  Hexagonal LuFeO$_3$ films have been studied using x-ray absorption and
optical spectroscopy. The crystal splittings of Fe$^{3+}$ are extracted as
$E_{e'}-E_{e"}$=0.7 eV and $E_{a_1'}-E_{e'}$=0.9 eV and a 2.0 eV optical band
gap is determined assuming a direct gap. First-principles calculations confirm
the experiments that the relative energies of crystal field splitting states do
follow $E_{a_1'}>E_{e'}>E_{e"}$ with slightly underestimated values and a band
gap of 1.35 eV.
