QUANTIZATION OF ANTI DE SITTER AND SYMMETRIC SPACES
5
2. Curvature deformations of rank one Hermitian symmetric spaces
and their associated UDF's
2.1. Preliminary set up and reminder. In [BiMs], a formal UDF for the
actions of the Iwasawa component R
0
:=
AN
of SU(1, n) is given in oscillatory
integral form.
It
has been observed in [BBM] that this type of UDF is actually
non-formal for proper actions on topological spaces. The precise framework and
statement are as follows. The group Ro is a one dimensional extension of the Heisen-
berg group No
:=
Hn. Through the natural identification R
0
=
SU(1, n)/U(n)
induced by the Iwasawa decomposition of SU(1, n), the group R
0
is endowed with
a (family of) left-invariant symplectic structure(s) w. Denoting by r0
:=
a0
x n0 its
Lie algebra, the map
(2.1) •o
-----+
Ro : (a, n)
~--+
exp(a) exp(n)
turns out to be a global Darboux chart on (R0 ,w). Setting n0
=
V x RZ with
table [(x,z), (x
1
,z1
)]
=
Ov(x,x1
)
Z, and ro
=
{(a,x,z)
I
,a,z
E
IR;x
E
V}, one has
THEOREM
2.1.
For all non-zero ()
E
IR,
there exists a Prechet function space
&th
Cgo(Ro)
c
&o
c
C
00
(Ro), such that, defining for all u, v
E
Cgo(Ro)
(2.2)
(u
*o
v)(ao, xo, zo)
:=
(Jdi~'R.o
{
cosh(2(al- a2)) [cosh(a2- a0
)
cosh(a0
-
al)
]dim'R.o-2
Jn.o x'R.o
2"
x exp (; { Sv ( cosh(a1 - a2)xo, cosh(a2 - ao)xt, cosh(ao- a1)x2)
- &
sinh(2(ao- a1))z2})
0,1,2
x u(at, Xt, z1) v(a2, x2, z2) da1da2dx1dx2dz1dz2 ;
where Sv(xo, X!, x2)
:=
Ov(xo, Xl)
+
nv(xl, x2)
+
nv(x2, xo) is the phase for the
W eyl product on ego (V) and
&
stands for cyclic summation3
,
one has:
0,1,2
(i) u
*o
v is smooth and the map cgo(Ro)
X
Cgo(Ro)
---+
C
00
(Ro) extends
to an associative product on
&o.
The pair
(&o,*o)
is a (pre-C*) Prechet
algebra.
(ii) In coordinates (a, x, z) the group multiplication law reads
L (
I I I) ( I
-a'
I
-2a'
I
1
n (
I)
-a')
(a,x,z)
a , x , z
=
a+ a , e x
+
x , e z
+
z
+ 2HV
x, x e .
The phase and amplitude occurring in formula {2.2) are both invariant
under the left action L : Ro
x
Ro
---+
Ro.
(iii) Formula (2.2) admits a formal asymptotic expansion of the form:
()
U*()V
rv
UV
+
2i{u,v} +0((}2
) j
where {, } denotes the symplectic Poisson bracket on C
00
(R0
)
associ-
ated with w. The full series yields an associative formal star product on
(Ro,w) denoted by
*o·
3In
(BiMs], the exponent dim Ro - 2 was forgotten in the expression of the amplitude of the
oscillating kernel.
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