QUANTIZATION OF ANTI DE SITTER AND SYMMETRIC SPACES

15

particular, every open orbit Mo of R in M containing Mphys is itself endowed

with a black hole structure.

Recall that if J denotes an element of Z(K) whose associated conjugation coin-

cides with the Cartan involution () then the R-orbit

Mo

in

9

/H.

of an element

uH.

with u

2

=

J is open and contains

Mphys,

see

[CD07].

Remark that the extension

lemma 3.8 yields an oscillatory integral UDF for proper actions of

R.

But here

the situation is simplified by the following observation - for convenience of the

presentation we write it below for

n

=

4 but the results are valid for any

n

2

3.

PROPOSITION 4.2.

The R-homogeneous space Mo admits a unique structure

of globally group type symplectic symmetric space. The latter is isomorphic to

(R0

,

w,

s) described in section 2.

For the proof, recall first (see

[CD07])

that the solvable part of the Iwasawa

decomposition of

.so(2, 3)

may be realized with as nilpotent part

n

=

{W, V, M,

L}

and Abelian

a=

{J1

,

h}

with the commutator table

[V, W] = M, [V, L] = 2W,

[J1, W] =

W,

[J2, V] =

V,

[J1,L] =

L,

[J2,L] =

-L,

[J1,M] =

M,

[J2,M] =

M.

Notice that W, J1

E

~'

and

h

E

q.

This decomposition is related to the one given

in (3.4) by

No =L

N1

=V

H1

=

J1

+

J2

H2

=

J1- J2.

We choose to study the orbit of the element {)

=

uH.

with

We denote by

Re

its stabilizer group in

R,

byte the Lie algebra of

Re;

t' is the

subalgebra of t generated by elements of t minus the generator of te and

R'

is the

analytic subgroup of

n

whose algebra is t'.

LEMMA 4.3.

The action ofR' on

U

is simply transitive,

i.e.

R'uH.

=

RuH..

PROOF. The first step is to prove that

Re

is connected and

U

simply connected

in order to prevent any double covering problem. The stabilizer of

uH.

is

(4.1) Re = {r En IT. uH. = uH.} ={TEn I Cu-l(T) E H.}.

Since

n

is an exponential group, we have te

=

{X

E

n

I

Ad(u- 1

)X E

~}

with

Re

=

exp te. The set te being connected,

Re

is connected too. A long exact

sequence argument using the fibration

Re---t

n

---t

u

shows that

H

0

(Re)

~

H

1

(U),

which proves that U is simply connected.

As an algebra, tis a split extension t

=

te

EBact

t'. Hence, as group,

R

=

ReR',

or equivalently

R

=

R'Re.

This proves that the action is transitive. The action is

even simply transitive because U is simply connected. D

Let us now find the algebra t

6

.

The Cartan involution

X

f---+

-Xt

is imple-

mented as

C

J

with

J

= (

-I2x2 I3x3) .

Using the relations u

2

=

J and

a(u)

=

u-I,

one sees that

Cu-l(T)

E

H.

if and only

if

a( Cu-lT) =Cu-lT.

This condition is equivalent to

()a(T)

=

T.

The involution

a