  The origin of irradiation and fluorescence of the 6.4 keV bright giant
molecular clouds surrounding Sgr A*, the central supermassive black hole of our
Galaxy, remains enigmatic. Testing the theory of a past active period of Sgr A*
requires X-ray polarimetry. In this paper, we show how modern imaging
polarimeters could revolutionize our understanding of the Galactic Center.
Through Monte Carlo modeling, we produce a 4-8 keV polarization map of the
Galactic Center, focusing on the polarimetric signature produced by Sgr B1, Sgr
B2, G0.11-0.11, Bridge E, Bridge D, Bridge B2, MC2, MC1, Sgr C3, Sgr C2, and
Sgr C1. We estimate the resulting polarization, include polarized flux dilution
by the diffuse plasma emission detected toward the GC, and simulate the
polarization map that modern polarimetric detectors would obtain assuming the
performances of a mission prototype. The eleven reflection nebulae investigated
in this paper present a variety of polarization signatures, ranging from nearly
unpolarized to highly polarized (about 77%) fluxes. A major improvement in our
simulation is the addition of a diffuse, unpolarized plasma emission that
strongly impacts soft X-ray polarized fluxes. The dilution factor is in the
range 50% - 70%, making the observation of the Bridge structure unlikely even
in the context of modern polarimetry. The best targets are the Sgr B and Sgr C
complexes, and the G0.11-0.11 cloud. An exploratory observation of a few
hundred kilo-seconds of the Sgr B complex would allow a significant detection
of the polarization and be sufficient to derive hints on the primary source of
radiation. A more ambitious program (few Ms) of mapping the giant molecular
clouds could then be carried out to probe with great precision the turbulent
history of Sgr A*, and place important constraints on the composition and
three-dimensional position of the surrounding gas.
