Let V be a finite-dimensional real vector space and K a compact simple Lie group with Lie algebra ř. Consider the Fréchet–Lie group G:= J0°(V; K) of i-jets at 0 e V of smooth maps V ! K, with Lie algebra g = 70° (V ; ř). Let P be a Lie group and write p:= Lie(P). Let a be a smooth P-action on G. We study smooth projective unitary representations p of G xœ P that satisfy a so-called generalized positive energy condition. In particular, this class captures representations that are in a suitable sense compatible with a KMS state on the von Neumann algebra generated by -ρ(G'). We show that this condition imposes severe restrictions on the derived representation dp of g ì p, leading in particular to sufficient conditions for X|g to factor through J0¡(V; K), or even through K.@en

In this dissertation, we study (projective) unitary representations of possibly infinite dimensional locally convex Lie groups, in the sense of Bastiani, that either satisfy a positive energy condition, or a KMS(Kubo-Martin-Schwinger) condition. Both of these are motivated by physics. The main purpose of this thesis is to gain general understanding for these classes of representations, and more specifically to develop general tools by which they can be studied in systematic fashion. These tools are consequently applied to specific cases of interest, demonstrating that these conditions are typically extremely restrictive, that the classification of these classes of representations is feasible in various cases, and that these tools can be effectively applied towards achieving such a classification....@en