Kirillov-Lipsman orbit method of a class of Gelfand pairs

Résumé

Abstract Let $G:=K\ltimes N$ be the semidirect product with Lie algebra $\mathfrak{g},$ where $N$ is a connected and simply connected nilpotent Lie group, and $K$ is a subgroup of the automorphism group, $Aut(N),$ of $N.$ We say that the pair $(K,N)$ is a nilpotent Gelfand pair when the set $L_K^1(N)$ of integrable $K$-invariant functions on $N$ forms an abelian algebra under convolution.According to Lipsman, the unitary dual $\widehat{G}$ of $G$ is in one-to-one correspondence with the space of admissible coadjoint orbits $\mathfrak{g}^\ddag/G$ of $G.$ Under some assumptions on the pair $(K,N)$ we will show in this work, that the quotient topology of $\mathfrak{g}^\ddag/G$ could be read from the usual topology of some parameters space $\mathcal{P},$ called Mackey's parameters space (will be defined below in section 3). Furthermore, we show that the Kirillov-Lipsman bijection $$\widehat{G}\simeq\mathfrak{g}^\ddagger/G$$is a homeomorphism for a class of Lie groups associated with the nilpotent Gelfand pairs $(K,N).$

Citer ce document

Accès au document

Texte intégral en lecture en ligne, réservé aux abonnés SPHAERO et aux membres de l'institution. Se connecter

Voir l'article sur le site de la revue

Auteur(s)

Statistiques

Consultations : 1

Téléchargements : 0