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 Fr\'echet-Lie group $G := J_0^\infty(V; K)$ of $\infty$-jets at $0 \in V$ of smooth maps $V \to K$, with Lie algebra $\mathfrak{g} = J_0^\infty(V; \mathfrak{k})$. Let $P$ be a Lie group and write $\mathfrak{p} := \textrm{Lie}(P)$. Let $\alpha$ be a smooth $P$-action on $G$. We study smooth projective unitary representations $\bar{\rho}$ of $G \rtimes_\alpha P$ 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 $\mathfrak{g} \rtimes \mathfrak{p}$, leading in particular to sufficient conditions for $\bar{\rho}\big|_{G}$ to factor through $J_0^2(V; K)$, or even through $K$.


Introduction
This paper is concerned with projective representations of groups and Lie algebras of jets. Let K denote a compact simple Lie group with Lie algebra k and let V be a finite-dimensional real vector space. Then we consider the Fréchet-Lie group G := J ∞ 0 (V ; K) with Lie algebra g := J ∞ 0 (V ; k) ∼ = R V * ⊗ k. These consist of ∞-jets at 0 ∈ V of smooth K-and k-valued functions, respectively. We are interested in smooth projective unitary representations of G which satisfy either a so-called positive energy, or a KMS condition.
To describe these, let P be a finite-dimensional Lie group with Lie algebra p. Assume that there is a smooth action α of P on G. A continuous projective unitary representation ρ : G ⋊ α P → PU(H ρ ) is of positive energy at the cone C ⊆ p if for every p ∈ C, there is a strongly continuous homomorphic lift t → e itH of t → ρ(e tp ) whose generator H satisfies Spec(H) ≥ 0. We say that ρ is KMS at p ∈ p if there is a normal state φ on the von Neumann algebra M := ρ(G) ′′ generated by ρ(G) such that φ satisfies the KMS condition for the one-parameter group R → Aut(M), t → Ad(ρ(e tp )). As we shall see, these two seemingly very different classes of representations exhibit similar behavior in certain respects. In particular, they both give rise to so-called generalized positive energy representations, a notion which relaxes the positive energy condition and is introduced in Section 4 below. We study these generalized positive energy representations and thereby also those which satisfy either the positive energy or the KMS condition.
The motivation for looking at the positive energy and KMS representations of the group G ⋊ α P originates in prior work by B. Janssens and K.-H. Neeb, who studied in [JN21] a class of projective unitary representations of the group of compactly supported gauge transformations G := Γ c (M ; Ad(K)) of a principal K-bundle K over M , where Ad(K) denotes the corresponding adjoint bundle. Suppose that the Lie group P acts smoothly on K by automorphisms of the principal bundle K. This induces a smooth action of P on the infinite-dimensional Lie group G. Their main result is: Thus, if there are no fixed points in M for C, then the problem of classifying the projective unitary positive energy representations of G ⋊ P is essentially reduced to the one-dimensional case, which has been extensively studied, see for example [PS86,Was98,Tan11,Kac90,KR87,GW84,TL99]. Moreover, if there are no one-dimensional P -equivariantly embedded submanifolds in M , one is essentially reduced to the case where ρ factors through the germs at the fixed point set Σ ⊆ M of the cone C ⊆ p. In the present paper, we address the setting where fixed points do exist and where ρ actually factors through the germs at a single fixed point.
Thus, let a ∈ M be a fixed point of the P -action on M and let V := T a (M ). If a smooth projective unitary representation ρ of G factors through the germs at a ∈ M , then the continuity of ρ implies that it must further factor through the ∞-jets J ∞ a (Ad(K)) ∼ = J ∞ 0 (V ; K) = G at a ∈ M , as is shown in Section A of the appendix. This brings us to groups of jets and motivates the study of smooth projective unitary representations of G ⋊ α P . Clearly, any smooth projective unitary representation of G ⋊ α P defines one of G ⋊ P via the jet homomorphism j ∞ a : G → J ∞ a (Ad(K)) ∼ = G. In this way, the present paper contributes to the understanding of positive energy and KMS-representations of gauge groups.
In [Sim23], KMS-representations were very recently studied in the context of finite-dimensional Lie groups, leading to complete characterization of such representations that generate a factor of type I. In relation to the unitary representation theory of gauge groups, let us also mention the papers [GGV77, AHK78,PS76] and [Ism76], in which unitary representations of gauge groups C ∞ c (M ; K) are constructed which are nonlocal in the sense that they do not factor through the restriction map C ∞ c (M ; K) → C ∞ c (N ; K) for some proper submanifold N ⊆ M . When dim(M ) ≥ 3, these are irreducible ( [Wal87] and [AHKT81]). They are also considered in [ADGV16] and [AHKM + 93]. Unitary representations of groups of jets have also been considered in [GG68] and [AT94].

Structure of the paper
The paper is divided in two parts. In Part I, we introduce both the (generalized) positive energy and the KMS condition. We start in Section 2 by briefly recalling the relation between continuous projective unitary representations, central extensions and the second Lie algebra cohomology H 2 ct (g, R). We move on to consider positive energy representations in Section 3 and discuss some of their properties. In Section 4 we relax the positive energy condition and introduce the so-called generalized positive energy condition. We show that for a projective unitary representation which of is generalized positive energy, its kernel is related to a particular quadratic form canonically associated to the corresponding class in H 2 ct (g, R). This observation will play a key role in Part II. In Section 5, we briefly recall the modular theory of von Neumann algebras and then proceed to define KMS representations. We consider a number of examples and discuss some of their properties, in particular making the important observation that KMS representations give rise to generalized positive energy ones. To some extent, this unifies the positive energy and KMS conditions, allowing their simultaneous study. We remark also that Part I is formulated in the general context of possibly infinite-dimensional locally convex Lie groups, which is the appropriate context within the larger program that studies the projective unitary representations of gauge groups.
In Part II, we return to the Fréchet-Lie group G := J ∞ a (V ; K) of ∞-jets at 0 ∈ V . After fixing our notation, we discuss in Section 7 a normal-form problem for the p-action on g = J ∞ a (V ; k). Using the general observations made in Part I together with the normal-form results obtained in Section 7, we proceed in Section 8 with the study of (generalized) positive energy representations of the Lie algebra g ⋊ D p, where D : p → der(g) denotes the p-action on g corresponding to α.

