1. INTRODUCTION 3
2.31, which turns out to apply in our situation as a direct consequence of Proposi-
tion 2.29, the main technical result of this chapter.
In chapter 3, we consider an arbitrary normal j-group B (i.e. a connected sim-
ply connected Lie group whose Lie algebra is a normal j-algebra—see Definition
3.1). The main result of this chapter, Theorem 3.35, shows that its square B × B
canonically underlies an admissible tempered pair (B × B,Scan). B When elementary,
every normal j-group has a canonical simply transitive action on a specific solv-
able symplectic symmetric space. The two-point function Scan B we consider here
comes from an earlier work of one of us. It consists of the sum of the phases
ScanjS
of the oscillatory kernels associated to invariant star-products on solvable sym-
plectic symmetric space [3, 8], in the Pyatetskii-Shapiro decomposition [22] of a
normal j-group B into a sequence of split extensions of elementary normal j-factors:
B = (... ((S1 S2) S3) . . . ) SN . The two-point phase function Scan
S
in the
elementary case, then consists of the symplectic area of the unique geodesic triangle
in S (viewed as a solvable symplectic symmetric space), whose geodesic edges admit
e, x and y as midpoints (e denotes the unit element of the group S):
Scan(x1,x2)
S
:= Area
(
ΦS
−1(e,
x1,x2)
)
,
with
ΦS :
S3

S3
, (x1,x2,x3)
(
mid(x1,x2), mid(x2,x3), mid(x3,x1)
)
,
where mid(x, y) denotes the geodesic midpoint between x and y in S (again uniquely
defined in our situation).
In chapter 4, we consider an arbitrary normal j-group B, and define the above-
mentioned oscillatory kernels simply by tensorizing oscillating kernels found in
[8] on elementary j-factors. The resulting kernel has the form
=
θ− dim B
mcan
B
exp
i
θ
Scan
B
,
where Scan
B
is the two-point phase mentioned in the description of chapter 2 above,
and mcan
B
= mcan
S1
⊗· · ·⊗mcan,
SN
where mcan
Sj
=
Jac1/21
ΦSj−
denotes the square root of the
Jacobian of the “medial triangle” map ΦS
−1.
In particular, it defines an oscillatory
integral on every symbol space of the type
Bμ(B
× B, Bν(B, E)). When valued in
a Fr´ echet algebra A, this yields a non-perturbative and associative star-product
θ
on the union of all symbol spaces
Bμ(B,
A).
In chapter 5, we consider any tempered action of a normal j-group B on a
Fr´ echet algebra A. By tempered action we mean a strongly continuous action
α of B by automorphisms on A, such that for every semi-norm .
j
there is a
weight (“tempered” for a suitable notion of temperedness) μj
α
such that αg(a)
j

μj α(g) a
j
for all a A and g B. In that case, the space of smooth vectors A∞
for α naturally identifies with a subspace of
Bˆ(B, μ
A∞) (where ˆ μ is affiliated to
μα
= {μj
α}j∈N)
through the injective map:
α :
A∞

Bˆ(B, μ A∞)
, a [g αg(a)] .
We stress that even in the case of an isometric action, and contrarily to the Abelian
situation, the map α always takes values in a symbol space
Bμ,
for a non-trivial
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