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

Abstract

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.

MODEL CATEGORIES AND PRO-p IWAHORI–HECKE MODULES

“…In fact, it is concentrated in degrees 0 ≤ i ≤ d and admits a functorial isomorphism C or c (X (•) ,F(M )) I ∼ = GP(M ) • (11) in Ch(H) (cf. [22], Remark 3.24). In particular, there is a functorial isomorphism H 0 (C or c (X (•) ,F(M )) I ) ∼ = M of H -modules (cf.…”

“…The corresponding complex C or c (X (•) ,F(M )) of oriented chains is I -exact (cf. [22], Proposition 2.9). In fact, it is concentrated in degrees 0 ≤ i ≤ d and admits a functorial isomorphism C or c (X (•) ,F(M )) I ∼ = GP(M ) • (11) in Ch(H) (cf.…”

“…On the other hand, the second author functorially associates with M ∈ Mod(H) a G-equivariant coefficient system F(M ) on the semisimple Bruhat-Tits building X of G (cf. [22], §3.2). The corresponding complex C or c (X (•) ,F(M )) of oriented chains is I -exact (cf.…”

“…Passing to Ho(Mod(H)) ∼ = GProj(H)/Proj(H) simplifies the situation further by getting rid of the objects of finite projective dimension. Finally, we explain in which way the objects of C(G) appear in the theory of equivariant coefficient systems on Bruhat-Tits buildings as studied in [22] and [28]. To this aim, we assume that H is associated to G = G(K) and I = I C as in Example 4.1 (iv).…”

“…On the other hand, we consider the complex C or c (X (•) ,F(M )) of oriented chains of the G-equivariant coefficient system F(M ) on the Bruhat-Tits building X of G associated with M by the second author (cf. [22], §3.2). In Proposition 5.13, we show that there is a functorial isomorphism of complexes…”