Overview of main results
Let us describe the main results of Section 8. We will first need to introduce some notation. We write X I for the Lie algebra of formal vector fields on V vanishing at the origin. The p-action D splits into a horizontal and a vertical part according to D(p) = −L v(p) + ad σ(p) , for some Lie algebra homomorphism v : p → X op I and a linear map σ : p → g satisfying the Maurer-Cartan equation For any p ∈ p, the formal vector field v(p) splits further as v(p) = v l (p) + v ho (p), into its linearization v l (p) and its higher order part v ho (p), which is a formal vector field on V vanishing up to first order at the origin. Let σ 0 (p) be the constant part of the formal power series σ(p). Let Σ p ⊆ C denote the additive subsemigroup of C generated by Spec(v l (p)). Let V C c (p) denote the span in V C of all generalized eigenspaces of v l (p) corresponding to eigenvalues with zero real part. Set V c (p) := V C c (p) ∩ V . If C ⊆ p is a subset, define V c (C) := p∈C V c (p), which we call the 'center subspace of V associated to C', in analogy with the center manifold of a fixed point of a dynamical system. Let V c (C) ⊥ ⊆ V * be its annihilator in V * . If π is a continuous projective unitary representation of g ⋊ D p, let C(π) be the set of all points p ∈ p for which π is of generalized positive energy at p. Write R := R V * := ∞ n=0 P n (V ) for the ring of formal power series on V , where P n (V ) denotes the set of degree-n homogeneous polynomials on V .
The first main result concerns positive energy representations. It states that unless the spectrum of v l (p) and σ 0 (p) happens to intersect non-trivially, any smooth projective unitary representation ρ of G ⋊ α P which is of positive energy at p ∈ p factors through the 2-jets J 2 0 (V ; K) ⋊ α P : Theorem 8.1. Let ρ be a smooth projective unitary representation of G ⋊ α P which is of positive energy at p ∈ p. Assume that Spec(ad σ0(p) ) ∩ Spec(v l (p)) = ∅. Then ρ factors through J 2 0 (V ; K) ⋊ α P . Moreover the image of −L v l (p) + ad σ0(p) in P 2 (V ) ⊗ k ⊆ J 2 0 (V ; K) is contained in ker ρ.
The second main result determines restrictions imposed by the generalized positive energy condition. If p ∈ C(π), then unless possibly when the "non-resonance condition" Spec(ad σ0(p) ) ∩ Σ p = ∅ is violated, it suffices to consider the case where all eigenvalues of v l (p) are purely imaginary. The precise statement is: Theorem 8.3. Let π be a continuous projective unitary representation of g ⋊ D p. Let C ⊆ C(π). Assume that Spec(ad σ0(p) ) ∩ Σ p = ∅ for all p ∈ C. Then RV c (C) ⊥ ⊗ k ⊆ ker π, and hence π| g factors through R V c (C) * ⊗ k.
Since R V c (C) * ⊗ k = k whenever V c (C) = {0}, Theorem 8.3 in particular gives sufficient conditions for π to factor through k, that depend only on the spectrum of σ 0 and v l (p). For the third main result, we consider the special case where p is non-compact and simple: Theorem 8.6. Assume that p is non-compact and simple. Suppose that v l defines a non-trivial irreducible p-representation on V . Let π be a continuous projective unitary representation of g ⋊ D p. Let C ⊆ C(π) be a P -invariant convex cone. Either C is pointed or π| g factors through k.
Remark 1.2. If ρ is a smooth projective unitary representation of G ⋊ α P which is of generalized positive energy at the cone C ⊆ p, then its derived representation dρ on the space of smooth vectors H ∞ ρ is so, too. Moreover, as we shall see in Lemma 6.3 below, the exponential map of G = J ∞ 0 (V ; K) restricts to a diffeomorphism from the pro-nilpotent ideal ker ev 0 : J ∞ 0 (V ; k) → k onto ker ev 0 : J ∞ 0 (V ; K) → K . Thus, the above results all have immediate analogous consequences on the group level.
Definition 2.2. If D is a complex vector space, denote by L(D) the Lie algebra of linear operators on D.
If D is a pre-Hilbert space with Hilbert space completion H, we also write For X ∈ L † (D) set X † := X * | D . Then (X † ) † = X and (−) † endows L † (D) with an involution [Sch90, Ch. 2]. If D is a pre-Hilbert space, define the Lie algebra Definition 2.3. Let D be a complex pre-Hilbert space.
-A unitary representation of the Lie algebra g on D is a Lie algebra homomorphism π : g → u(D). A projective unitary representation is a Lie algebra homomorphism π : g → pu(D) := u(D)/iRI.
We say that ρ lifts ρ. Suppose ρ 1 and ρ 2 are two projective unitary representations, inducing by pull-back the lifts ρ 1 : of ρ 1 and ρ 2 , respectively. Then ρ 1 and ρ 2 are unitarily equivalent if and only if there is an isomorphism Φ : Analogously, any projective unitary g-representation π with domain D can be lifted to a unitary representation π : The continuous central extensions of g by R are up to isomorphism classified by H 2 ct (g, R), the continuous second Lie algebra cohomology with trivial coefficients [JN19, Def. 6.2, Prop. 6.3]. Thus, to study the projective unitary representations of g up to equivalence, one may first determine H 2 ct (g, R), choose for each class [ω] ∈ H 2 ct (g, R) a representative ω and then proceed to determine the equivalence classes of unitary representations π of the central extension R⊕ ω g satisfying π(1, 0) = iI. We will also write c := (1, 0) ∈ R⊕ ω g for the central generator.
Remark 2.5. In the literature, one encounters the notion of the level of a unitary representation π of Rc ⊕ ω g, which is the number l ∈ R such that π(c) = ilI (see e.g. [PS86,sec. 9.3]). Let us briefly clarify how such representations are included in the program outlined above, even though π(c) = iI is always assumed. Simply notice that such a representation of level l factors through the map Rc ⊕ ω g → Rc ⊕ l·ω g induced by multiplication by l on the central factor. The corresponding representation π 2 of Rc ⊕ l·ω g satisfies π 2 (c) = iI. Notice that Rc⊕ ω g → R⊕ l·ω g is an isomorphism of Lie algebras whenever l = 0, but not as central extensions unless l = 1, because a morphism of central extensions is required to be the identity on the central component. For 1 = l ∈ R, the cocycles ω and l · ω are not equivalent unless [ω] = 0 in H 2 ct (g, R). Remark 2.6. If a projective unitary representation ρ of G is smooth, then the corresponding central T-extension

Positive Energy Representations
Let us introduce the class of positive energy representations. After defining the notion, some immediate consequences are considered that will be relevant in Part II. Let G be a regular locally convex Lie group with Lie algebra g. If c ∈ R, D is a pre-Hilbert space and X ∈ L † (D) satisfies X † = X, we write X ≥ c if ψ, Xψ ≥ c ψ 2 for every ψ ∈ D.
Definition 3.1. Let C ⊆ g be a convex cone and D be a pre-Hilbert space.
-A continuous unitary representation π of g on D is said to be of positive energy (p.e.) at ξ ∈ g if −iπ(ξ) ≥ 0. It is of p.e. at C if it is of p.e. at every ξ ∈ C. Write C(π) = { ξ ∈ g : π is of p.e. at ξ }.
-Let π be a continuous projective unitary representation of g on D with lift π : • g → u(D). Then π is of p.e. at ξ if there exists • ξ ∈ C(π) ⊆ • g covering ξ. Write D(π) for the set of all such ξ. We say that π is of p.e. at C if C ⊆ C(π).
-A smooth (projective) unitary representation of G on H ρ is said to be of p.e. at ξ ∈ g if the corresponding derived (projective) unitary representation of g on H ∞ ρ is so. It is said to be of p.e. at C if it is so at every ξ ∈ C.
Remark 3.2. Let ρ be a smooth unitary representation of G. Then C := C(dρ) is always a closed, G-invariant convex cone. Consequently, C ∩ −C and C − C are ideals in g, called the edge and span of C, respectively. If ξ ∈ C ∩ −C then ξ ∈ ker dρ, so by passing to the quotient g/ ker dρ one may always achieve that C is pointed.
Next, we define the notion of a semibounded representation.
The representation ρ is said to be semibounded if W ρ contains an interior point, where Remark 3.4. For finite-dimensional Lie groups, the class of semibounded representations has been subject to detailed study in [Nee00]. In particular, they are highest weight representations [Nee00, Def. X.2.9, Thm. X.3.9]. For a consideration of semibounded representations in the context of infinite-dimensional Lie groups, we refer to [Nee17] and [Nee10b].
In the finite-dimensional context, the semiboundedness condition turns out to be extremely restrictive, which in turn has consequences for arbitrary positive energy representations. The following result, Theorem 3.5, is based on the results in the monograph [Nee00].
Theorem 3.5. Assume that G is connected and locally exponential. Take d ∈ g and let a = d ⊳ g be the closed ideal in g generated by d. Assume that dim(a) < ∞ and that a is stable, in the sense that Ad G (a) ⊆ a. Let A ⊳ G be a connected normal Lie subgroup integrating a. Let (ρ, H ρ ) be a smooth unitary G-representation which is of p.e. at d ∈ g. Write h := a/ ker dρ. The following assertions hold: 1. a = C − C, where C ⊆ g is the closed G-invariant convex cone in g generated by d.
2. The closure of C + ker dρ in h is a pointed, closed, generating and G-invariant convex cone. Thus C ∩ −C ⊆ ker dρ. 5. Let a n denote the maximal nilpotent ideal of a. Then [a, [a n , a n ]] ⊆ ker dρ.

