Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups

Neeb, K. H.

Annales de l'Institut Fourier, Tome 64 (2014) no. 5, pp. 1823-1892.

Une représentation unitaire $\pi$ d’un groupe de Lie $G$ est dite semi-borné, si les opérateurs $i\mathtt{d}\pi \left(x\right)$ de la représentation derivée sont semi-bornés uniformément sur une partie ouverte de l’algèbre de Lie $𝔤$ de $G$. Nous déterminons toutes les représentations irréductibles semi-bornées des groupes ${\widehat{ℒ}}_{\phi }\left(K\right)$ qui sont extensions doubles du groupe ${ℒ}_{\phi }\left(K\right)$, où $K$ est un groupe de Lie hilbertien et $\phi$ est une automorphisme de $K$ d’ordre fini qui mène à l’un des $7$ systèmes de racines affines irréductibles localement finis. Pour atteindre cet objectif, nous étendons la méthode d’induction holomorphe aux certaines classes de groupes de Lie-Fréchet.

Il s’agit du premier papier traitant des aspects globaux des groupes de Lie dont l’algèbre de Lie est une algèbre de Kac–Moody à rang infini.

A unitary representation $\pi$ of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\mathtt{d}\pi \left(x\right)$ from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra $𝔤$ of $G$. We classify all irreducible semibounded representations of the groups ${\widehat{ℒ}}_{\phi }\left(K\right)$ which are double extensions of the twisted loop group ${ℒ}_{\phi }\left(K\right)$, where $K$ is a simple Hilbert–Lie group (in the sense that the scalar product on its Lie algebra is invariant) and $\phi$ is a finite order automorphism of $K$ which leads to one of the $7$ irreducible locally affine root systems with their canonical $\mathbb{Z}$-grading. To achieve this goal, we extend the method of holomorphic induction to certain classes of Fréchet–Lie groups and prove an infinitesimal characterization of analytic operator-valued positive definite functions on Fréchet–BCH–Lie groups.

This is the first paper dealing with global aspects of Lie groups whose Lie algebra is an infinite rank analog of an affine Kac–Moody algebra. That positive energy representations are semibounded is a new insight, even for loops in compact Lie groups.

Classification : 22E65, 22E45
Keywords: infinite dimensional Lie group, unitary representation, semibounded representation, Hilbert–Lie algebra, Hilbert–Lie group, Kac–Moody group, loop group, double extension, positive definite function
Mot clés : groupe de Lie de dimension infinie, représentation unitaire, algèbre de Lie hilbertienne, groupe de Lie hilbertien, groupe de Kac–Moody, groupe de lacettes, double-extension, fonction de type positif

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