Letters in Mathematical Physics 13 (1987) 7-15. 9 1987 by D. Reidet Publishing Company. Unitary Positive Energy Representations of the Gauge Group BRUNO TORRESANI ~r Centre de Physique TMorique, CNRS, Luminy, Case 907, Fl3288 Marseille Cedex 9, France (Received: 5 June 1986) Abstract. We investigate the positive energy representations (also called highest weight representations) of the gauge group C oo (T v, Go), Go being a compact simple Lie group, and discuss their unitarity, using the technique of Verma modules constructed from generalized loop algebras (a simple generalization of Kac-Moody affine Lie algebras). We show that the unitarity of the representation imposes severa restrictions on it. In particular, we show, as a part of a more general result, that the gauge group does not admit faithful unitary positive energy representations. O. Introduction The gauge group is defined to be the set of smooth functions from the spacetime M (Riemannian manifold) into the symmetry group Go (compact Lie group), with a pointwise product. In this Letter, we will assume that the local behaviour of the gauge theory is independent of the boundary conditions. We are then allowed to assume periodic boundary conditions and consider G = C~176 ~, Go) as a gauge group. More- over, when we make the radii oft v tend to infinity, the structures we will study possess a limit [6]. Essentially, two types of representations of gauge groups are currently under study: the energy representation [3] and the highest weight representations, which we are interested in (see also [1 ]), also called the positive energy representations. What we are going to do here is to study the positive energy representations from the infinitesimal point of view, i.e., representations of the gauge algebra P (T ~, go), where go is the Lie algebra of Go and P(T ~, go) means the set of functions from T v into go, with finite Fourier series. We will investigate the generalized Verma modules [2], from which any positive energy representation can be obtained, and impose unitarity. We will then get a simple classification; in particular, a very important result is the lack of any faithful representation. The Letter is organized as follows: in Section 1, we recall some basic definitions and some of the results of papers [4] and [5] about the root system of generalized loop algebras. We also give the general construction of the generalized Verma module and of the contravariant Hermitian form. In Section 2, we introduce the nilpotent-type representations and give a necessary and sufficient condition for a positive energy representation to be of the nilpotent type. In Section 3, we recall some results of Kac * Allocataire du MRT.