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On the cuspidal cohomology of arithmetic groups
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 131, Number 5, October 2009
- pp. 1431-1464
- 10.1353/ajm.0.0073
- Article
- Additional Information
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It is the main objective of this paper to show a vanishing result
for cuspidal cohomology of arithmetic groups in classical groups~$G$
defined over some number field~$k$.
Our approach relies on the fact that cuspidal automorphic forms are nonsingular
in the sense of Howe. This result puts a strong constraint on the
(archimedean components of) irreducible cuspidal automorphic representations of $G$
that can possibly contribute to the cuspidal cohomology. Combining this with the
classification of Vogan-Zuckerman of unitary representations with nonzero cohomology
provides a constant $r_0(G/k)$, only depending on $G/k$, below which the cuspidal
cohomology vanishes. We will give a formula for this constant for each classical
group of type (I) in the classification scheme due to Weil. We conclude with making
this result explicit for some split classical groups over a totally real algebraic
number field.