Semibounded Unitary Representations of Double Extensions of Hilbert--Loop Groups

A unitary representation of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\dd\pi(x)$ 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 $\hat\cL_\phi(K)$ which are double extensions of the twisted loop group $\cL_\phi(K)$, 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 $\Z$-grading. To achieve this goal, we extend the method of holomorphic induction to certain classes of Fr\'echet-Lie groups and prove an infinitesimal characterization of analytic operator-valued positive definite functions on Fr\'echet--BCH--Lie groups.