  We study the capabilities of the Fermi-LAT instrument for identifying
particle Dark Matter properties as mass, annihilation cross section and
annihilation channels, with gamma-ray observations from the Galactic Center.
For the potential Dark Matter signal, besides the prompt gamma-ray flux
produced in Dark Matter annihilations, we also take into account the flux
produced by inverse Compton scattering of the electrons and positrons generated
in Dark Matter annihilations off the interstellar photon background. We show
that the addition of this contribution is crucial in the case of annihilations
into e^+ e^- and mu^+ mu^- pairs. In addition to the diffuse galactic and
extragalactic background, we also consider the full catalog of high-energy
gamma-ray point sources detected by Fermi. The impact of the degeneracies
between the different Dark Matter annihilation channels has been studied. We
find that for Dark Matter masses below ~200 GeV and for typical thermal
annihilation cross sections, it will be possible to obtain stringent bounds on
the Dark Matter properties.