ρ|
Proof. For the first point, let a ′ be the closure of C − C in g. As a ′ is a closed ideal in g containing d, we have a ⊆ a ′ . On the other hand, we know that Ad G (d) ⊆ a because a is stable. Thus C ⊆ a and hence a ′ ⊆ a. So a ′ = a. In particular dim(a ′ ) < ∞ and so C − C = a ′ = a. Next we prove the second statement. Take ξ ∈ (C + ker dρ) ∩ −(C + ker dρ). Then dρ(ξ) ≥ 0 and dρ(ξ) ≤ 0, in view of Remark 3.2, and hence Spec(dρ(ξ)) = {0}. As dρ(ξ) is essentially skew-adjoint, it follows that ξ ∈ ker dρ. Thus C + ker dρ is pointed in h. As C is G-invariant and convex, it is clear that the same holds for the cone C + ker dρ in h. The latter is also generating in h because a = C − C. Next we show that ρ| A is semibounded. As a is spanned by C and dim a < ∞, it follows that C ⊆ a has interior points. As C ⊆ W ρ , this implies that W ρ has interior points. Hence ρ| A is semibounded. For the remaining points, we use the results in [Nee00]. Corollary 3.6. Let G, d, a and a n be as Theorem 3.5. Let (ρ, H ρ ) be a smooth projective unitary representation of G. Suppose that ρ is of p.e. at d ∈ g. Then [a, [a n , a n ]] ⊆ ker dρ.
Proof. Let ρ : • a n onto a and a n , respectively.
The following simple lemma will also be useful. Proof. Let dρ : The assumptions imply that for every ξ ∈ g there exists E ξ ∈ R s.t. −idρ(ξ) ≥ E ξ . As this holds in particular for both ξ and −ξ, dρ(ξ) is a bounded operator for any ξ ∈ g. As dim(g) < ∞, one finds by choosing a basis (e µ ) of g that there exists

Generalized Positive Energy Representations
Let G denote a regular locally convex Lie group with Lie algebra g. The class of p.e. representations can be generalized by relaxing the condition −idρ(ξ) ≥ 0 in Definition 3.1. We define a suitable relaxed notion, the generalized positive energy condition, and show that it can still be very restrictive. In Section 5, we will encounter a class of representations which are not of p.e. but are of generalized positive energy.
-Let π : h → pu(D) be a continuous projective unitary representation of h on D with lift π : Then π is said to be of generalized positive energy at ξ ∈ h if ξ ∈ C(π). Similarly, we say it is of g.p.e. at C ⊆ h if C ⊆ C(π).
-Let ρ : G → U(H ρ ) be a smooth unitary representation of G. Then ρ is said to be of g.p.e. at ξ ∈ g if its derived representation dρ on H ∞ ρ is so.
-Let ρ : G → PU(H ρ ) be a smooth projective unitary representation of G with lift ρ : Then ρ is of g.p.e. at ξ ∈ g if ρ is of g.p.e. at some Remark 4.2. If π is a (projective) continuous unitary representation of g, then the set C(π) ⊆ g is always an Ad G -invariant cone.
An important observation for the class of g.p.e. representations is the following one: Lemma 4.3. Let π : g → u(D) be a continuous unitary representation of g on the pre-Hilbert space D. Let ξ ∈ C(π). Suppose that η ∈ g satisfies [[ξ, η], η] ∈ Z(g). Then for every ψ in some dense subspace D 0 we have: (2) The equations (2) follows from the fact that at 2 + bt + c ≥ 0 for all t ∈ R if and only if a, c ≥ 0 and b 2 ≤ 4ac.
In the projective context, this sets up a relation between ker π and the class [ω] ∈ H 2 ct (g; R) defined by the corresponding central R-extension • g of g. This is exploited in Section 8.   Corollary 4.6. Let p and g be locally convex Lie algebras. Let D : p → der(g) be a homomorphism for which the corresponding action p × g → g is continuous. Let D be a complex pre-Hilbert space and let π : g ⋊ D p → pu(D) be a continuous projective unitary representation of g ⋊ D p on D. Let [ω] ∈ H 2 ct (g ⋊ D p; R) be the corresponding class in H 2 ct (g ⋊ D p; R). Let η ∈ g, p ∈ C(π) ∩ p and assume that [D(p)η, η] = 0. Then ω(D(p)η, η) ≥ 0 and ω(D(p)η, η) = 0 ⇐⇒ π(D(p)η) = 0.

KMS Representations
In the following, we introduce the class of KMS representations. We will see in particular that these give rise to generalized positive energy representations. Consequently, they can be studied using the results of Section 4. Its definition makes use of the modular theory of von Neumann algebras, which we recall first.

Modular Theory of von Neumann Algebras
Let us recall the modular condition and the notion of a KMS state on a von Neumann algebra M, whilst fixing our conventions and notation. We refer to  -φ is said to satisfy the modular condition for σ if the following two conditions are satisfied: 2. For every x, y ∈ M, there exists a bounded continuous function F x,y : St → C which is holomorphic on St and s.t. for every t ∈ R: -φ is said to be KMS w.r.t. σ at inverse temperature β > 0 if it satisfies the modular condition for t → σ −βt . In that case, we also say that φ is σ-KMS at inverse-temperature β. If β = 1 we simply say that φ is a σ-KMS state.  In [BEK80] and [BEK86], the set K β of normal σ-KMS states at inverse temperature β is studied in the setting of C * -dynamical systems.
5. As a consequence of the previous points, if M is a factor and φ, ψ ∈ S(M) are both σ-KMS, then φ = ψ, so that two distinct normal states can not share the same modular automorphism group.
Then σ φ t = τ −t by uniqueness of the modular automorphism group (and the minus sign in the definition of KMS states). As Ω φ is separating for M φ , it follows that σ φ t (π φ (x)) = π φ (σ −t (x)).

Example 5.5 (Gibbs States). Let
The modular automorphism group corresponding to φ is given by Thus φ satisfies the modular condition for σ −βt and is therefore KMS at inverse-temperature β w.r.t. σ t .
Gibbs states φ(x) = 1 Z β Tr(e −βH x) constitute the simplest class of examples of KMS states. We will encounter a variety of different KMS states in Section 5.2.2 below.

KMS Representations
In the following, let G be a regular locally convex Lie group with Lie algebra g. Let N ⊆ G be an embedded Lie subgroup.
Definition 5.6. Let (ρ, H ρ ) be a continuous unitary G-representation. Let N := ρ(N ) ′′ ⊆ B(H ρ ) be the von Neumann-algebra generated by ρ(N ). For φ ∈ N * , define the function φ : If the subgroup N is clear from the context, we drop N from the notation and simply write KMS(ρ, ξ) and KMS(ρ, ξ) ∞ . We then also say that ρ is KMS at ξ if it is so relative to N .
Remark 5.7. For any fixed ξ ∈ g satisfying Ad(e tξ )N ⊆ N for all t ∈ R, one may as well consider the semidirect product N ⋊ α R, where α : R → Aut(N ) is defined by α t := Ad(e tξ ) N . Definition 5.6 additionally allows for the situation where ρ is KMS at multiple ξ I ∈ g, relative to possibly distinct subgroups N I ⊆ G, where I ∈ I for some indexing set I. We will see an example of this in Example 5.21 below.
In the following, let (ρ, H ρ ) be a continuous unitary G-representation and let N : Consider the left action of G on S(N ) defined by (g.φ)(x) := φ(ρ(g) −1 xρ(g)) for x ∈ N . Notice that this action leaves S(N ) ∞ invariant.
Proof. Write N g := ρ(g)N ρ(g) −1 . Let φ ∈ KMS(ρ, ξ, N ). As e tξ N e −tξ ⊆ N it follows that e tAdg(ξ) normalizes gN g −1 for every t ∈ R. Define the following automorphism groups: In order to show g.φ ∈ KMS(ρ, Ad g (ξ), gN g −1 ), we must verify that g.φ satisfies the modular condition for the automorphism group η ξ −t of N g . Notice that as isomorphisms N g → N we have As φ ∈ KMS(ρ, ξ, N ), we know that φ • σ ξ t = φ for all t ∈ R. It then follows immediately from (3) that Next, take x, y ∈ N g . Then x = ρ(g)x ′ ρ(g) −1 and y = ρ(g)y ′ ρ(g) −1 for some x ′ , y ′ ∈ N . Let the function F x ′ ,y ′ : St → C be continuous and bounded, holomorphic on St and satisfy Then F x,y satisfies the conditions of Definition 5.2 for η ξ −t . Indeed, notice using Equation Notice that N ∞,φ and D φ are invariant under the left N -and N ⋊ α R-actions, respectively.
where the last equality follows by Remark 5.4.
vectors by definition, the final observation follows.

