One of the most well known classes of K\"ahler-Einstein manifolds, i.e. manifolds
which carry a Kahler metric $g$ such that $Ric_{g}= \lambda \cdot g$ $c\in\mathbb{R}$, are the **generalized flag manifolds**
$$G^{\mathbb{C}}/P\cong G/K$$
 of a compact connected simple Lie group.
Here $P$ is a parabolic subgroup of the complexification $G^{\mathbb{C}}$ of $G$, and $K=p\cap G$ is the centralizer of a torus $S\subset G$, i.e. $K=C(S)$. If  $S=T=$maximal torus, then we obtain a full flag manifold $G/T$.

In the family of generalized flag manifolds, we find a very important subfamily of K\"ahler-Einstein manifolds, the **(isotropy irreducible) Hermitian symmetric spaces** $M=G/K$ of compact type
(i.e. compact symmetric spaces endowed with a Hermitian structure invariant under the  symmetries. In particular, this Hermitian structire is Kahler) .  It well know that such a space $M=G/K$ admits a unique (as isotropy irreducible) K\"ahler-Einstein metric. We mention here two facts for  isotropy irreducible Hermitian symmetric spaces $M=G/K$:

1) The isotropy subgroup $K$ has an 1-dimensional center. 

2) They are the only generalized flag manifolds which are the same time symmetric spaces.

Now, a (generalized) flag manifold is also a homogenepous K\"ahler manifold (the K\"ahler structure on these manifolds arising form the fact that any flag manifold is an adjoint orbit of an element in the Lie algebra of $G$).   In particular, flag manifolds exhaust all compact   simply connected Homogeneous Kahler manifolds corresponding to (compact, connected) simple Lie groups.  Their classification is based on the  painted Dynkin diagrams.  

Now, any  $M=G^{\mathbb{C}}/P=G/C(S)$ admits a finite number  of invariant complex structures. More-ever for any such complex structure we can define (a unique)  homogeneous K\"ahler--Einstein metric, which given in terms of representation theory by computing the Koszul form $2\delta_{\frak{m}}$.  Thus a flag manifolds admits a finite number of K\"ahler-Einstein metrics, Note that if some of the invariant complex structures are equivalent, then, the K\"ahler-Einstein metrics corresponding to these complex structures would be isometric. 

More information about the geometry of flag manifolds, painted Dynkin diagrams,  K\"ahler-Einstein metrics, etc, can be found  in the following articles:


 D. V. Alekseevsky:  **Flag manifolds**,
in Sbornik Radova, 11th Jugoslav. Geom. Seminar. Beograd 6 (14) (1997) 3--35.

 
  D. V. Alekseevsky and A. M. Perelomov: 
 **Invariant K\"ahler-Einstein metrics on compact homogeneous spaces**, 
Funct. Anal. Appl. 20 (3)  (1986)  171--182. 

Y. Sakane: 
  **Homogeneous Einstein metrics on flag manifolds**,
 Lobachevskii J. Math.   (4) (1999)  71--87.
 
  
 M. Kimura: 
 **Homogeneous Einstein metrics on certain K\"ahler C-spaces**,
  Adv. Stud. Pure Math.  18-I  (1990)  303--320. 

 M. Bordeman, M. Forger and H. R\"omer: 
  **Homogeneous K\"ahler manifolds: paving the way towards new supersymmetric sigma models**, 
 Comm. Math. Phys. 102 (1986)  604--647.
  
   
  A. Borel and F. Hirzebruch: 
 **Characteristics classes and homogeneous spaces I**,
  Amer. J. Math. 80  (1958)  458--538.
  
 A. Arvanitoyeorgos and I. Chrysikos: 
  **Invariant Einstein metrics on generalized flag manifolds with two isotropy summands**, 
 J. Aust. Math. Soc. 90 (2) (2011) 237--251.
 
  
 A. Arvanitoyeorgos and I. Chrysikos: 
  **Invariant Einstein metrics on generalized flag manifolds with four isotropy summands**, 
Ann. Glob. Anal. Geom. 37 (2) (2010) 185-219. 
 
   
  
   M. Nishiyama: 
 **Classification of invariant complex structures on irreducible compact simply connected coset spaces**,
  Osaka J. Math. 21 (1984) 39--58.