PROM PRO-p IWAHORI-HECKE MODULES TO (φ, Γ)-MODULES, I

Let o be the ring of integers in a finite extension K of Q(p), and let k be its residue field. Let G be a split reductive group over Q(p), and let T be a maximal split torus in G. Let H (G, I-0) be the pro-p Iwahori-Hecke o-algebra. Given a semi-infinite reduced chamber gallery (alcove walk) C-(center dot) in the T-stable apartment, a period phi is an element of N(T) of C-(center dot) of length r, and a homomorphism tau : Z(p)(x) -> T compatible with phi, we construct a functor from the category Mod(fin)(H(G, I-0)) of finite-length H (G, I-0)-modules to etale (phi(r), Gamma)-modules over Fontaine's ring O-epsilon. If G = GL(d+1) (Q(p)), then there are essentially two choices of (C-(center dot), phi, tau) with r = 1, both leading to a functor from Mod(fin)(H(G, I-0)) to etale (phi, Gamma)-modules and hence to Gal(Qp)-representations. Both induce a bijection between the set of absolutely simple supersingular H (G, I-0) circle times(0) k-modules of dimension d +1 and the set of irreducible representations of Gal(Qp) over k of dimension d + 1. We also compute these functors on modular reductions of tamely ramified locally unitary principal series representations of G over K. For d = 1, we recover Colmez's functor (when restricted to o-torsion GL(2)(Q(p))-representations generated by their pro-p Iwahori invariants).