Restrictions Imposed by the KMS Condition
Let us next determine some consequences of the KMS condition. Most notably, we will show that representations ρ which are smoothly-KMS give rise to generalized positive energy representations ρ φ on the GNS-Hilbert space H φ of the corresponding state φ.
We continue in the notation of Section 5.2. Fixing a Lie subgroup N ⊆ G and some element ξ ∈ g satisfying Ad(e tξ )N ⊆ N for all t ∈ R, we may as well suppose that G = N ⋊ α R for some smooth R-action α on N by automorphisms. Let g := Lie(G), n := Lie(N ) and write D ∈ der(n) for the derivation on n corresponding to α. Thus g = n ⋊ D Rd, where d := 1 ∈ R denotes the standard basis element. Assume that ρ is KMS at d relative to N , and let φ ∈ KMS(ρ, d, N ). We extend the N -representation ρ φ = π φ • ρ on the GNS- A first observation is the following: As φ satisfies the modular condition for σ φ , so does ψ for the automorphism group Hence it is so for A φ . In view of Remark 5.3(1) this implies that the modular automorphism group σ ψ on A φ is uniquely determined by the modular condition. Thus for all t ∈ R. As A φ is Abelian, we know by [BR97,Prop. 5.3.28] that for all a ∈ A and t ∈ R. If N is a factor, then φ is faithful and π φ is injective by Remark 5.3(1,3). Thus Let us illustrate Proposition 5.11 with the following noteworthy consequence for loop groups: Corollary 5.12. Let K be a compact 1-connected simple Lie group with Lie algebra k. Define LK := C ∞ (S 1 ; K) and Lk := C ∞ (S 1 ; k). Let α denote the T-action on LK by rotations, with corresponding derivation D := d dθ on Lk. Consider the Lie group G := LK ⋊ α T with Lie algebra g := Lk ⋊ D Rd, where d := 1 ∈ R. Suppose that the smooth unitary G-representation ρ is KMS at d ∈ g relative to LK. Assume that ρ(LK) ′′ is a factor. Then LK ⊆ ker ρ.
Proof. Suppose T ⊆ K is a maximal torus with Lie algebra t. Then LT ⊆ LK is an Abelian α-invariant subgroup. By Proposition 5.11 it follows that dρ(DLt) = {0}. As any X ∈ k is contained in a maximal torus, it follows that dρ( df dθ ⊗ X) = 0 for any f ∈ C ∞ (S 1 ) and X ∈ k. Consequently dρ(Dg) = {0} and hence Dg C ⊆ ker dρ, where we have extended dρ : g → L † (H ∞ ρ ) C-linearly to the complexification g C . As ker dρ is an ideal in g C and Lk C = DLk C + [DLk C , DLk C ], it follows that Lk C ⊆ ker dρ. Notice that LK is connected because K is 1-connected. It is also locally exponential by [Nee01, Thm. II.1]. It follows that LK ⊆ ker ρ.
Thus, one necessarily has to pass to a non-trivial central T-extension We now proceed with the observation that KMS representations give rise to generalized positive energy representations on the GNS-Hilbert space corresponding to the KMS state: In particular the representation Proof.
The assertion follows.
Proof of Theorem 5.13: As a consequence of Theorem 5.13, we find that the observations of Section 4 impose restrictions on KMS representations. Let us illustrate this with the following immediate consequence: Let ρ φ be the corresponding projective unitary representation of N ⋊R on H φ , where R acts on N by Ad(e tξ ) N . By Theorem 5.13, ρ φ is of g.p.e. at d ∈ • n⋊Rd and so ρ φ is of g.p.e. at d. It follows from Proposition 4.4 that ω([ξ, η], η) ≥ 0 and ω([ξ, η], η) = 0 ⇐⇒ dρ φ ([ξ, η]) = 0. As N is a factor, the KMS state φ ∈ S(N ) is faithful and the corresponding GNS-representation π φ : Remark 5.16. A related notation is that of a passive state, which is usually considered in the context of a C * -dynamical system (A, σ), where A is a C * -algebra and σ : where U 0 (A) denotes the identity component of the group U(A) of unitary elements in A. In this case, φ is necessarily σ-invariant [PW78, Thm. 1.1], so that σ is canonically implemented by a strongly-continuous unitary 1-parameter group t → e itH φ on the GNS-Hilbert space H φ . Let π φ : A → B(H φ ) be the GNSrepresentation of A associated to φ and let Ω φ ∈ H φ be the corresponding cyclic vector. Then (7) becomes which is similar to equation (1). It was moreover shown in [PW78] that any ground-or σ-KMS state is necessarily passive (cf. [BR97, Thm. 5.3.22]), which is analogous to the observation that both positive energy and KMS representations provide examples of generalized positive energy ones, in view of Theorem 5.13. We refer to [PW78] and [BR97] for more information on (completely) passive states.

