Notations and conventions

Given a Lie group G, with Lie algebra g, we denote by dG(g) a left invariant

Haar measure. In the non-unimodular case, we consider the modular function ΔG,

defined by the relation:

dG(g)ΔG(g) :=

dG(g−1)

.

Unless otherwise specified,

Lp(G),

p ∈ [1, ∞], will always denote the Lebesgue space

associated with the choice of a left-invariant Haar measure made above. We also

denote by D(G) the space of smooth compactly supported functions on G and by

D

(G) the dual space of distributions.

We use the notations L and R , for the left and right regular actions:

Lgf(g ) :=

f(g−1g

) , Rgf(g ) := f(g g) .

By X and X, we mean the left-invariant and right-invariant vector fields on G

associated to the elements X and −X of g:

X :=

d

dt

t=0

RetX , X :=

d

dt

t=0

LetX .

Given a element X of the universal enveloping algebra U(g) of g, we adopt the same

notations X and X for the associated left- and right-invariant differential operators

on G. More generally, if α is an action of G on a topological vector space E, we

consider the infinitesimal form of the action, given for X ∈ g by:

Xα(a)

:=

d

dt

t=0

αetX (a) , a ∈

E∞

,

and extended to the whole universal enveloping algebra U(g), by declaring that the

map U(b) → End(A∞), X → Xα is an algebra homomorphism. Here, E∞ denotes

the set of smooth vectors for the action:

E∞

:= a ∈ E : [g → αg(a)] ∈

C∞(G,

E) .

Let ΔU be the ordinary co-product of U(g). We make use of the Sweedler

notation:

ΔU (X) =

(X)

X(1) ⊗ X(2) ∈ U(g) ⊗ U(g) , X ∈ U(g) ,

and accordingly, for f1,f2 ∈

C∞(G)

and X ∈ U(g), we write

(2) X(f1f2) =

(X)

(

X(1) f1

)(

X(2) f2

)

, X(f1f2) =

(X)

(

X(1) f1

)(

X(2) f2

)

.

7