Unitary positive energy representations of the gauge group (Q1089114)

Let G be a group of smooth functions from a Riemannian manifold (the space-time) M into the compact Lie group K, and \({\mathfrak g}\) its Lie algebra. The author constructs the generalized Verma modules of \({\mathfrak g}\). The so-called elementary and semi-elementary representations of \({\mathfrak g}\) are introduced by means of these modules. Let \(\omega\) be the involutive antilinear antiautomorphism of \({\mathfrak g}\). A representation \(\pi\) of \({\mathfrak g}\) is said to be unitarizable with respect to \(\omega\) if there exists a positive definite contravariant Hermitian form H, such that \(H(\pi (g)u,v)=H(u,\pi (\omega g)v)\) for every \(g\in {\mathfrak g}\). It is proved that the only unitarizable representations of \({\mathfrak g}\) with highest weight are either elementary or semi-elementary ones. The algebra \({\mathfrak g}\) does not admit faithful unitarizable representations with highest weight.