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

2012/05/23 by Neeb, Karl-Hermann
#22E45 #22E65 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1205.5201

Abstract

A unitary representation of a, possibly infinite dimensional, Lie group G is called semibounded if the corresponding operators i\ddπ(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 \cL_ϕ(K) which are double extensions of the twisted loop group \cL_ϕ(K), where K is a simple Hilbert--Lie group (in the sense that the scalar product on its Lie algebra is invariant) and ϕ 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 Fréchet-Lie groups and prove an infinitesimal characterization of analytic operator-valued positive definite functions on Fréchet--BCH--Lie groups.

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