Some Examples of KMS Representations
Let us consider a variety of examples of KMS representations, thereby showing in various situations that a well-known σ-KMS state φ on a von Neumann algebra N admits some underlying smooth structure. More precisely, we construct a continuous unitary representation ρ of a (typically infinite-dimensional) Lie group G such that N = ρ(N ) ′′ , φ ∈ KMS(ρ, ξ, N ) ∞ and σ t = Ad(ρ(e tξ )) for some ξ ∈ g and Lie subgroup N of G.
In particular, in this case the 1-parameter group σ on N implements the R-action t → Ad(e tξ ) N on the Lie subgroup N of G.
Let us begin with the simplest class of examples, which correspond to Gibbs states, as in Example 5.5: for any x ∈ N . Consequently, φ ∈ KMS(ρ, βξ) and so ρ is a KMS representation at βξ ∈ g. If in addition φ : G → C is smooth, then ρ is smoothly-KMS at βξ ∈ g. By Lemma 5.9, ρ is also KMS at any element in the adjoint orbit of βξ. In view of Example 5.5, the representation In fact, Proposition 5.18 below entails that any KMS representation ρ for which N is a factor of type I is of the form described in Example 5.17. Moreover a complete characterization of such representations was very recently obtained in the context where N is a finite-dimensional Lie group [Sim23].
Proposition 5.18. Let ξ ∈ g and β > 0. Suppose that ρ| N is irreducible and that φ ∈ KMS(ρ, βξ, N ). Let By assumption, φ satisfies the modular condition for the automorphism group t → Ad(ρ(e −tβξ )) =: σ −βt . On the other hand, as φ is faithful, there exists by Remark 5.3(1) a unique automorphism group σ φ t of N for which φ satisfies the modular condition. It follows xρ(e tβξ ) for every x ∈ N . As Z(N ) = CI and both t → δ it and t → ρ(e tβξ ) are strongly continuous unitary 1-parameter groups, it follows that there is some continuous homomorphism c : Since Tr(δ) = 1, we have Z β = Tr(e −βH ) = Tr(e −(βH+µ) e µ 1) = Tr(δe µ 1) = e µ φ(1) = e µ < ∞. It follows that For more interesting examples, one has to consider a Lie subgroup N of G which is not of type I, so that the von Neumann algebra N need not be type I.
Example 5.19 (Powers' factors). Define G n := n k=1 SU(2) and let η n : G n ֒→ G n+1 be defined by Write g n := Lie(G n ) and L(η n ) := Lie(η n ). The direct limit G := lim − →n (G n , η n ) consists of sequences (u k ) in SU(2) with u k = 1 for all but finitely many values of k. It can be equipped with the structure of a regular Lie group that is modeled on the locally convex inductive limit g := lim Consider the following R-action α on G defined by (α t (u)) k := e tξ u k e −tξ for u ∈ G. The corresponding action 1). Using the so-called Powers' factors, we define unitary representations ρ of G ♯ which are smoothly-KMS at d relative to G ⊳ G ♯ and for which ρ(G) ′′ is a factor of type III λ for arbitrary λ ∈ (0, 1). Define the finite-dimensional C * -algebra define a directed system of Hilbert spaces, and the algebraic direct limit lim − →n H φn becomes naturally a pre-Hilbert space. Let H denote its Hilbert space completion. Let ι n : H φn ֒→ H denote the canonical inclusion. For every n ∈ N, there is a * -representation π n of M n on H defined for x = x 1 ⊗ · · · ⊗ x n ∈ M n by which is well-defined because u = (u k ) ∈ G is a sequence in SU(2) with u k = 1 for all k sufficiently large.
Since ρ(βt) = ∆ −it φ∞ and ρ(G) ′′ = M ∞ , it follows that ρ is KMS at βd ∈ g ♯ relative to G ⊳ G ♯ . To see that φ ∞ : G → C is smooth, it suffices to show that its restriction to G n is smooth for every n ∈ N, using the universal property of the smooth manifold structure on G = lim − →n G n [Glö05, Thm. 3.1]. This is the case, as Ω, ρ(u)Ω = n k=1 φ(u k ) for any u ∈ G n , which is smooth G n → C. Thus Ω ∈ H ∞ ρ and so ρ is smoothly-KMS at βd ∈ g ♯ relative to G ⊳ G ♯ .
A standard subspace is a closed R-linear subspace K ⊆ H which is both cyclic and separating. We show that any standard real subspace gives rise to a smooth KMS representation. Let K ⊆ H be a standard real subspace. Write δ K for the corresponding modular operator on H, which is generally unbounded, positive and self-adjoint, see e.g. [NO17,Sec. 3]. Then t → δ it K is a strongly-continuous unitary 1-parameter group on H satisfying in particular δ it K K ⊆ K. We first pass to the R-smooth vectors K ∞ to obtain a regular Lie group H(K ∞ , ω) ⋊ R. We then construct a KMS representation thereof using second-quantization. The details are given below.
Let K ∞ denote the set R-smooth vectors in K. Then K ∞ is dense in K and R-invariant. It moreover carries a Fréchet topology which is finer than the one inherited as a subspace of K and for which the action R × K ∞ → K ∞ is smooth [Nee10a,Thm. 4.4, Lem. 5.2]. As ω : K ∞ × K ∞ → R is bilinear and continuous w.r.t. this topology, it is smooth. Thus the generalized Heisenberg group N := H(K ∞ , ω) is a Lie group. (Notice that ω| K ∞ may be degenerate.) It is as a subgroup of G generated by K ∞ . As K ∞ is a Frèchet space, it is Mackey complete by [KM97,Thm. I.4.11], which implies using [Nee06, Thm. V.1.8] that N is regular. Write n := Lie(N ). By construction R acts smoothly on N by δ it K , so that N ♯ := N ⋊ R is a regular Lie group. Let n ♯ := n ⋊ Rd denote its Lie algebra. We construct a representation of N ♯ which is smoothly KMS at d ∈ n ♯ relative to N ⊳ N ♯ . Let us recall the standard representation of H(H, ω) on the Bosonic Fock space F (H). Equip the symmetric algebra S • (H) with the inner product v 1 · · · v n , w 1 · · · w n = σ∈Sn n j=1 v j , w σj .
In view of (9), W and F together define a representation ρ of the Lie group N ♯ by ρ(n, t) := W (n)F (δ it K ). Let N := W (N ) ′′ . As K is a standard real subspace and K ∞ is dense in K, it follows that Ω is cyclic and separating for N [NO17, Lem. 6.2]. Let φ denote the faithful vector state on N defined by φ(x) = Ω, xΩ . Using [NO17,Prop. 6.10] we have ∆ it φ = F (δ it K ) for all t ∈ R. Consequently ρ is KMS at −d ∈ n ♯ relative to N ⊳ N ♯ (notice the minus sign in Definition 5.2). To see it is smoothly KMS, observe that φ : N → C is smooth because it is given by The following provides an example where ρ is smoothly-KMS at various ξ I ∈ g, relative to distinct subgroups N I ⊆ G, where I ∈ I for some indexing set I: Recall that SU(1, 1) acts on S 1 . Explicitly, for g = α β β α ∈ SU(1, 1) with α, β ∈ C satisfying |α| 2 −|β| 2 = 1, define g(z) := αz+β βz+α for z ∈ C with |z| = 1. With g as above, define the unitary action of SU(1, 1) on the complex Hilbert space L 2 (S 1 ; C) by (u(g)f )(z) := (α − βz) −1 f (g −1 (z)) for f ∈ L 2 (S 1 ; C). Let H 2 + (S 1 ; C) be the closed subspace of L 2 (S 1 ; C) spanned by the non-negative Fourier modes. Let H 2 − (S 1 ; C) be its orthogonal complement in L 2 (S 1 ; C). Notice that SU(1, 1) leaves these subspaces invariant. Consider the complex Hilbert space V := H 2 + (S 1 ; C) ⊕ H 2 − (S 1 ; C), where H 2 − (S 1 ; C) denotes the Hilbert space complex-conjugate to H 2 − (S 1 ; C). Let V R = L 2 (S 1 ; C) denote the real vector space underlying V . Define the real Fréchet space V ∞ R := C ∞ (S 1 ; C) and consider the symplectic vector space We construct a unitary representation ρ of G which is of p.e. at r ∈ su(1, 1) and which is KMS at ξ I ∈ su(1, 1) relative to N I , for every I ∈ I. The details are given below.
As the SU(1, 1)-action u on L 2 (S 1 ; C) leaves both H 2 + (S 1 ; C) and H 2 − (S 1 ; C) invariant, we obtain a unitary representation u of SU(1, 1) on V = H 2 + (S 1 ; C) ⊕ H 2 − (S 1 ; C) which is by construction of p.e. at r ∈ su(1, 1). As in Example 5.20, let W denote the standard representation of the real Heisenberg group H(V, Im −, − ) on the Fock space F (V ). Letting SU(1, 1) act on F (V ) by second quantization, we obtain a smooth unitary representation ρ of G on F (V ) which is of p.e. at r ∈ su(1, 1). Explicitly, ρ is given by ρ(v, g) = W (v)F ( u(g)) for v ∈ H(V ∞ R , ω) and g ∈ SU(1, 1). It follows from [Was98, Sec. II.14] that V I ⊆ V is a standard real subspace for any interval I ∈ I. Let δ it I denote the corresponding modular 1-parameter group, as in Example 5.20. The assignment I → V I , called a net of standard subspaces, satisfies I 1 ⊆ I 2 =⇒ V I1 ⊆ V I2 (isotony), V g.I = u(g)V I for g ∈ SU(1, 1) (SU(1, 1)-covariance) and I 1 ∩ where ∆ I denotes the modular operator on F (V ) defined from N I using the cyclic and separating vector Ω := 1 ∈ F(V ). Let φ = Ω, • Ω be the corresponding state on N I . Then (10) shows that φ : N I → C is smooth. Thus ρ is smoothly-KMS at 2πξ I ∈ su(1, 1) relative to N I ⊆ G, for any I ∈ I. For more details on the Bisognano-Wichmann property and nets of standard subspaces, see e.g. [Mor18] or [Mun01].

Generalized Positive Energy Representations of Jet Lie Groups and Algebras
We now depart from the general context of Part I. Using the observations made in Part I, we study projective unitary representations of jet Lie groups and algebras that are of generalized positive energy. Let us first fix our notation, which is kept throughout Part II.

