Mod p equivariant cohomology of homogeneous spaces

Let (G,K) be a compact Lie pair. Then K acts on the homogeneous space G/K by left translation. In [3], Shiga has studied the condition on the pair (G,K) that the fibration G/K → EK×KG/K → BK is totally non-cohomologous to zero with respect to the real field from the viewpoint of the equivariant de Rham cohomology and its model. We generalize the main theorem in [3] on the ordinary cohomology theory with coefficients in any field.