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

Abstract

LetGGbe the group of rational points of a split connected reductive group over a nonarchimedean local field of residue characteristicpp. LetIIbe a pro-ppIwahori subgroup ofGGand letRRbe a commutative quasi-Frobenius ring. IfH=R[I∖G/I]H=R[I\backslash G/I]denotes the pro-ppIwahori-Hecke algebra ofGGoverRRwe clarify the relation between the category ofHH-modules and the category ofGG-equivariant coefficient systems on the semisimple Bruhat-Tits building ofGG. IfRRis a field of characteristic zero this yields alternative proofs of the exactness of the Schneider-Stuhler resolution and of the Zelevinski conjecture for smoothGG-representations generated by theirII-invariants. In general, it gives a description of the derived category ofHH-modules in terms of smoothGG-representations and yields a functor to generalized(φ,Γ)(\varphi ,\Gamma )-modules extending the constructions of Colmez, Schneider and Vignéras.