Notation
Let V be a finite-dimensional real vector space and K a 1-connected compact simple Lie group with Lie algebra k. For any n ∈ N ≥0 , we denote by P n (V ) ⊆ R the space of homogeneous polynomials on V of degree n. Let R := R V * := ∞ n=0 P n (V ) denote the ring of formal power series on V with coefficients in R, equipped with the product topology. Let I = (V * ) be the maximal ideal of R, containing those elements with vanishing constant term. We write ev 0 : R → R ∼ = R/I for the corresponding quotient map. Let g be the R-module g := R ⊗ k of formal power series on V with coefficients in k. Then g is a topological Lie algebra with the Lie bracket defined by We also write f X instead of f ⊗ X for f ∈ R and X ∈ k. Define R k := R/I k+1 , I k := I/I k+1 and g k := g/(I k+1 .g) for k ∈ N ≥0 . Then R = lim ← − R k and g = lim ← − g k as topological vector spaces and Lie algebras, respectively. For k ∈ N ≥0 , let G k = J k 0 (V ; K) be the unique 1-connected Lie group integrating the finite-dimensional Lie algebra g k . Let G := J ∞ 0 (V ; K) := lim ← − G k be the corresponding projective limit, which is a pro-Lie group with topological Lie algebra g = lim ← −k g k . (See e.g. [HM07] for a detailed consideration of pro-Lie groups). Write X I for the Lie algebra of formal vector fields on V vanishing at the origin. Identify X I ∼ = der(I) using the Lie derivative v → L v . Notice further that der(I) ∼ = I ⊗ V . Define similarly X I k := X I /(I k+1 X I ) ∼ = der(I k ).
Let p be a finite-dimensional Lie algebra acting on g by the homomorphism D : p → der(g). Using the fact that all derivations of k are inner, by Whitehead's first Lemma [Jac79, III.7. Lem. 3], it follows from [Kac90, Ex. 7.4] that D(p) splits into a horizontal and vertical part according to D(p) = −L v(p) + ad σ(p) , where v : p → X op I is a homomorphism of Lie algebras and where σ : p → g is a linear map that necessarily satisfies the following Maurer-Cartan equation: Remark 6.1. As we shall see in Section 7.3 below, Equation (11) can be written as δσ + 1 2 [σ, σ] = 0 in the differential graded Lie algebra ( • p * )⊗g, whose differential is that of the Chevalley-Eilenberg complex, where g is considered as p-module according to p → −L v(p) .
We will refer to D as a lift of the p-action on R to g and we call σ the vertical twist of the lift D. We remark also that D(p) satisfies the following Leibniz rule: We will denote by j k various k-jet projections R → R k , g → g k and X I → X I k . It should be clear from the context which map is being used. Also, we will freely identify the quotient g 0 ∼ = k with the Lie subalgebra k ⊆ g of formal power series having only a non-trivial constant term. Similarly, we identify j 1 X I = X I1 ∼ = gl(V ) with the subalgebra gl(V ) ⊆ X I of linear vector fields on V .
A first observation is the fact that G = J ∞ 0 (V ; K) is not just a pro-Lie group, but actually a regular Lie group modeled on the Fréchet space g = J ∞ 0 (V ; k).
Proposition 6.2. Both G and G ⋊ α P are regular Fréchet-Lie groups.
Proof. It is clear that the Lie algebras g and g⋊ D p are Fréchet. For every n ∈ N, the Lie group G n = J n 0 (V ; K) is 1-connected, because K is so. Then also G is 1-connected, since π k (G) = π k lim ← −n G n = lim ← −n π k (G n ) for every k ∈ N, see e.g. [Hat02,Prop. 4.67]. Thus G is the unique 1-connected pro-Lie group with Lie(G) = g, which is locally contractible by [HN09, Theorem 1.2]. Then [HN09, Theorem 1.3, Prop. 5.7] entails that G is a regular Lie group. As the action α : P × G → G is smooth, also G ⋊ α P is a Lie group and it is regular by [Nee06, Thm. V.I.8], because both G and P are so.
Moreover, we have the following useful fact: Lemma 6.3. The exponential map exp G : g → G restricts to a diffeomorphism from I ⊗ k = ker ev 0 : g → k onto ker ev 0 : G → K .
Proof. For k ∈ N ≥0 , let H k := ker ev 0 : G k → K ⊳ G k be the maximal nilpotent normal subgroup of G k . Then h k := Lie(H k ) = ker ev 0 : g k → k = (I k ⊗ k) ⊳ g k .

Normal Form Results
By choosing suitable local coordinates, one may attempt to simplify the vector fields v(p) and the vertical twist σ(p) of the lift D(p) = −L v(p) + ad σ(p) simultaneously. One might for example try to show that there are local coordinates in which the formal vector fields v(p) are linear for every p ∈ p simultaneously, thereby linearizing the formal p-action. Similarly, one might aim to show that in suitable coordinates, σ(p) ∈ k ⊆ R⊗ k is constant for all p ∈ p, so that σ is a Lie algebra homomorphism p → k. In the following, this 'normal form problem' is considered. The results of Section 8 will depend on the availability of suitable normal forms, whose existence we study in the present section.
In Section 7.1, we briefly recall the transformation behavior of v and σ under suitable automorphisms of g. We proceed in Section 7.2 to recollect some known results regarding normal forms for Lie algebras of vector fields with a common fixed point. Finally, we consider in Section 7.3 the vertical twist σ.
-A formal diffeomorphism of V is an automorphism h of R. An automorphism of g is said to be horizontal if it is of the form h ⊗ id k for some h ∈ Aut(R). We write h.ξ or h(ξ) instead of (h ⊗ id k )(ξ) for ξ ∈ g.
-A gauge transformation is an automorphism of g of the form e adξ for some ξ ∈ g.
Remark 7.2. Any formal diffeomorphism h ∈ Aut(R) preserves the maximal proper ideal I and is determined by its restriction h| V * , which can be regarded as an element h of I ⊗ V for which To determine the transformation behavior of D : p → der(g), we have to consider the adjoint action of Aut(g) on der(g). Instead of considering arbitrary automorphisms of g, we will specialize to horizontal ones and to gauge transformations. For h ∈ Aut(R) and v ∈ X op I , we write h.v for the action of Aut(R) on X op I obtained from the adjoint action of Aut(R) on der(R) ∼ = X I ∼ = X op I . The following two proofs are due to K.-H. Neeb and B. Janssens. They appear in the presently unpublished article [JN].

Lemma 7.3 ([JN]
). Let D ∈ der(g) and ξ ∈ g. Then Proof. Let k ∈ N be arbitrary. Consider the continuous path γ : I → der(g) defined by γ(t) = e tad ξ De −tad ξ . Notice that j k • γ : I → der(g k ) is smooth for all k and consequently so is γ. Moreover where the last step uses that α • ad η = ad α(η) • α for any α ∈ Aut(g). Thus
-Two homomorphisms v, w : p → X op I are said to be formally-equivalent if there is a formal diffeomorphism h ∈ Aut(R) such that h.v(p) = w(p) for all p ∈ p.
-Two linear maps σ, η : p → R ⊗ k satisfying the Maurer-Cartan equation (11) are called gauge-equivalent if there is some ξ ∈ g such that In this case, we write σ ∼ η and say that σ and η are related by the gauge transformation e adξ .

Lie Algebras of Formal Vector Fields with a Common Fixed Point
The normal form problem for vector fields near a fixed point has been subject to extensive study. Let us first gather some relevant known results.

The case of a single vector field
Naturally, the special case which has been considered most is the case where p is simply R, in which case one is looking for normal forms of dynamical systems near a fixed point, in the formal context. This case is already quite interesting. Let us recollect some relevant results. For more information, we refer to [Arn88].
Let v be a vector field on V . Write v = v l + v ho , where v l = j 1 0 (v) ∈ gl(V ) ⊆ X I is the linearization of v at 0 ∈ V and v ho ∈ X I 2 is a vector field vanishing up to first order at 0 ∈ V . Let v l = v l,s + v l,n be the Jordan decomposition of v l over C, where v l,s is semisimple and v l,n is nilpotent. Write V C := V ⊗ R C. Let (e j ) d j=1 be a basis of eigenvectors of v l,s in V C with dual basis (x j ) d j=1 of V * . Let (µ j ) d j=1 denote the corresponding eigenvalues.
Definition 7.6. Let n ∈ N d ≥0 be a multi-index. A monomial vector field x n ∂ xj with |n| ≥ 2 is called resonant if n, µ = µ j , where n, µ := d i=1 n i µ i . Identifying v l,s with the linear vector field d j=1 µ j x j ∂ xj on C d , this is equivalent to [v l,s , x n ∂ xj ] = 0.

