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L-modules and the Conjecture of Rapoport and Goresky-MacPherson

Leslie Saper

Published 2001-12-22, updated 2005-05-13Version 4

Consider the middle perversity intersection cohomology groups of various compactifications of a Hermitian locally symmetric space. Rapoport and independently Goresky and MacPherson have conjectured that these groups coincide for the reductive Borel-Serre compactification and the Baily-Borel-Satake compactification. This paper describes the theory of L-modules and how it is used to solve the conjecture. More generally we consider a Satake compactification for which all real boundary components are equal-rank. Details will be given elsewhere (math.RT/0112251). As another application of L-modules, we prove a vanishing theorem for the ordinary cohomology of a locally symmetric space. This answers a question raised by Tilouine.

Comments: 16 pages, 4 figures, AMS-LaTeX, smfart.cls, uses xypic 3.7 package; v2: minor typos fixed, definitions of D_P(V) and n_P(V) corrected; v3: various references added in footnotes; v4: updated bibliography, revised and translated abstract, minor typos fixed
Journal: Formes Automorphes (I) -- Actes du Semestre du Centre \'Emile Borel, printemps 2000 (J. Tilouine, H. Carayol, M. Harris, and M.-F. Vign\'eras, eds.), Ast\'erisque 298 (2005), pp. 319-334
Categories: math.RT, math.DG
Subjects: 11F75, 22E40, 32S60, 55N33, 14G35, 22E45
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