Is classification of compact isotropy irreducible homogeneous Kaehler(-Einstein) manifolds known? Here, a homogeneous space is called isotropy irreducible if the isotropy representation is irreducible.
I have browsed the book "Einstein Manifolds" by Besse, and the paper "On isotropy irreducible Riemannian manifolds" by Wang and Ziller, but I cannot find the classification under the Kaehlerian condition.
Theorem (Borel, Remmert): Let $X$ be a connected, compact, homogeneous Kaehler manifold. Then $X$ is biholomorphic to the product $Y\times A_X$ where $Y$ is a flag manifold and $A_X$ is the Albanese torus of $X$. This gives you the complex structure.
By the Bochner trick, any Einstein metric on $X$ is invariant under translation of the torus. I don't know how else the Einstein condition affects the story. I would guess that the Kaehler--Einstein metric and isotropy irreducibility would force the torus factor to be trivial and the flag variety to be a compact Hermitian symmetric space.
According to Wang, M., Ziller, W. (1986), "Einstein metrics with positive scalar curvature", in: Shiohama, K., Sakai, T., Sunada, T. (eds) "Curvature and Topology of Riemannian Manifolds", Lecture Notes in Mathematics, vol. 1201, Springer, Berlin, Heidelberg, every homogeneous Kaehler-Einstein metric that is isotropy irreducible is Hermitian symmetric.
This fact is due to A. Lichnerowicz, "Variétés pseudokählériennes à courbure de Ricci non nulle; application aux domaines bornés homogènes de $C^r$", C. R. Acad. Sci. Paris 235 (1952), 12–14 (in French).
Hence every compact isotropy irreducible homogeneous Kaehler--Einstein manifold is an irreducible Hermitian symmetric space of compact type.
