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 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
)
.
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