Coefficient Systems on the Bruhat-Tits Building and Pro-𝑝 Iwahori-Hecke Modules

Let G G be the group of rational points of a split connected reductive group over a nonarchimedean local field of residue characteristic p p . Let I I be a pro- p p Iwahori subgroup of G G and let R R be a commutative quasi-Frobenius ring. If H = R [ I ∖ G / I ] H=R[I\backslash G/I] denotes the pro- p p Iwahori-Hecke algebra of G G over R R we clarify the relation between the category of H H -modules and the category of G G -equivariant coefficient systems on the semisimple Bruhat-Tits building of G G . If R R is a field of characteristic zero this yields alternative proofs of the exactness of the Schneider-Stuhler resolution and of the Zelevinski conjecture for smooth G G -representations generated by their I I -invariants. In general, it gives a description of the derived category of H H -modules in terms of smooth G G -representations and yields a functor to generalized ( φ , Γ ) (\varphi ,\Gamma ) -modules extending the constructions of Colmez, Schneider and Vignéras.