L2-cohomology of locally symmetric spaces, I

Let X be a locally symmetric space associated to a reductive algebraic group G defined over Q. C-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification (X) over cap; they were introduced in [33]. That paper also introduced the micro-support of an,C-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 (X) over cap [33], as well as the ordinary cohomology of X [36]. In this paper we extend the theory so that it covers L-2-cohomology. In particular we construct an L-module Omega((2)) (X, E) whose cohomology is the L-2-cohomology H-(2)(X;E) and we calculate its micro-support. As an application we obtain a new proof of the conjectures of Borel and Zucker.