418 Submanifolds and Holonomy
is a homogeneous G-space. Let t
X
be the intersection of all maximal abelian subal-
gebras of g containing X and T
X
the torus in G with Lie algebr a t
X
. Then the isotropy
subgroup of G at X is Z
G
(T
X
), the centralizer of T
X
in G, and therefore
G ·X = G/Z
G
(T
X
).
In particular, if X is a regular element of g, that is, if there is a unique maximal
abelian subalgebra t of g containing X,thenG ·X = G/T ,whereT is the maximal
torus in G with Lie algebra t. Any orbit G ·X of the adjoint representation of G is
called a complex flag manifold or C-space. The latter notion is used mor e frequently
in earlier papers on this topic. In the special case of G = SU
n
one obtains the flag
manifolds of all possible flags in C
n
in this way. In particular, when T is a maximal
torus of SU
n
,thenSU
n
/T is the flag manifold of all full flags in C
n
, that is, of all
possible arrangements {0}⊂V
1
⊂ ... ⊂ V
n−1
⊂ C
n
,whereV
k
is a k-dimensional
complex linear subspace of C
n
.
The importance of complex flag manifolds becomes clear from the following
facts. Each orbit G ·X admits a canonical complex structure, which is also integrable.
If G is simple, there exists a unique (up to homothety) G-invariant K¨ahler-Einstein
metric on G ·X with positive scalar curvature and compatible with the canonical com-
plex structure on G ·X. Moreover, any K¨ahler-Einstein metric on G ·X is homoge-
neous under its own group of isometries and is obtained from a G-invariant K¨ahler-
Einstein metric via some automorphism of the complex structure. Conversely, any
simply connected, compact, homogeneous K¨ahler manifold is isomorphic as a com-
plex homogeneous manifold to some orbit G ·X of the adjoint representation of G,
where G = I
o
(M) and X ∈g. Note that each compact homogeneous K¨ahler manifold
is the Riemannian product of a flat complex torus and a simply connected, compact,
homogeneous K¨ahler manifold.
Real flag manifolds
A real flag manifold is an orbit of an s-representation. Real flag manifolds are
also known as R-spaces, a terminology that is used more f requently in earlier papers
on this topic. Note that the s-representation of a Riemannian symmetric space of
noncompact type is the same as the one of the corresponding dual symmetric space.
Thus, in order to classify and study real flag manifolds, it is sufficient to consider just
one type of symmetric spaces.
Let M = G/K be a semisimple Riemannian symmetric space of noncompact type
with G = I
o
(M), o ∈ M and K the isotropy group of G at o. Note that K is connected
as M is simply connected and G is connected. We consider the corresponding Cartan
decomposition g = k ⊕p of the real semisimple Lie algebra g.Let0= X ∈ p and
K ·X the orbit of K through X via the s-representation. For each k ∈ K we have
k ·X = d
o
k(X)=Ad(k)X and therefore
K ·X = K/K
X
with K
X
= {k ∈K : Ad(k)X = X}.
Let a
X
be the intersection of all maximal abelian subspaces a of p with X ∈a.Wesay
that X is regular if a
X
is a maximal abelian subspace of p, or equivalently, if there