L²-cohomology of locally symmetric spaces. I

Journal Article

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 L²-cohomology. In particular we construct an L-module whose cohomology is the L²-cohomology of X and we calculate its micro-support. As an application we obtain a new proof of the conjectures of Borel and Zucker.

Full Text

Duke Authors

Cited Authors

  • Saper, L

Published Date

  • 2005

Published In

Volume / Issue

  • 1 / 4

Start / End Page

  • 889 - 937

International Standard Serial Number (ISSN)

  • 1558-8599

Digital Object Identifier (DOI)

  • 10.4310/PAMQ.2005.v1.n4.a9