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
;
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