QUANTIZATION OF ANTI DE SITTER AND SYMMETRIC SPACES 9

actually lies in the semisimple part .s[(b). For any

x

E

.s[(b), we denote by

x

xs

+

xN, xs, xN

E

.s[(b), its abstract Jordan-Chevalley decomposition. Observe

that, denoting by .s[(b) = .s[+ EB.sL the decomposition in

(±1)-~-

eigenspaces, one

has that

a

C

.s[_. Also, the algebra

aN:= {aN}aEa

is an Abelian subalgebra in .s[_

commuting with

a.

Setting

as:= {as}aEa,

we have

DEFINITION

3.3. A gAs is called weakly nilpotent if Jb(as) C 3b(aN) and7

aN

c

Der(b).

Let be =: EBaEP ba be the weight space decomposition with respect to the

action of

as.

Note that for all

a,

one has

aN.ba

C

b0

.

Moreover, for all

Xa

E

ba

and

as

E

as,

one has

a(as.Xa)

=

a(as)a(Xa)

=

aasa- 1aXa

= ~(as).a(Xa) =

-as.a(Xa)·

Therefore,

-a

E

If and aba = b-a· Note in particular that abo= bo.

DEFINITION

3.4. An involutive Lie algebra (g, a) is called weakly nilpotent

if there exists a sequence of subalgebras { ai}oir of g such that

(i)

a0

is a weakly nilpotent gAs of g with associated supplementary ideal

b(O).

(ii) ai+l is a weakly nilpotent gAs of 3bil ( ai) (0 :::; i :::;

r -

1) where, for

i ~ 1, b(i) denotes the a-stable ideal of

Jb(i-lJ

(ai-d associated with ai.

(iii) 3brl ( ar) is Abelian.

PROPOSITION

3.5.

Assume the iLa (g, a) to be weakly nilpotent. Then there

exists a (complex) subalgebra

.s

of

g

such that the restriction prp

Is :

.s -----. p

is a linear

isomorphism.

PROOF.

Let

a

be a weakly nilpotent gAs of g and set

Va

:= ba EB La for

every

a

E

If. Note that

V0

= b0 and that each subspace

Va

is then a-stable and

one sets

Va

= ta EB Pa for the corresponding eigenspace decomposition. Choose a

partition8 of lf\{0} as lf\{0} =: q+ U

q,-

with the properties that -IJ+ =

q,-

and

that if

a,

(3

E

q+ with

a+

(3

E

If then

a+

(3

E

q+. One has b = EBaEP+

Va

EB bo.

We set b+ := EBaEP+ ba and p+ := prp (b+). It turns out that the restriction map

prplb+ : b+-----. p+ is a linear isomorphism. Indeed for

X

E

be:= b n t and

a E

as

one has

a(a.X)

=

aaaaX

=

~(a).X

=

-a.X;

hence as.be

C

p. Therefore, for

all

X

E

ba n t

a

-=/=-

0, one can find

a

E

as

such that

a.X

=

X

E

p n

£;

thus

ban t = 0 as soon as

a

-=/=-

0, yielding ker(prplb+) = 0. The last condition of

Definition 3.3 implies that

as

acts by derivations on b, hence the usual argument

yields that b+ is a subalgebra normalized by

a

EB b0

.

Moreover the first condition

7The

last condition is automatic when b is Abelian. Observe also that it is satisfied when

p( a)

is contained in a Levi factor of the derivation algebra Der( b).

8Such

a partition can be defined as follows. Let ~be a Cartan subalgebra of sl(b) containing

as and let A E

~*

denote the set of weights of the representation of sl(b) on b. Note that

the restriction map

.

-+

.las

from A to

as

is surjective onto 1. Let

~

=

~IR

Ell

i~JR

be a real

decomposition such that the restriction of the Killing form to

~IR

is positive definite. Every weight

in A is then real valued when restricted to

~IR

[KnaOl].

Now, any choice of a basis of

~IR

defines a

partial ordering on A with the desired properties. To pass to the set of weights cl, consider the

~JR-

components,

a~,

of as viewed as a vector subspace

of~-

The restriction map p: cl--+ .PlaiR is then a

s

bijection. Indeed, for. E ker(p), one has, by ((linearity,

.(a+ia') =.(a) +i.(a') =OVa, a'

E a~.

Hence.= 0 as an element of 1. Therefore, an order on A induces on cl an order having the same

properties.