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On the index of Dirac operators on arithmetic quotients

Published online by Cambridge University Press:  17 April 2009

Anton Deitmar
Anton Deitmar
Affiliation:
Mathematisches InstitutUniversitat HeidelbergIm Neuenheimer Feld 288D-69120 HeidelbergGermany
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Abstract

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The aim of this note is to show how the trace formula of Arthur-Selberg can be used to derive index theorems for noncompact arithmetic manifolds. Of special interest is the question, under which circumstances there is an index formula without error term, that is, of the same shape as in the compact case. We shall thus present evidence for the hypothesis that the error term for the Euler operator vanishes in the case that the rational rank is smaller than the real rank.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 3 , December 1997 , pp. 489 - 497
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Arthur, J., ‘The invariant trace formula II. Global theory.’, J. Amer. Math. Soc. 1 (1988), 501–554.CrossRefGoogle Scholar
[2]Atiyah, M., ‘Elliptic operators, discrete groups and Von Neumann algebras’, in Colloque “Analyse et Topologie” en l'Honneur de Henri Cartan (Orsay, 1974), Astérisque 32–33 (Soc. Math. France, Paris, 1976), pp. 43–72.Google Scholar
[3]Atiyah, M. and Schmid, W., ‘A geometric construction of the discrete series for semisimple Lie groups’, Invent. Math. 42 (1977), 1–62.CrossRefGoogle Scholar
[4]Barbasch, D., ‘Fourier inversion for unipotent invariant integrals’, Trans. Amer. Math. Soc. 249 (1979), 51–83.CrossRefGoogle Scholar
[5]Barbasch, D. and Moscovici, H., ‘L 2-index and the Selberg trace formula’, J. Funct. Anal. 53 (1983), 151–201.CrossRefGoogle Scholar
[6]Connes, A. and Moscovici, H., ‘The L 2-index theorem for homogeneous spaces of Lie groups’, Ann. of Math. 115 (1982), 291–330.CrossRefGoogle Scholar
[7]Deitmar, A., ‘Higher torsion zeta functions’, Adv. Math. 110 (1995), 109–128.CrossRefGoogle Scholar
[8]Harder, G., ‘A Gauss-Bonnet formula for discrete arithmetically denned groups’, Ann. Sci.École Norm. Sup. 4 (1971), 409–455.CrossRefGoogle Scholar
[9]Harish-Chandra, , ‘Discrete series for semisimple Lie groups II’, Acta Math. 116 (1966), 1–111.CrossRefGoogle Scholar
[10]Harish-Chandra, :, ‘Harmonic analysis on real reductive groups I. The theory of the constant term’, J. Funct Anal. 19 (1975), 104–204.CrossRefGoogle Scholar
[11]Hirzebruch, F., ‘Hilbert modular surfaces’, Enseign. Math. 19 (1973), 183–281.Google Scholar
[12]Labesse, J.P., ‘Signature des variétés modulaires de Hilbert et representations diédrales’, in Cohomology of arithmetic groups and automorphic forms (Luminy-Marseille, 1989), Lecture Notes in Mathematics 1447 (Springer-Verlag, Berlin, Heidelberg, New York, 1990), pp. 249–260.CrossRefGoogle Scholar
[13]Labesse, J.P., ‘Pseudo-coefficients trés cuspidaux et K-théorie’, Math. Ann. 291 (1991), 607–616.CrossRefGoogle Scholar
[14]Müller, W., ‘Signature defects of cusps of Hilbert modular varieties and values of L-series at s = 1’ J. Differential Geom. 20 (1984), 55–119.CrossRefGoogle Scholar
[15]Parthasarathy, R., ‘Dirac operators and the discrete series’, Ann. of Math. 103 (1976), 375–394.Google Scholar
[16]Stern, M., ‘L 2-index theorems on locally symmetric spaces’, Invent. Math. 96 (1989), 231–282.CrossRefGoogle Scholar
[17]Stern, M., ‘Lefschetz formulae for arithmetic varieties’, Invent. Math. 115 (1994), 241–296.CrossRefGoogle Scholar