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

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 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.