  Let X be a locally symmetric space associated to a reductive algebraic group
G defined over Q. L-modules are a combinatorial analogue of constructible
sheaves on the reductive Borel-Serre compactification of X; they were
introduced in [math.RT/0112251]. That paper also introduced the micro-support
of an L-module, a combinatorial invariant that to a great extent characterizes
the cohomology of the associated sheaf. The theory has been successfully
applied to solve a number of problems concerning the intersection cohomology
and weighted cohomology of the reductive Borel-Serre compactification
[math.RT/0112251], as well as the ordinary cohomology of X [math.RT/0112250].
In this paper we extend the theory so that it covers L^2-cohomology. In
particular we construct an L-module whose cohomology is the L^2-cohomology of X
and we calculate its micro-support. As an application we obtain a new proof of
the conjectures of Borel and Zucker.
