


On the Cohomology of Certain NonCompact Shimura Varieties (AM173) 
This book studies the intersection cohomology of the Shimura varieties associated to unitary groups of any rank over Q. In general, these varieties are not compact. The intersection cohomology of the Shimura variety associated to a reductive group G carries commuting actions of the absolute Galois group of the reflex field and of the group G(Af) of finite adelic points of G. The second action can be studied on the set of complex points of the Shimura variety. In this book, Sophie Morel identifies the Galois actionat good placeson the G(Af)isotypical components of the cohomology. Morel uses the method developed by Langlands, Ihara, and Kottwitz, which is to compare the GrothendieckLefschetz fixed point formula and the ArthurSelberg trace formula. The first problem, that of applying the fixed point formula to the intersection cohomology, is geometric in nature and is the object of the first chapter, which builds on Morel's previous work. She then turns to the grouptheoretical problem of comparing these results with the trace formula, when G is a unitary group over Q. Applications are then given. In particular, the Galois representation on a G(Af)isotypical component of the cohomology is identified at almost all places, modulo a nonexplicit multiplicity. Morel also gives some results on base change from unitary groups to general linear groups. "This book is a research monograph, yet the author takes care in recalling in detail the relevant notation and previous results instead of just referring to the literature. Also, explicit calculations are given, making the book readable not only for experts but also for interested advanced students."Eva Viehmann, Mathematical Reviews Preface vii Series:
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