On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

[Vecteurs analytiques pour les représentations unitaires des groups de Lie à dimension infinie]

Neeb, Karl-H.

Annales de l'Institut Fourier, Tome 61 (2011), p. 1839-1874 / Harvested from Numdam

Résumé

Soit $G$ un groupe de Lie–Banach connexe et simplement connexe. Sur l’algèbre enveloppante complexe de son algèbre de Lie $\mathfrak{g}$ nous définissons la notion de fonctionnelle analytique et montrons que chaque fonctionnelle analytique positive $\lambda $ est integrable au sens où elle est de la forme $\lambda (D) = \langle {\tt d}\pi (D)v, v\rangle $ pour un vecteur analytique $v$ d’une représentation unitaire de $G$. Dans la preuve de ce résultat nous obtenons des critères pour l’integrabilité des $*$-representations des algèbres de Lie en représentations de groupe unitaires.

Pour le coefficient matriciel $\pi ^{v,v}(g) = \langle \pi (g) v, v \rangle $ d’un vecteur $v$ d’une représentation unitaire d’un groupe de Lie–Fréchet analytique $G$ nous montrons que $v$ est un vecteur analytique si et seulement si $\pi ^{v, v}$ est analytique dans un voisinage de l’identité. En combinant ce résultat à ceux sur les fonctionnelles analytiques positives nous obtenons que chaque fonction analytique de type positive locale sur un group de Lie–Fréchet–BCH simplement connexe s’étend en une fonction analytique globale.

Abstract

Let $G$ be a connected and simply connected Banach–Lie group. On the complex enveloping algebra of its Lie algebra $\mathfrak{g}$ we define the concept of an analytic functional and show that every positive analytic functional $\lambda $ is integrable in the sense that it is of the form $\lambda (D) = \langle {\tt d}\pi (D)v, v\rangle $ for an analytic vector $v$ of a unitary representation of $G$. On the way to this result we derive criteria for the integrability of $*$-representations of infinite dimensional Lie algebras of unbounded operators to unitary group representations.

For the matrix coefficient $\pi ^{v,v}(g) = \langle \pi (g)v,v\rangle $ of a vector $v$ in a unitary representation of an analytic Fréchet–Lie group $G$ we show that $v$ is an analytic vector if and only if $\pi ^{v,v}$ is analytic in an identity neighborhood. Combining this insight with the results on positive analytic functionals, we derive that every local positive definite analytic function on a simply connected Fréchet–BCH–Lie group $G$ extends to a global analytic function.

Détail

Publié le : 2011-01-01

MR 2961842 | Zbl 1241.22023 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2660
Classification:  22E65,  22E45
Mots clés: groupe de Lie de dimension infinie, représentation unitaire, fonction de type positif, vecteur analytique, integrabilité d’une représentation d’une algèbre de Lie

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