A resolution of the Steinberg module of a parabolic subgroup of GL_n

Abstract

Let F be a field, G = GL_n(F), and let St be the Steinberg module of G. For a parabolic subgroup P of G, let St_P be its Steinberg module. There is a natural map rho_P: St -> St_P corresponding to intersection of a modular symbol with the Borel--Serre boundary. We construct a ZP-resolution Sh_P of St_P, the P-sharbly resolution. We also construct a P-equivariant chain map from the sharbly resolution of St to the P-sharbly resolution of St_P which enables us to compute the map induced on Steinberg homologies by rho_P. In the case when F is a number field and Gamma subset G is a torsion-free arithmetic subgroup, rho_P induces the restriction map H^(Gamma, E) -> H^(Gamma cap P, E) via Borel--Serre duality, for any coefficient module E which is flat over the base ring. If E is a vector space over Z/pZ this provides a computation of the restriction map on cohomology from Gamma to Gamma cap P in terms of sharblies, which enables the study of this restriction map in characteristic p, where Eisenstein series cannot be used.