Let $G$ be a compact Lie group and $a\in\mathfrak{g}^*$ (dual of Lie algebra of Lie group $G$). Then let $\mathcal O_a$ be a coadjoint orbit. Then every co-adjoint orbit is Kähler manifold and also projective variety. How can we compute the Kodaira dimension of co-adjoint orbit as projective variety?

Motivation: The Kodaira dimension of co-adjoint orbits are important, because we can classify these type of projective varieties by Kodaira dimension which is birationally invariant.

In fact I am looking for

$$\kappa(\mathcal O_a)=\limsup_{m\to \infty}\frac{\log\text{dim}H^0(\mathcal O_a, K_{\mathcal O_a}^{\otimes m})}{\log m}$$

nothyperKähler varieties. In fact, they are homogeneous and have positive Ricci curvature. $\endgroup$ – Robert Bryant Oct 29 '14 at 13:56