  We consider compact homogeneous spaces G/H, where G is a compact connected
Lie group and H is its closed connected subgroup of maximal rank. The aim of
this paper is to provide an effective computation of the universal toric genus
for the complex, almost complex and stable complex structures which are
invariant under the canonical left action of the maximal torus T^k on G/H. As
it is known, on G/H we may have many such structures and the computations of
their toric genus in terms of fixed points for the same torus action give the
constraints on possible collections of weights for the corresponding
representations of T^k in the tangent spaces at the fixed points, as well as on
the signs at these points. In that context, the effectiveness is also
approached due to an explicit description of the relations between the weights
and signs for an arbitrary couple of such structures. Special attention is
devoted to the structures which are invariant under the canonical action of the
group G. Using classical results, we obtain an explicit description of the
weights and signs in this case. We consequently obtain an expression for the
cobordism classes of such structures in terms of coefficients of the formal
group law in cobordisms, as well as in terms of Chern numbers in cohomology.
These computations require no information on the cohomology ring of the
manifold G/H, but, on their own, give important relations in this ring. As an
application we provide an explicit formula for the cobordism classes and
characteristic numbers of the flag manifolds U(n)/T^n, Grassmann manifolds
G_{n,k}=U(n)/(U(k)\times U(n-k)) and some particular interesting examples.
