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On the cohomology of uniform arithmetically defined subgroups in SU*(2n)

Published online by Cambridge University Press:  18 July 2011

JOACHIM SCHWERMER  and
CHRISTOPH WALDNER
JOACHIM SCHWERMER
Affiliation:
Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria. e-mail: Joachim.Schwermer@univie.ac.at
CHRISTOPH WALDNER
Affiliation:
Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria. e-mail: Joachim.Schwermer@univie.ac.at
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Abstract

We study the cohomology of compact locally symmetric spaces attached to arithmetically defined subgroups of the real Lie group G = SU*(2n). Our focus is on constructing totally geodesic cycles which originate with reductive subgroups in G. We prove that these cycles, also called geometric cycles, are non-bounding. Thus this geometric construction yields non-vanishing (co)homology classes.

In view of the interpretation of these cohomology groups in terms of automorphic forms, the existence of non-vanishing geometric cycles implies the existence of certain automorphic forms. In the case at hand, we substantiate this close relation between geometry and automorphic theory by discussing the classification of irreducible unitary representations of G with non-zero cohomology in some detail. This permits a comparison between geometric constructions and automorphic forms.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 3 , November 2011 , pp. 421 - 440
Copyright
Copyright © Cambridge Philosophical Society 2011

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