbb{F}}_p$-sheaves
• A semisimple mod $p$ Langlands correspondence in families for $G{L}_2({\mathbb{Q}}_p)$