Theorem 7.7 (Poincaré-Dulac Theorem [Dul04] [Arn88, ch.5]).
There exists w ho ∈ X I 2 which is a C-linear combination of resonant monomials s.t. v is formally equivalent to w = v l + w ho ∈ X I . In particular [v l,s , w ho ] = 0 in X I . Corollary 7.8 (Poincaré [Poi79]). If there are no resonances, that is to say, if n, µ = µ j for all n ∈ N d ≥0 with |n| ≥ 2 and j ∈ {1, · · · , d}, then the vector field v can be formally linearized, so that v is formally equivalent to the linear vector field v l .

The case of actions by semisimple Lie algebras
Next, we move to Lie algebra representations by formal vector fields of semisimple Lie algebras. As nicely explained in [FM04] and first observed by Hermann in [Her68], in the formal setting the obstructions to being able to linearize a Lie algebra of vector fields simultaneously lie in various first Lie algebra cohomology groups H 1 (p, W ) for suitable finite-dimensional p-modules W . In view of Whitehead's First Lemma [Jac79, III.7.

Lem. 3], this results in:
Theorem 7.10 ( [Her68]). Let p be a semisimple Lie algebra and v : p → X op I be a Lie algebra homomorphism. Then v is formally equivalent to its linearization j 1 v : p → gl(V ) ⊆ X op I around the origin. Remark 7.11. Corresponding statements of Theorem 7.10 in the setting of germs of smooth/analytic vector fields and diffeomorphisms have been proven in [GS68] and [FM04] under additional assumptions. They are false in general without suitable extra conditions, as was shown in [GS68].

Normal Form Results for the Vertical Twist
Let us next consider the vertical twist σ : p → g of the lift D(p) = −L v(p) + ad σ(p) to g of the p-action −L v(p) on R, which has to satisfy the Maurer Cartan equation (11). We fix the horizontal part v : p → X op I and act by gauge transformations. The main results of this section are the following two theorems, whose proof comprises the remainder of the section. The reader who is eager to consider the projective unitary g.p.e. representations of g can proceed to Section 8 after reading Theorem 7.12 and Theorem 7.13 below.
Let us also remark that the methods used in this section to prove Theorem 7.12 and Theorem 7.13 were communicated to the author by B. Janssens and K.H. Neeb and appear in similar form in their presently unpublished work [JN], albeit in a more specific context. The author has placed their approach in a more general context and extracted the two theorems below.
Theorem 7.12. Assume that p is semisimple. Let the linear map σ : p → g satisfy the Maurer-Cartan equation (11). Then σ is gauge-equivalent to σ 0 := ev 0 •σ : p → k. If p has no non-trivial compact ideals, then σ is gauge-equivalent to 0. The next result concerns the case p = R, in which case we identify v with v(1) ∈ X I and σ with σ(1) ∈ g. In this case, the Maurer-Cartan equation (11) is trivially satisfied for any σ ∈ g and v ∈ X I . Theorem 7.13. Assume that p = R. Let σ ∈ g and v ∈ X I . Let v l := j 1 v ∈ gl(V ) be the linearization of v at 0 ∈ V . Assume w.l.o.g. that σ 0 := ev 0 (σ) ∈ t for some maximal torus t ⊆ k. The following assertions hold: 1. Assume that n, µ = α(σ 0 ) for any root α ∈ it * of k and n ∈ N d ≥0 with |n| ≥ 1. Then σ is gauge-equivalent to some σ ′ ∈ R ⊗ t.

Suppose that
Remark 7.14. Suppose that K → M is a principal fiber bundle with compact simple structure group K. Let α : T → Aut(K) be a smooth action on K by bundle automorphisms. Suppose that a ∈ M is a fixed point of the induced T-action on M and set V := T a (M ). By Theorem 7.9, the T-action on M is linear in suitable local coordinates around a ∈ M . Passing to J ∞ a (M ) ∼ = R and J ∞ a (Ad(K)) ∼ = g, one obtains a T-action on both R and g. The corresponding derivations are given by −L v and D = −L v + ad σ respectively, for some linear semisimple vector field v on V and some σ ∈ g. This is the setting of the third item in Theorem 7.13, according to which we may further assume that σ ∈ t, where t ⊆ k is a maximal torus, by acting with gauge transformations.
The remainder of this section is devoted to the proof of Theorem 7.12 and Theorem 7.13.

Reformulation using differential graded Lie algebras
In order to classify the equivalence classes of vertical twists σ : p → g, we interpret equation (11) as the Maurer Cartan equation in the differential graded Lie algebra (DGLA) L := L R := ( • p * )⊗ g. As a cochain complex, L is the Chevalley-Eilenberg complex associated to the p-module g, where p acts on g by p.ψ = −L v(p) ψ. Explicitly, the differential δ is given by where as usual, the arguments with a caret are to be omitted. The graded Lie bracket on L is the unique considered as an expression in L ⋊ RD, where F (w) = − ∞ n=0 1 (n+1)! w n = − 1 0 e tw dt. Let us check that the above is indeed well-defined, even though L is not a nilpotent DGLA. Since G = lim ← −k G k is a regular Lie group, it has an exponential map and so the automorphism e ad ξ := Ad(e ξ ) on g is defined. Consequently, so is for any p ∈ p. Thus the expression in equation (15) makes sense. Notice further that for σ ∈ L 1 , the above reduces precisely to the transformation behavior (14) of the vertical twist. In accordance with Definition 7.5, we say that the MC-elements σ, σ ′ ∈ L 1 are gauge-equivalent if they satisfy σ ′ = ξ.σ for some ξ ∈ L 0 , in which case we write σ ∼ σ ′ . Our goal is to study the MC-elements in L 1 up to gauge-equivalence.
Let n ∈ N ≥0 . Define analogously the following DGLAs, where we consider P n (V ) as p-module by identifying P n (V ) with I n /I n+1 for n ∈ N ≥0 , so that p.f = −L v l (p) f for p ∈ p and f ∈ P n (V ):

Shifted DGLAs
It will be beneficial to split off the constants terms of the k-valued formal power series, because contrary to L R , L I is a projective limit of nilpotent DGLAs. We discuss next how this can be done.
For any MC-element χ ∈ L 1 R0 = p * ⊗ k ⊆ L R of degree 1, define the "shifted" DGLA L χ R , which agrees with L R as a graded Lie algebra but has a shifted differential given by δ χ (σ) := δ(σ) + [χ, σ]. The differential δ χ agrees with the Chevalley-Eilenberg differential of • p * ⊗ g if g is considered as p-module with the twisted action p.σ := −L v(p) + [χ, −]. In particular, δ 2 χ = 0. Let us write R ⊗ χ k for this module structure to distinguish it form the usual one on g = R ⊗ k, which was given by p.ξ = −L v(p) ξ. Define also the extended DGLA L χ R ⋊ RD χ , where [D χ , σ] = δ χ (σ). Define in analogous fashion L χ In , L χ I and L χ P n , where we have used that the p-action on R ⊗ χ k leaves I n ⊗ k invariant for every n, so that P n (V ) ⊗ k ∼ = (I n ⊗ k)/(I n+1 ⊗ k) is naturally a p-module. The following is a standard result: Lemma 7.15.

Let σ ∈ L 1 R . Then χ + σ is a MC-element in L R if and only if σ is a MC-element in
R be a MC-element. Then ψ = χ + σ for some degree-1 MC-elements σ ∈ L χ I and χ ∈ L R0 . Proof.

As χ is a MC-element and [σ,
Thus, for any ξ ∈ g we have 3. Since R = R 0 ⊕ I as a vector space, we can write ψ = χ + σ, where χ = j 0 (ψ) ∈ L R0 and σ ∈ L χ I . As j 0 is a morphism of DGLAs, it is clear that χ = j 0 (ψ) is a MC-element in L R0 ⊆ L R . By the first point it follows that σ is a MC-element in L χ I ⊆ L χ R .

