Let G be a connected real Lie group and let [gfr ] be
its Lie algebra. We shall denote by [qfr ] ⊂ [gfr ] the radical of [gfr ].
Let [gfr ] = [qfr ] [ltimes ] [sfr ] (where [sfr ] is semisimple or 0) be a Levi
decomposition of [gfr ] (cf. [11]). When [sfr ] ≠ 0
we can apply the Iwasawa decomposition on [sfr ] (cf. [8])
[sfr ] = [nfr ] [oplus ] [afr ] [oplus ] [kfr ], where [nfr ] is nilpotent and [afr ]
is abelian and normalizes [nfr ] so that [nfr ] [oplus ] [afr ] is a soluble algebra.
Since [nfr ] [oplus ] [afr ] normalizes [qfr ] it is clear that
[rfr ] = [qfr ] [oplus ] [nfr ] [oplus ] [afr ] is a soluble Lie algebra of [gfr ].
By Lie's theorem (cf. [11]) we can find a basis on
[rfr ]c = [rfr ] [otimes ] C for which the adjoint
action of [rfr ] on [rfr ]c takes a triangular form. Let us denote by
λ1(x); λ2(x), …,
λn(x), x ∈ [rfr ] the corresponding
eigenvalues. The λj's can
be identified with elements of Homℝ([rfr ], C)
and are called the roots of the adjoint action of [rfr ]. Let us denote by
[Lscr ] = {L1, …, Lk}
the set of the non zero real parts of the λj's.
We shall say that the group G is a B-group if [Lscr ] ≠ &0slash; and if there exist
α1, …, αk [ges ] 0,
[sum ]kj=1 αj = 1, such that
[sum ]kj=1 αjLj = 0. Otherwise we say that G is an
NB-group. It can be shown that the above definition is independent of the particular
choice of the Levi and Iwasawa decompositions that are used (cf. [13]).We shall denote by dlg = dg
(resp. drg) the left (resp. right) Haar measure on
G and by m(g)
= drg/dlg
the modular function.Let [Xscr ] = {X1, X2, …,
Xn} be left invariant fields on G
that verify the Hörmander condition (cf. [15]) and let
Δ = −[sum ]X2j be the corresponding
sub-Laplacian. Δ is formally self adjoint on the Hilbert space
L2(G, drg) and the
spectral gap of Δ is defined byformula here
The Ado theorem for finite Lie conformal algebras with Levi decomposition
