  The next-to-leading order contribution $\d\P\omn (\o ,\vc q )$ to the
polarization function of the hot gluon system is analysed at non-zero wave
vectors $\vc q \,$. Using Braaten-Pisarski resummation and general covariant
gauges, $\d\P\omn$ is found to be gauge-fixing independent and transverse on
the longitudinal mass-shell. The real part of the longitudinal component
$\d\P_\ell$ is UV and IR stable (for real $q$). At imaginary $q$ it is IR
singular, and at the point $\o=0$, $q^2=-3m^2$ it coincides with the result of
Rebhan for next-to-leading order Debye screening. When $q$ approaches the
lightcone, $\d\P_\ell $ diverges like $1 / \wu {\o^2 - q^2} $, reflecting the
breakdown of the Braaten-Pisarski decomposition scheme in this limit.
