  Spin-dependent tunneling through an indirect bandgap barrier like the
GaAs/AlAs/GaAs heterostructure along [001] direction is studied by the
tight-binding method. The tunneling is characterized by the proportionality of
the Dresselhaus Hamiltonians at $\Gamma $ and $X$ points in the barrier and by
Fano resonances. The present results suggest that large spin polarization can
be obtained for energy windows that exceed significantly the spin splitting. We
also formulate two conditions that are necessary for the existence of energy
windows with large polarization.
