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.