Let be the ring of integers in a finite extension of , and let be its residue field. Let be a split reductive group over , and let be a maximal split torus in . Let be the pro- Iwahori–Hecke -algebra. Given a semi-infinite reduced chamber gallery (alcove walk) in the -stable apartment, a period of of length , and a homomorphism compatible with , we construct a functor from the category of finite-length -modules to étale -modules over Fontaine’s ring . If , then there are essentially two choices of with , both leading to a functor from to étale -modules and hence to -representations. Both induce a bijection between the set of absolutely simple supersingular -modules of dimension and the set of irreducible representations of over of dimension . We also compute these functors on modular reductions of tamely ramified locally unitary principal series representations of over . For , we recover Colmez’s functor (when restricted to -torsion -representations generated by their pro- Iwahori invariants).
"From pro- Iwahori–Hecke modules to -modules, I." Duke Math. J. 165 (8) 1529 - 1595, 1 June 2016. https://doi.org/10.1215/00127094-3450101