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 by

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