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$L_2$-cohomology and intersection homology of locally symmetric varieties, II

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L2-cohomology and intersection homology of locally symmetric varieties, II COMPOSITIO MATHEMATICA STEVEN ZUCKER L2-cohomology and intersection homology of locally symmetric varieties, II Compositio Mathematica, tome 59, no 3 (1986), p. 339-398. <> © Foundation Compositio Mathematica, 1986, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: // implique l’accord avec les conditions gé- nérales d’utilisation ( Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 339 L2-COHOMOLOGY AND INTERSECTION HOMOLOGY OF LOCALLY SYMMETRIC VARIETIES, II Steven Zucker Compositio Mathematica 59 (1986) 339-398 © Martinus Nijhoff Publishers, Dordrecht - Printed in the Netherlands Introduction In this paper, we continue our study of the L2-cohomology (with the customary coefficients E, and natural metrics) of quotients FNX of symmetric spaces by arithmetic groups. Though it bears the title of [16], it is actually the sequel to [15]. (In fact, [16] is but a condensed account of [15].) The latter was conceived in an attempt to better understand the L2-cohomology of itself, and to obtain better injectivity results than were then available for the mapping of it into ordinary cohomology. However, it became almost immediately apparent that it was the conjecture [15, (6.20)] (see (3.2) here) - which indeed would imply sharp isomorphism ranges for the mapping - that was attracting most of the attention. The conjecture asserts, for X Hermitian, the natural isomorphism between the L2-cohomology of 1’BX and the (middle perversity) intersection homology [10] of its Baily-Borel-Satake compactification

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