  With a treatment of the 4$f$ states of EuB$_6$ based on LDA+U method, the
mixing of Eu $f$ states with B $p$ states around the X point of the Brillouin
zone is shown to have unexpected consequences for the effective exchange
interactions. We analyze in detail the orbital character of electronic states
close to the Fermi level and discuss the effective exchange between the
itinerant electrons and the local $4f$ moments. The analysis suggests that the
ordered phase may provide the first example of a {\it half metallic semimetal},
and that the physics of EuB$_6$ should be described in terms of a two band
Kondo lattice model with parallel (ferromagnetic) coupling of the conduction
electrons and antiparallel (antiferromagnetic) coupling of the valence
electrons to the local $4f$ moments.
