On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

Neeb, Karl-H.

Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 1839-1874.

Soit $G$ un groupe de Lie–Banach connexe et simplement connexe. Sur l’algèbre enveloppante complexe de son algèbre de Lie $𝔤$ 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 $\lambda \left(D\right)=\langle \mathtt{d}\pi \left(D\right)v,v\rangle$ 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 ${\pi }^{v,v}\left(g\right)=\langle \pi \left(g\right)v,v\rangle$ 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.

Let $G$ be a connected and simply connected Banach–Lie group. On the complex enveloping algebra of its Lie algebra $𝔤$ 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 $\lambda \left(D\right)=\langle \mathtt{d}\pi \left(D\right)v,v\rangle$ 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 ${\pi }^{v,v}\left(g\right)=\langle \pi \left(g\right)v,v\rangle$ 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.

Classification : 22E65, 22E45
Keywords: Infinite dimensional Lie group, unitary representation, positive definite function, analytic vector, integrability of Lie algebra representations.
Mot 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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