Transformation de Fourier des distributions de type positif sur un groupe de Lie unimodulaire

Représentations monomiales des groupes de Lie nilpotents

“…On recrit ainsi la formule de Plancherel abstraite pour la representation cylcique monomiale (r, a τ ): THέORέME 1 ( [5], [17] 4 . Soit / = e* relativement a la base duale de g* et prenons ί) = Re 2 θ R^4 dans S(f, g).…”

“…On s'interessera a la formule de Plancherel abstraite, due a Penney [17] et a Bonnet [5], appliquee a la representation cyclique (T, a τ ), oύ a τ e (^~0 0 ) // ' X/ est donne, pour φ dans Jt? τ°°, ce qui entraine que φ est une fonction C 00 sur G (cf.…”

“…[18]), par a τ (φ) = φ(e). Lorsque ΐ) appartient a M(f 9 g), T est irreductible et Γon connalt le resultat du a Howe [12] (1) (τ(φK,« τ > -ί Σ (^(φ)^,^)^(^) (cf [5]). …”

The Plancherel formula for the horocycle spaces and generalizations, II

“…More precisely, given a Lie group G and a closed subgroup H, let us assume that the quasi-regular representation T = X G .H -Ind^ 1 is type I. Then there is a unique direct integral decomposition /•© (1.1) r = / nĴ d(H) [2] The Plancherel formula for the horocycle spaces and generalizations, II 195 Here G(H) denotes the irreducible unitary representation classes of G that are weakly contained in r, a closed subset of the unitary dual G. The Plancherel theory that is derived in the previously cited papers includes a specific analytic formula that provides detailed information not only on the structure of G(H), the multiplicity function n n and the Plancherel measure /x, but also on an intertwining operator that effects the direct integral decomposition. This is done in [9] and [10] in various cases that manifest finite multiplicity (that is, n w < oo, /z-a.e.…”

The Plancherel formula for homogeneous spaces with polynomial spectrum

“…3.2]). When n τ is infinite on a set of positive measure, one must employ the Bonnet form of the Plancherel formula [2] to overcome the difficulties indicated in the opening paragraph of part I.…”