QUANTIZATION OF ANTI DE SITTER AND SYMMETRIC SPACES 19

F

being the partial Fourier transform with respect to the central variable

z:

:F(u)(a,x,~)

:=

J

e-i~zu(a,x,z)dz;

and

Po

the one parameter family of diffeomorphism(s):

1 1 .

/o(a, x,

~)

=

(a,

0

x, -()

smh(B~)).

cosh( 2 ~)

Set J

:=

I(¢-

1)*Jactl-!

and observe that for all u

E

c=

n

L

2

,

the function

J (¢-

1)*u

belongs to L

2

.

Indeed, one has

J

IJ

(¢-

1

)*ul 2

=

J

14* JI 2 1Jactllul 2

=

J

lul 2

·

Therefore, a standard density argument yields the following isometry:

To:

L2 (R0 )--+ L2 (Ro):

u

~---+

F-

1

o

mJ

o

(¢-

1)*

o :F(u),

where

mJ

denotes the multiplication by J. Observing that To

=

F-1

o

mJ

o :F o

T,

one has

*~

2

)

=

F-1

o

mJ

o

F(*o).

A straightforward computation (similar to the

one in

[Bie02])

then yields the announced formula. 0

REMARK

4.7. Let us point out two facts with respect to the above formulas:

(i) Note the cyclic symmetry of the oscillating three-point kernel

A e:JJ-

8

.

(ii)

The above oscillating integral formula gives rise to a strongly closed,

symmetry invariant, formal star product on the symplectic symmetric

space ( Ro,

w,

s).

PROPOSITION

4.8. The space L

2

(Ro)

00

of smooth vectors in L

2

(R0

)

of the left

regular representation closes as a subalgebra of

(L

2

(R0

),

*~ 2 )).

PROOF.

First, observe that the space of smooth vectors may be described as

the intersection of the spaces {Vn} where Vn+l

:=

(Vnh, with V0

:=

L

2

(R0

)

and

(Vnh is defined as the space of elements a of Vn such that, for all X

E

to, X.a

exists as an element of Vn (we endow it with the projective limit topology).

Let thus a, bE V1. Then, (X.a)

*

b +a* (X.b) belongs to V0

.

Observing that

D

c

V1 and approximating a and b by sequences {an} and { bn} in D, one gets (by

continuity oh): (X.a)*b+a*(X.b)

=

lim(X.an*bn+an*(X.bn))

=

limX.(an*bn)

=

X.(a

*b).

Hence a* b belongs to V1. One then proceeds by induction. 0

4.2.2. Twisted L

2

-spinors and deformations of the Dirac operator. We now fol-

low in our four dimensional setting the deformation scheme presented in

[BDSR]

in the three-dimensional BTZ context.

At the level of the (topologically trivial) open R-orbit, the spin structure over

Mo

and the associated spinor C 2-bundle - restriction of the spinor bundle on

AdSn - are trivial. The space of (smooth) spinor fields may then be viewed as

S

:=

c=(R0

,

C2

),

on which the (isometry) group

R

0

acts on via the left regular

representation. In this setting, the restriction to

Mo

of the Dirac operator

D

on

AdSn may be written as D

=

L:i

"Yi (Xi

+

ri),

where

• {Xi} denotes an orthonormal basis of to

=

T.9(M

0 )

(w.r.t. the adS-

metric at the base point {} of

M

0

);

• for X

E

to,

X

denotes the associated left-invariant vector field on R

0

;

F

being the partial Fourier transform with respect to the central variable

z:

:F(u)(a,x,~)

:=

J

e-i~zu(a,x,z)dz;

and

Po

the one parameter family of diffeomorphism(s):

1 1 .

/o(a, x,

~)

=

(a,

0

x, -()

smh(B~)).

cosh( 2 ~)

Set J

:=

I(¢-

1)*Jactl-!

and observe that for all u

E

c=

n

L

2

,

the function

J (¢-

1)*u

belongs to L

2

.

Indeed, one has

J

IJ

(¢-

1

)*ul 2

=

J

14* JI 2 1Jactllul 2

=

J

lul 2

·

Therefore, a standard density argument yields the following isometry:

To:

L2 (R0 )--+ L2 (Ro):

u

~---+

F-

1

o

mJ

o

(¢-

1)*

o :F(u),

where

mJ

denotes the multiplication by J. Observing that To

=

F-1

o

mJ

o :F o

T,

one has

*~

2

)

=

F-1

o

mJ

o

F(*o).

A straightforward computation (similar to the

one in

[Bie02])

then yields the announced formula. 0

REMARK

4.7. Let us point out two facts with respect to the above formulas:

(i) Note the cyclic symmetry of the oscillating three-point kernel

A e:JJ-

8

.

(ii)

The above oscillating integral formula gives rise to a strongly closed,

symmetry invariant, formal star product on the symplectic symmetric

space ( Ro,

w,

s).

PROPOSITION

4.8. The space L

2

(Ro)

00

of smooth vectors in L

2

(R0

)

of the left

regular representation closes as a subalgebra of

(L

2

(R0

),

*~ 2 )).

PROOF.

First, observe that the space of smooth vectors may be described as

the intersection of the spaces {Vn} where Vn+l

:=

(Vnh, with V0

:=

L

2

(R0

)

and

(Vnh is defined as the space of elements a of Vn such that, for all X

E

to, X.a

exists as an element of Vn (we endow it with the projective limit topology).

Let thus a, bE V1. Then, (X.a)

*

b +a* (X.b) belongs to V0

.

Observing that

D

c

V1 and approximating a and b by sequences {an} and { bn} in D, one gets (by

continuity oh): (X.a)*b+a*(X.b)

=

lim(X.an*bn+an*(X.bn))

=

limX.(an*bn)

=

X.(a

*b).

Hence a* b belongs to V1. One then proceeds by induction. 0

4.2.2. Twisted L

2

-spinors and deformations of the Dirac operator. We now fol-

low in our four dimensional setting the deformation scheme presented in

[BDSR]

in the three-dimensional BTZ context.

At the level of the (topologically trivial) open R-orbit, the spin structure over

Mo

and the associated spinor C 2-bundle - restriction of the spinor bundle on

AdSn - are trivial. The space of (smooth) spinor fields may then be viewed as

S

:=

c=(R0

,

C2

),

on which the (isometry) group

R

0

acts on via the left regular

representation. In this setting, the restriction to

Mo

of the Dirac operator

D

on

AdSn may be written as D

=

L:i

"Yi (Xi

+

ri),

where

• {Xi} denotes an orthonormal basis of to

=

T.9(M

0 )

(w.r.t. the adS-

metric at the base point {} of

M

0

);

• for X

E

to,

X

denotes the associated left-invariant vector field on R

0

;