18

P.

BIELIAVSKY, L. CLAESSENS, D. STERNHEIMER, ANDY. VOGLAIRE

PROOF. The algebra 1 can be written 1

=

JRA EBact JRD EBact

Span{B,

C},

a

split extension, hence a general element reads

r

1

(a,

{3, "(,

8)

=

eAeoD e!3W+"YM.

One

can use Campbell-Baker-Hausdorff formula to split it into a factor in Ro and one

in

R'

(where

f

and

g

are some auxiliary functions):

(4.9)

r1(a, {3,"(,

8)

=

~ehe(f(o)+!3)w(g(o)+"Y)M

ERe surjective on R.'

0

The conclusion is that

R

1

is the group

R

that we were searching for. To

summarize, the structure is as follows.

(i) The AdS space is decomposed into a family of cells: the orbits of a sym-

plectic solvable Lie group

R

as in Proposition 4.2 above. Note that these

cells may be viewed as the symplectic leaves of the Poisson generalized

foliation associated with the left-invariant symplectic structure on

n.

(ii) The open 'k-orbit

M

0

,

endowed with a black hole structure, identifies

with the group manifold

'k.

4.2. Deformation triples for Mo.

4.2.1. Left-invariant Hilbert function algebras on Ro. In this section, we present

a modified version of the oscillatory integral product (2.2) leading to a left-invariant

associative algebra structure on the space of square integrable functions on

R

0 .

THEOREM 4.6. Let u and v be smooth compactly supported functions on R

0

.

Define the following three-point functions:

(4.10)

and

S :=Sv(

cosh(a1- a2)xo, cosh(a2- ao)x1, cosh(ao- a1)x2)

- &

sinh (2(ao- a1))z2;

0,1,2

A:= [cosh (2(a1- a2)) cosh(2(a2- ao)) cosh(2(ao- al))

1

cosh a1 - a2 cosh a2 - ao cosh ao - a1 .

[ ( ) ( ) ( )J

dimR.o-2] 2

Then the formula

(4.11)

(2)

1

1

liS

u

*

0

v :

=

Odim Ro

A e

e

u

0

v

R.o xR.o

extends to

L

2

(R0

)

as a left-invariant associative Hilbert algebra structure. In par-

ticular, one has the strong closedness11 property:

J

u

*~ 2 )

v

=

J

uv .

PROOF. The oscillatory integral product (2.2) may be obtained by intertwining

the Weyl product on the Schwartz spaceS (in the Darboux global coordinates (2.1))

by the following integral operator

[BiMs]:

T

:= ;:- 1 o

(¢01)*

oF,

11The notion of strongly closed star product was introduced in

[CFS]

in the formal context.