4 STRUCTURE AND CLASSIFICATION OF SYMMETRIC SPACES
THEOREM
2. Let X be á Symmetrie space and G = G(X) its group of displace-
ments. Then
(i) X is flat if and only if G is Ábelian.
(ii) X is semisimple if and only if G is semisimple.
(iii) X is reduetive if and only if G is reduetive.
(iv) X is irreducible if and only if either X has dimension one or g is simple or g
is the direct sum of two isomorphic simple Ideals, g = g
1
8 g
1
, and ó( ×, Õ) =
(Õ, X) for all X, Õ e gx.
In particular, we see that if X is flat, then X is isomorphic to
Ur
X
Js
for some
nonnegative integers r and s, where r 4- s = dim(X) and Ô = {ae e C | \z\ = 1}.
It also follows that if X is reduetive and simply connected, then X is, as a
Symmetrie space, the direct product of
Us
for some s with a simply connected
semisimple Symmetrie space, which again is the product of irreducible semisimple
Symmetrie Spaces.
The canonical 2-form (the Ricci-form) on a semisimple Symmetrie space makes
it in a canonical way into a pseudo-Riemannian Symmetrie space. Also if (G, H)
is a Symmetrie pair with G semisimple, then the corresponding Symmetrie space
X = G/H is semisimple and G acts on X by isometries w.r.t. the canonical
pseudo-Riemannian strueture. However, G need not be equal to the group of
displacements. But if we define Z(H) = {g e G\gx = ÷ Vx e X), then G(X)
= G/Z(H), and one can show that G{X) is the connected component of the
identity of the isometry group of X. By considering X = (G/Z(H))/(H/Z(H))
we could assume that G = G(X). This is not always convenient, but we always
assume, by possibly factoring out the connected component of the identity of
Z(H), that G and G( X) have the same Lie algebra.
Let now X = G/H be a semisimple Symmetrie space. Then one can show that
G has a Cartan-involution è which commutes with ó, and that è is unique up to
//-conjugacy. Let Ê be fixed points of È. Then Ê is a maximal compact
subgroup of G, except when G has infinite center Z. In that case K/Z is maximal
compact in G/Z. Notice, in particular, that if G is compact then è is trivial.
If we consider X_= Ê/Ê Ð Ç as a subsymmetric space of X = G/H, then the
restriction of * to X_ is nondegenerate and negative definite. One should think of
X_ as a maximal compact subsymmetric space containing the basepoint. (All such
are of the form h · X_ for some h e H.) X__ is a reduetive Symmetrie Space in its
own right. Notice that the canonical pseudo-Riemannian strueture on X has
index (r, s) = (dim A^, dim Z_), when X+ is defined below.
One can actually realize X as a fiber bündle over X_: Let g = i) + q = f + i)
be the decompositions of g aecording to eigenvalues + 1 and -1 for ó and 0,
respectively. Since ó and 0 commute, we have the direct sum decomposition
8 = iinf + i)nt) + q n l + qni.
Let t)", the associatedSymmetrie subalgebra to i), be defined as the fixed points of
the involution óè\ i.e.
Õ = ß ) ç f + q ç ñ.
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