Generic and Mod p Kazhdan-Lusztig Theory for GL 2

Let F be a non-archimedean local field with residue field Fq and let G = GL 2/F. Let q be an indeterminate and let H (1) (q) be the generic prop Iwahori-Hecke algebra of the p-adic group G(F). Let V G be the Vinberg monoid of the dual group G. We establish a generic version for H (1) (q) of the Kazhdan-Lusztig-Ginzburg antispherical representation, the Bernstein map and the Satake isomorphism. We define the flag variety for the monoid V G and establish the characteristic map in its equivariant K-theory. These generic constructions recover the classical ones after the specialization q = q ∈ C. At q = q = 0 ∈ Fq, the antispherical map provides a dual parametrization of all the irreducible H (1) Fq (0)-modules. When F = Qp with p ≥ 5, we relate our space of mod p Satake parameters to Emerton-Gee's space of semisimple mod p two-dimensional representations of the Galois group Gal(Q p /Qp), thereby arriving at a version in families of Breuil's semisimple mod p Langlands correspondence for GL2(Qp)