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Geometric cycles with local coefficients and the cohomology of arithmetic subgroups of the exceptional group G 2

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Abstract

We study the cohomology of a compact locally symmetric space attached to an arithmetic subgroup of a rational form of a group of type G 2 with values in a finite dimensional irreducible representation E of G 2. By constructing suitable geometric cycles and parallel sections of the bundle \({\tilde{E}}\) we prove non-vanishing results for this cohomology. We prove all possible non-vanishing results compatible with the known vanishing theorems regarding unitary representations with non-zero cohomology in the case of the short fundamental weight of G 2. A decisive tool in our approach is a formula for the intersection numbers with local coefficients of two geometric cycles.

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Authors and Affiliations

  1. Department of Mathematics, University of Vienna, Nordbergstrae 15, 1090, Vienna, Austria

    Christoph Waldner

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  1. Christoph Waldner
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Correspondence to Christoph Waldner.

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The author was supported by the Austrian Science Fund (FWF), project no. P 21090 at the University of Vienna, when the results of this paper were obtained.

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Waldner, C. Geometric cycles with local coefficients and the cohomology of arithmetic subgroups of the exceptional group G 2 . Geom Dedicata 151, 9–25 (2011). https://doi.org/10.1007/s10711-010-9516-5

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