Let $M$ be compact $G$-homogeneous manifold, equipped with the equivariant complex structure, when $G$ is a semi-simple algebraic group. The obvious example is every flag manifold. In that case, all non-vanishing cohomology classes lives in $\Omega^{(k,k)}$, for some $k \leq $dim$M$. Does that hold generally?

Sketch of a possible proof: Every cohomology class has an $G$-equivariant representative. Hence, we must just to look for equivariant elements in $ {\cal T}^{(p,q)}(M)$. But by a "rep theory argument", such elements only exists if $p=q$, hence, we have non-vanishing classes only in $\Omega^{(k,k)}$.

Is proof correct? (Does it hold its water!) I would like help for the "fleshing-out" of the "rep theory argument".

A lot of thank you's!

Let $M$ be compact $G$-homogeneous manifold, equipped with the equivariant complex structure, when $G$ is a semi-simple algebraic group. The obvious example is every flag manifold. In that case, all non-vanishing cohomology classes lives in $\Omega^{(k,k)}$, for some $k \leq $dim$M$. Does that hold generally?

Sketch of a possible proof: Every cohomology class has an $G$-equivariant representative. Hence, we must just to look for equivariant elements in $ {\cal T}^{(p,q)}(M)$. But by a "rep theory argument", such elements only exists if $p=q$, hence, we have non-vanishing classes only in $\Omega^{(k,k)}$.

Is proof correct? (Does it hold its water!) I would like help for the "fleshing-out" of the "rep theory argument".

A lot of thank you's!

P.S. Do the cohomology rings of such manifolds have a presentation analogous to Schubert calculus of flag manifolds?