Generalized Positive Energy Representations of Groups of Jets

Let $V$ be a finite-dimensional real vector space and $K$ a compact simple Lie group with Lie algebra $\mathfrak{k}$. Consider the Fréchet-Lie group $G:=J_0^{\mathrm{\infty }}(V;K)$ of $\mathrm{\infty }$-jets at $0\in V$ of smooth maps $V\to K$, with Lie algebra $\mathfrak{g}=J_0^{\mathrm{\infty }}(V;\mathfrak{k})$. Let $P$ be a Lie group and write $\mathfrak{p}:=\text{Lie}(P)$. Let $\alpha$ be a smooth $P$-action on $G$. We study smooth projective unitary representations $\bar{\rho }$ of $G{⋊}_{\alpha }P$ 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 $\bar{\rho }(G)$. We show that this condition imposes severe restrictions on the derived representation $d\bar{\rho }$ of $\mathfrak{g}⋊\mathfrak{p}$, leading in particular to sufficient conditions for $\bar{\rho }{\mid }_G$ to factor through $J_0^2(V;K)$, or even through $K$.