# Generalized positive energy representations of groups of jets

### Milan Niestijl

Delft University of Technology, Netherlands

## Abstract

Let $V$ be a finite-dimensional real vector space and $K$ a compact simple Lie group with Lie algebra $k$. Consider the Fréchet–Lie group $G:=J_{0}(V;K)$ of $∞$-jets at $0∈V$ of smooth maps $V→K$, with Lie algebra $g=J_{0}(V;k)$. Let $P$ be a Lie group and write $p:=Lie(P)$. Let $α$ be a smooth $P$-action on $G$. We study smooth projective unitary representations $ρˉ $ of $G⋊_{α}P$ 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 $ρˉ (G)$. We show that this condition imposes severe restrictions on the derived representation $dρˉ $ of $g⋊p$, leading in particular to sufficient conditions for $ρˉ ∣_{G}$ to factor through $J_{0}(V;K)$, or even through $K$.

## Cite this article

Milan Niestijl, Generalized positive energy representations of groups of jets. Doc. Math. 28 (2023), no. 3, pp. 709–763

DOI 10.4171/DM/920