The Lyndon-Hochschild-Serre spectral sequence for a parabolic subgroup of [formula omitted

Abstract

Let Γ be a congruence subgroup of level N in GLn(Z). Let P be a maximal Q-parabolic subgroup of GLn/Q, with unipotent radical U, and let Q=(P∩Γ)/(U∩Γ). Let p>dimQ⁡(U(Q))+1 be a prime number that does not divide N. Let M be a (U,p)-admissible Γ-module. Consider the Lyndon-Hochschild-Serre spectral sequence arising from the exact sequence 1→U∩Γ→P∩Γ→Q→1, which abuts to H⁎(P∩Γ,M). We show that if M is a trivial U∩Γ-module, then certain classes in the E2 page survive to E∞. We use this to obtain information about classes in H⁎(P∩Γ,M) even if M is not a trivial U∩Γ-module. This information will be used in future work to prove a Serre-type conjecture for sums of two irreducible Galois representations.