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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: [email protected]
CHRISTOPH WALDNER
Affiliation:
Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria. e-mail: [email protected]
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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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References

REFERENCES

[1]Borel, A.Compact Clifford–Klein forms of symmetric spaces. Topology 2 (1963), 111–122.CrossRefGoogle Scholar
[2]Borel, A. and Harish–Chandra, Arithmetic subgroups of algebraic groups. Ann. of Math. 75 (1962), 485–535.CrossRefGoogle Scholar
[3]Borel, A. and Serre, J.-P.Corners and arithmetic groups. Comment. Math. Helvet. 48 (1973), 436–491.CrossRefGoogle Scholar
[4]Borel, A. and Wallach, N.Continuous Cohomology, Discrete Subgroups and Representations of Reductive Groups. Ann. Math. Studies 94 (Princeton University Press, 1980).Google Scholar
[5]Farell, F., Ontaneda, P. and Raghunathan, M.S.Non-univalent harmonic maps homotopic to diffeomorphisms, J. Differential Geometry 54 (2000), 227–253.Google Scholar
[6]Greub, W., Halperin, S. and Vanstone, R. Connections, Curvature, and Cohomology. Volume III: Cohomology of principal bundles and homogeneous spaces. Pure and Applied Mathematics, Vol. 47-III (Academic Press 1976).Google Scholar
[7]Helgason, S.Differential Geometry, Lie Groups, and Symmetric Spaces (Academic Press, 1962).Google Scholar
[8]Knus, M.-A., Merkurjev, A., Rost, M. and Tignol, J.-P. The book of involutions. AMS, Colloquium Publications, Volume 44 (1998).CrossRefGoogle Scholar
[9]Knapp, A. W. and Vogan, D. A. Cohomological induction and unitary representations. Princeton Mathematical Series 45 (Princeton Univ Press, 1995).CrossRefGoogle Scholar
[10]Matsushima, Y.On Betti numbers of compact, locally symmetric Riemannian manifolds, Osaka Math. J. 14 (1962), 1–20.Google Scholar
[11]Millson, J.On the first Betti number of an hyperbolic manifold. Ann. Math. 104 (1976), pp. 235–247.CrossRefGoogle Scholar
[12]Millson, J. and Raghunathan, M. S.Geometric construction of cohomology for arithmetic groups I. In: Geometry and Analysis (Paper dedicated to the memory of Patodi), pp. 103–123, Indian Academy of Sciences, Bangalore, 1980.Google Scholar
[13]Parthasarathy, R.A generalization of the Enright-Vardarajan modules. Comp. Math. 36 (1978), 53–73.Google Scholar
[14]Rohlfs, J.Arithmetisch definierte Gruppen mit Galoisoperation. Invent. Math. 48 (1978), 185–205.CrossRefGoogle Scholar
[15]Rohlfs, J.The Lefschetz number of an involution on the space of classes of positive definite quadratic forms, Comment. Math. Helv. 56 (1981), 272–296.CrossRefGoogle Scholar
[16]Rohlfs, J. and Schwermer, J.Intersection numbers of special cycles, J. Amer. Math. Soc. 6 (1993), 755–778.CrossRefGoogle Scholar
[17]Salamanca Riba, S.On the unitary dual of some classical Lie groups. Comp. Math. 68 (1988), 251–303.Google Scholar
[18]Schwermer, J.Geometric cycles, arithmetic groups and their cohomology. Bull. Amer. Math. Soc. 47 (2010), no. 2, 187–279.CrossRefGoogle Scholar
[19]Schwermer, J. Geometric cycles, Albert algebras and related cohomology classes for arithmetic groups, to appear in Groups, Geometry and Dynamics.Google Scholar
[20]Vogan, D.Unitarizability of certain series of representations. Ann. of Math. 120 (1984), 141–187.CrossRefGoogle Scholar
[21]Vogan, D.Cohomology and group representations. In: Proc. Symp. Pure Maths. 61 (Amer. Math. Soc., 1997), pp. 219–243.CrossRefGoogle Scholar
[22]Vogan, D. and Zuckerman, G. J.Unitary representations with non-zero cohomology. Comp. Math. 53 (1984), 51–90.Google Scholar
[23]Waldner, C.Geometric cycles and the cohomology of arithmetic subgroups of the exceptional group G 2. J. Topology 3 (2010), 81–109.CrossRefGoogle Scholar
[24]Wallach, N. R.Real Reductive Groups, I. Pure Appl. Math. 132 (Academic Press, 1988).Google Scholar