12 P. BIELIAVSKY, L. CLAESSENS, D. STERNHEIMER, ANDY. VOGLAIRE
g · eS
C
8
(r)S
=
I(aA
+
eaxv
+
zZ)S, where
Cs
denotes the action by con-
jugation, C
8
(g)
=
sgs-
1
.
The map
¢ :
aA'
+
v
+
zZ
f--+
aA
+
ea.Xv
+
zZ is a
diffeomorphism from t' to t
0
,
and the corresponding map
ci
=
I
o ¢ o
I'-\
or
sr
f--+
C8
(r), is a diffeomorphism from
R'
to
R
0
.
We now observe
PROPOSITION
3.9.
(i)
For g
=
sr
E
'R', we have
cioL9
=
LP(g) oC8
oci.
(ii) The kernel K on R
0
is invariant under the conjugations by S.
(iii) Denoting by
w
and
w'
the left-invariant 2-forms with value -oZ* at the
identity on
R
0
and
R'
respectively, we have
ci*
w
=
w'.
PROOF.
(i) For all g
=
sr, g'
=
s'r'
E
R',
we have
ci o
L
9
(g')
=
ci(
ss'
C8
1-1
(r
)r')
=
Css' ( C8
1-1
(r
)r')
=
LP(g)
o
Cs
o ci(g').
(ii) Recall that for s
=
ea'x,
C
8
(I(aA
+
v
+
zZ))
=
I(aA
+
ea'Xv
+
zZ)
and that
X
E
Sp(V, Ov ). Therefore in the kernel
(2.2)
the amplitude is
independent of
v,
and in its phase, the function Sv is invariant under the
symplectomorphisms of
V.
The kernel as a whole is thus also invariant
under Sp(V, Ov).
(iii) By left-invariance, the condition is
We o (LP(g)-1 o ci o L
9
Le
=
w~. Using
the first property of
ci
above and the invariance of
We
under Sp(V, Ov ),
we get
We
0
(LP(g)-1
0
ci
0
Lg)*e =We
0
(Cs
0
cite =We
0
¢*0'
which is
readily seen to be equal to
w~.
Defining a kernel
K': R'
X
R'
X
R'---
conn'
by
K'
=
cp* K,
we now have
PROPOSITION
3.10. The kernel
K'
is
(i)
invariant under the diagonal left action of'R',
(ii)
associative.
D
Together with the functional space
ci*
£o, it thus defines a
WKB
quantization of
R'.
A quantization of these groups can be obtained by the same method as in
[BiMs].
So let us first quickly review that method. On a connected simply con-
nected symplectic solvable Lie group
(R, w),
with
w
a left-invariant exact symplectic
form (so that the action by left translations is strongly Hamiltonian), one chooses a
global Darboux chart I : t---
R
for which the Moyal star product is covariant, i.e.
if for X
E
\7,
.x
E
coo (
t) denotes the (dual) moment map in these coordinates,
and
*r
the Moyal product on t, one has
[.x,
Ay]*r
=
2B{Ax,
Ay }. Such charts
always exist on these groups (see
[Puk90], [AC90]).
Covariance of the Moyal product implies that po : t --- End(
coo (
t)
[[B]]) :
X
f--+
[.x, ·]*M
is a representation oft by derivations of the algebra (
C00
(t)
[[B]],
*r).
In
e
order to find an invariant product on t, one then tries to find an invertible operator
To which intertwines this action and that by fundamental vector fields, i.e. such
that T0
-
1 op0 (X)oT0 =X*. Those found up to now were all integral operators of the
type F-
1
o
¢0oF,
where
F
is a partial Fourier transform and
¢
0
a diffeomorphism
(see the proof of Theorem 4.6 for the precise form of the one in
[BiMs]).
An
invariant product *o on tis then defined by u *o v
=
T0-
1
(Tou *'f Tov), where *'f
is the Weyl product on t. Modulo some work on the function spaces, this gives
a WKB invariant quantization of
n.
In our case, choosing as Darboux chart the
map I(aA+v+zZ)
=
exp(aA) exp(e-axv+zZ), one checks that the same integral
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