Weighted cohomology of arithmetic groups

Let G be a semisimple algebraic group defined over the rational numbers, K a maximal compact subgroup of G = G(R), and Γ ⊂ G(Q) a neat arithmetic subgroup. LetX = Γ\G/K be the locally symmetric space associated to Γ, and E the local system on X constructed out of a finite-dimensional, irreducible, algebraic representation E of G. Fix a maximally Q-split torus S in G; S is assumed to be nontrivial, so that X is noncompact. Let A = S(R)0 and, by a slight abuse of notation, let X∗(A) denote the group of rational characters of S. Choose a minimal rational parabolic subgroup P0 ⊃ S; the choice determines a notion of positivity in X∗(A) and a set of positive roots among the roots of S in G. Let ρ0 be the half-sum of the positive roots. Weighted cohomology is an invariant of Γ introduced by M. Goresky, G. Harder, and R. MacPherson [12] in the study ([15], [16], [12], [14]) of the trace of Hecke operators in the cohomology of Γ. For each p ∈ X∗(A)⊗Q there are weight profiles p and p (upper and lower p; see 1.5) and groups W pH i(Γ, E) and W H i(Γ, E). (The notion of profile here differs from (but is equivalent to) that of [12]; see Remark 1.2 below.) If p+q = −2ρ0 then the profiles p and q are dual and the corresponding groups are in Poincaré duality. For very positive p one gets the compactly supported cohomology H i c(X,E) and dually, for very negative p, the full cohomology H i(X,E). The definition of the weighted cohomology groups is somewhat involved, so I will attempt to give some idea of the motivation behind it; detailed definitions are recalled in Section 1. Associated to p and p are complexes of sheaves WpC•(E) and WC•(E) on the reductive Borel-Serre compactification X of X. Let P be a rational parabolic subgroup, NP its unipotent radical, SP the split centre of P/NP and XP the P-boundary component in X. The idea behind the construction in [12] is as follows: The stalk cohomology of the direct image Ri∗E (by i : X →֒ X) at x ∈ XP is