L^2-cohomology of locally symmetric spaces, I

Leslie Saper

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arXiv · math/0412353

L^2-cohomology of locally symmetric spaces, I

Leslie Saper

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Abstract

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.

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BibTeX RIS

Leslie Saper. 2006-01-11. L^2-cohomology of locally symmetric spaces, I. https://arxiv.org/abs/math/0412353

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