L-modules and micro-support
Abstract
L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of a locally symmetric space. We define the micro-support of an L-module; it is a set of irreducible modules for the Levi quotients of the parabolic Q-subgroups associated to the strata. We prove a vanishing theorem for the global cohomology of an L-module in term of the micro-support. We calculate the micro-support of the middle weight profile weighted cohomology and the middle perversity intersection cohomology L-modules. (For intersection cohomology we must assume the Q-root system has no component of type D_n, E_n, or F_4.) Finally we prove a functoriality theorem concerning the behavior of micro-support upon restriction of an L-module to the pre-image of a Satake stratum. As an application we settle a conjecture made independently by Rapoport and by Goresky and MacPherson, namely, that the intersection cohomology (for either middle perversity) of the reductive Borel-Serre compactification of a Hermitian locally symmetric space is isomorphic to the intersection cohomology of the Baily-Borel-Satake compactification. We also obtain a new proof of the main result of Goresky, Harder, and MacPherson on weighted cohomology as well as generalizations of both of these results to general Satake compactifications with equal-rank real boundary components. An overview of the theory of L-modules and the above conjecture, as well as an application to the cohomology of arithmetic groups, can be found in math.RT/0112250 .
Cite
@article{arxiv.math/0112251,
title = {L-modules and micro-support},
author = {Leslie Saper},
journal= {arXiv preprint arXiv:math/0112251},
year = {2007}
}
Comments
88 pages, AMS-LaTeX, uses xypic 3.7 package; v2: minor typos fixed, definitions of D_P(V) in 10.2 and of n_P(V) in 24.1 corrected and their usage revised; v3: minor typos fixed, proof of Prop. 16.2 simplified, minor improvements in exposition