L-modules are mixed

Abstract

Let X be the locally symmetric space associated to a reductive $\mathbb Q$-group G and an arithmetic subgroup $\Gamma$. An L-module M is a combinatorial model of a constructible complex of sheaves on $\widehat X$, the reductive Borel-Serre compactification of X whose strata $X_P$ are indexed by $\Gamma$-conjugacy classes of parabolic $\mathbb Q$-subgroups P of G. We show that any L-module M is "mixed" in the sense it is an iterated mapping cone of maps to or from shifted weighted cohomology L-modules on strata $X_P$ of $\widehat X$ with coefficients in V, an irreducible regular $L_P$-module. These weighted cohomology "building blocks" are indexed (up to multiplicity) by V in the weak micro-support of M which is a computable local invariant. As an application we prove that the intersection cohomology of $\widehat X$ is isomorphic to the weighted cohomology of $\widehat X$, at least excluding $\mathbb Q$-types D, E, and F.