Abstract: Based on the singular cohomology, RO(G)-graded cohomology theory have been investigated from the homotopy theoretic route, and some results for the trivial group action are obtained.We generalize the results to the finite group action, and define the Eilenberg-Mac Lane spectrum.In addition, we build a model of the Eilenberg-Mac Lane spectrum.By using this model one can build the Deligne cohomology theory on real varieities, which generalizes the equivariant cohomology and Cech cohomology.