CHAPTER 1

Introduction

The general idea of deforming a given theory by use of its symmetries goes back

to Drinfel’d. One paradigm being that the data of a Drinfel’d twist based on a bi-

algebra acting on an associative algebra A, produces an associative deformation

of A. In the context of Lie theory, one considers for instance the category of

module-algebras over the universal enveloping algebra U(g) of the Lie algebra g of

a given Lie group G. In that situation, the notion of Drinfel’d twist is in a one

to one correspondence with the one of left-invariant formal star-product

ν

on the

space of formal power series

C∞(G)[[ν]],

see [14]. Disposing of such a twist, every

U(g)-module-algebra A may then be formally deformed into an associative algebra

A[[ν]]. It is important to observe that, within this situation, the symplectic leaf B

through the unit element e of G in the characteristic foliation of the (left-invariant)

Poisson structure directing the star-product ν, always consists of an immersed

Lie subgroup of G. The Lie group B therefore carries a left-invariant symplectic

structure. This stresses the importance of symplectic Lie groups (i.e. connected Lie

groups endowed with invariant symplectic forms) as semi-classical approximations

of Drinfel’d twists attached to Lie algebras.

In the present memoir, we address the question of designing non-formal Drin-

fel’d twists for actions of symplectic Lie groups B that underly negatively curved

K¨ ahlerian Lie groups, i.e. Lie groups that admit a left-invariant K¨ ahlerian structure

of negative curvature. These groups exactly correspond to the normal j-algebras

defined by Pyatetskii-Shapiro in his work on automorphic forms [22]. In particular,

this class of groups contains all Iwasawa factors AN of Hermitian type simple Lie

groups G = KAN.

Roughly speaking, one looks for a smooth one-parameter family of complex

valued smooth two-point functions on the group, {Kθ}θ∈R ⊂ C∞(B × B, C), with

the property that, for every strongly continuous and isometric action α of B on a

C∗-algebra

A, the following formula

(1) a

α

θ

b :=

B×B

Kθ(x, y) αx(a) αy(b) dx dy ,

defines a one-parameter deformation of the

C∗-algebra

A.

The above program was realized by Marc Rieffel in the particular case of the

Abelian Lie group B =

R2n

in [26]. More precisely, Rieffel proved that for any

strongly continuous and isometric action of

R2n

on any Fr´ echet algebra A, the

associated Fr´ echet subalgebra

A∞

of smooth vectors for this action, is deformed

by the rule (1), where the two-point kernel there, consists of the Weyl symbol

composition kernel:

Kθ(x, y) :=

θ−2n

exp

i

θ

ω0(x,

y) ,

1