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On the cuspidal cohomology of arithmetic groups


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.