  We investigate the electronic dispersion and transport properties of
graphene/WSe$_{2}$ heterostructures in the presence of a proximity-induced
spin-orbit coupling $\lambda_{v}$, sublattice potential $\Delta$, and an
off-resonant circularly polarized light of frequency $\Omega$ that renormalizes
$\Delta$ to $\bar{\Delta}_{\eta p} = \Delta +\eta p \Delta_{\Omega} $ with
$\eta$ and $p$ the valley and polarization indices, respectively, and $
\Delta_{\Omega} $ the gap due to the off-resonant circularly polarized light.
Using a low-energy Hamiltonian we find that the interplay between different
perturbation terms leads to inverted spin-orbit coupled bands. At high $\Omega$
we study the band structure and dc transport using the Floquet theory and
linear response formalism, respectively. We find that the inverted band
structure transfers into the direct band one when the off-resonant light is
present. The valley-Hall conductivity behaves as an even function of the Fermi
energy in the presence and absence of this light. At $\Delta_{\Omega}$ =
$\lambda_{v}$ - $\Delta$ a transition occurs from the valley-Hall phase to the
anomalous Hall phase. In addition, the valley-Hall conductivity switches sign
when the polarization of the off-resonant light changes. The valley
polarization vanishes for $\Delta_{\Omega}$ = 0 but it is finite for
$\Delta_{\Omega}$ $\neq$ 0 and reflects the lifting of the valley degeneracy of
the energy levels, for $\Delta_{\Omega} \neq 0$, when the off-resonant light is
present. The corresponding spin polarization, present for $\Delta_{\Omega}$ =
0, increases for $\Delta_{\Omega}$ $\neq$ 0. Further, pure $K$ or $K^{\prime}$
valley polarization is generated when $\Delta_{\Omega}$ changes sign. Also, the
charge Hall conductivity is finite for $\Delta_{\Omega}\neq 0$ and changes sign
when the handedness of the light polarization changes.
