  We calculate the magneto-optical conductivity and electronic density of
states for silicene, the silicon equivalent of graphene, and similar crystals
such as germanene. In the presence of a perpendicular magnetic field and
electric field gating, we note that four spin- and valley-polarized levels can
be seen in the density of states and transitions between these levels lead to
similarly polarized absorption lines in the longitudinal, transverse Hall, and
circularly polarized dynamic conductivity. While previous spin and
valley-polarization predicted for the conductivity is only present in the
response to circularly polarized light, we show that distinct spin- and
valley-polarization can also be seen in the longitudinal magneto-optical
conductivity at experimentally attainable energies. The frequency of the
absorption lines may be tuned by the electric and magnetic field to onset in a
range varying from THz to the infrared. This potential to isolate charge
carriers of definite spin and valley label may make silicene a promising
candidate for spin- and valleytronic devices.
