SEMIBOUNDED UNITARY REPRESENTATIONS OF DOUBLE EXTENSIONS OF HILBERT-LOOP GROUPS

A unitary representation 7r of a, possibly infinite dimensional, Lie group G is called semibounded if the corresponding operators idir(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 Io(K) which are double extensions of the twisted loop group Lrp(K), where K is a simple Hilbert Lie group (in the sense that the scalar product on its Lie algebra is invariant) and 0 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 Frechet Lie groups and prove an infinitesimal characterization of analytic operator-valued positive definite functions on Frechet 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.