We give in this paper an explicit construction of the covariant quantization of the rank-two massless tensor field on de Sitter space (linear covariant quantum gravity on a de Sitter background). The main ingredient of the construction is an indecomposable representation of de Sitter group. We here make the choice of a specific simple gauge fixing. We show that our gauge fixing eliminates any infrared divergence in the two-point function for the traceless part of this field. But it is not possible to do the same for the pure trace part (conformal sector). We describe the related Krein space structure and covariant field operators. This work is in the continuation of our previous ones concerning the massless minimally coupled scalar fields and the massive tensor field on de Sitter.
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