  We report results of our infrared and optical spectroscopy study of a
half-metallic ferromagnet Mn$_5$Ge$_3$. This compound is currently being
investigated as a potential injector of spin polarized currents into germanium.
Infrared measurements have been performed over a broad frequency (50 - 50000
cm$^{-1}$) and temperature (10 - 300 K) range. From the complex optical
conductivity $\sigma(\omega)$ we extract the electron self-energy
$\Sigma(\omega)$. The calculation of $\Sigma(\omega)$ is based on novel
numerical algorithms for solution of systems of non-linear equations. The
obtained self-energy provides a new insight into electron correlations in
Mn$_5$Ge$_3$. In particular, it reveals that charge carriers may be coupled to
bosonic modes, possibly of magnetic origin.