Study of MC-elements
In view of Lemma 7.15, let us first study the classification problem of gauge-orbits of MC-elements in L 1

R0
and then, for each MC-element χ ∈ L 1 R0 consider the orbits in L χ I under the gauge-action. 2. The gauge-action of X ∈ k = L 0 R0 on L R0 is given by X.χ = e X χ. Proof. Notice that δ(X) = 0 for any X ∈ k ⊆ g, because −L v(p) X = 0 for any p ∈ p. So p acts trivially on k = g 0 = j 0 g. Thus the Maurer-Cartan condition reads simply χ([p 1 , p 2 ]) − [χ(p 1 ), χ(p 2 )] = 0 for all p 1 , p 2 ∈ p, proving the first statement. The second statement follows at once from the definition (15), using once more that the p-action on k is trivial.
Next, we fix a homomorphism χ : p → k and turn to the MC-elements of the twisted DGLAs L χ I . Consider the following diagram of DGLAs: Any MC-element in L χ I projects to one in L χ I k for any k ∈ N, and all maps in the above diagram are equivariant w.r.t. the gauge-actions. Notice further that each L χ I k is nilpotent. To study the MC-elements in L χ I , we consider lifts of MC-elements from L χ I k to L χ I k+1 , so as to solve the problem step-by-step. This can be done using the following central extension of nilpotent DGLAs, where L χ I0 = {0} is trivial: 1. Suppose that σ L ∈ L projects to σ M . Then h := δσ L + 1 2 [σ L , σ L ] ∈ K 2 is closed and [h] ∈ H 2 (K) is independent of the lift σ L of σ M . Moreover, there is some η ∈ K 1 such that σ L + η is a MC-element in L 1 if and only if [h] = 0 in H 2 (K).

If σ L and σ ′
L are two lifts of σ M that are both MC-elements in L 1 , then ∆ := σ ′ L − σ L ∈ K 1 is closed. Conversely, if ∆ ∈ K 1 is closed and σ L is a lift of σ M which is a MC-element, then σ ′ L := σ L + ∆ is also a lift of σ M which is a MC-element. Moreover, the class [∆] ∈ H 1 (K) vanishes if and only if σ L and σ L ′ are related by a gauge transformation of some element ξ ∈ K 0 .
3. If σ L is any lift of σ M which is a MC-element, then the map ∆ → σ L + ∆ induces a bijection between H 1 (K) and K 0 -orbits of MC-elements in L 1 lifting σ M . Proof.
1. It is clear that h ∈ K 2 as it projects to zero in M 2 . Since δh = [δσ L , σ L ] (by the graded Leibniz rule), we find using δσ where the second term vanishes by the graded Jacobi identity and the first term vanishes because h ∈ K is central. Thus h is closed. Suppose that σ ′ L is some other lift of σ M and define h ′ := δσ ′ L + 1 2 [σ ′ L , σ ′ L ] ∈ K 2 . Then ∆ := σ ′ L − σ L ∈ K lies in the center, so that h ′ = h + δ∆. It follows that [h] ∈ H 2 (K) does not depend on the lift. If there is some η ∈ K 1 such that σ L + η is a MC-element in L 1 , then , then there exists η ∈ K 1 such that h + δη = 0. Then σ L + η is a MC-element, by the same computation.
Next, we apply Lemma 7.17 to the exact sequences (17).
Lemma 7.18. For every sequence (ξ k ) k∈N of degree-0 elements in L χ I with ξ k ∈ P k (V ) ⊗ χ k for every k ∈ N, there exists η ∈ I ⊗ χ k such that j n (η.σ) = ξ n .ξ n−1 . · · · .ξ 1 .σ for every σ ∈ L χ I and n ∈ N. Proof. Consider the Lie subgroup H := ker ev 0 : G → K ⊳ G with Lie algebra h := ker ev 0 : g → k = I ⊗ k. Similarly, for n ∈ N let H n := ker ev 0 : G n → K and h n := Lie(H n ). Recall that the exponential map exp : h → H is a global diffeomorphism, by Lemma 6.3. Write log : H → h for its inverse. From j n • exp = exp •j n : h → H n we obtain that log •j n = j n • log : H → h n for any n ∈ N. As [ξ k , I ⊗ k] ⊆ I k+1 ⊗ k and the ξ k are of increasing order, we claim that the limit η := lim N →∞ log N k=1 e ξ k exists in I ⊗ k w.r.t. the projective limit-topology, where k increases from right to left in the expression. Indeed, to see this it suffices to show that for each n ∈ N the sequence (j n η N ) ∞ N =1 stabilizes for large enough values of N , where η N := log N k=1 e ξ k . This is the case because for N ≥ n we have j n η N = j n log N k=1 e ξ k = log N k=1 e j n (ξ k ) = log n k=1 e j n (ξ k ) = j n η n where it was used that j n (ξ k ) = 0 for all k > n, because ξ k ∈ I k . Thus η = lim N η N is well-defined and satisfies j n η = j n η n for all n ∈ N. Let σ ∈ L χ I . Using the fact that it follows by induction that for any n ∈ N, the equality η n .σ = ξ n .ξ n−1 · · · ξ 1 .σ is valid. We thus get: j n (η.σ) = j n (η n .σ) = j n (ξ n .ξ n−1 · · · ξ 1 .σ), ∀n ∈ N.
Proposition 7.19. Assume that H 1 (p, P k (V ) ⊗ χ k) = 0 for every k ∈ N. Then every degree-1 MC-element in L χ I is gauge-equivalent to 0 in L χ I . Proof. Fix a MC-element σ ∈ L χ I . Recall that j n (ζ.σ) is again a MC-element in L χ In for any n ∈ N and ζ ∈ L χ I of degree 0. Notice also that j 0 σ = 0. As H 1 (p, P k (V ) ⊗ χ k) = 0 for every k ∈ N, it follows using Lemma 7.17 and the exact sequences (17), by induction on n ∈ N, that we can find a sequence of degree-0 elements (ξ k ) k∈N in L χ I with ξ k ∈ P k (V ) ⊗ k such that j n (ξ n .ξ n−1 · · · ξ 1 .σ) = 0 for every n ∈ N. It follows from Lemma 7.18 that there is some η ∈ I ⊗ χ k such that j n (η.σ) = ξ n .ξ n−1 . · · · .ξ 1 .σ = 0 in L χ I for all n ∈ N. Thus σ ∼ 0.
Lemma 7.20. Let p be a semisimple Lie algebra with no nontrivial compact ideals. If k is a compact semisimple Lie algebra, then there are no non-trivial homomorphisms p → k.
Proof. Let χ : p → k be a homomorphism. Then p/ ker χ is isomorphic to a subalgebra of k and is therefore compact. As p is semisimple and has no nontrivial compact ideals, it also has no non-trivial compact quotients. Thus p/ ker χ = {0} or equivalently p = ker χ, so χ is trivial.
Proposition 7.21. Assume that p is semisimple. Let χ : p → k be a homomorphism and let σ ′ ∈ L χ I . Suppose that σ := χ + σ ′ is a degree-1 MC-element in L R . Then σ is gauge-equivalent to χ in L R .
Proof. Since σ ∈ p * ⊗ g is a MC-element in L R , there is some degree-1 MC-element σ ′ ∈ L σ0 I such that σ = σ 0 + σ ′ , by Lemma 7.15. Then Proposition 7.21 implies that σ is gauge-equivalent to σ 0 . By Lemma 7.16 we further know that σ 0 : p → k is a homomorphism of Lie algebras. Thus, if p has no non-compact ideals then σ 0 is trivial by Lemma 7.20.
8 Projective Unitary G.P.E. Representations of (R ⊗ k) ⋊ D p Having obtained the normal form results Theorem 7.12 and Theorem 7.13, we now proceed with the study of continuous projective unitary representation of jet Lie groups and algebras that are of generalized positive energy.
Let us begin by briefly recalling the setting and our notation. We have that V is a finite-dimensional real vector space, R = R V * := ∞ n=0 P k (V ) is the ring of formal power series on V with coefficients in R and equipped with the product topology. Moreover, g denotes the topological Lie algebra g = R ⊗ k, where k is a compact simple Lie algebra and p is a finite-dimensional real Lie algebra acting on g by the homomorphism D : p → der(g), which splits into a horizontal and a vertical part according to D(p) = −L v(p) + ad σ(p) , where v : p → X op I is a homomorphism and where the linear map σ : p → g satisfies the Maurer Cartan equation (11). Let P and K be the 1-connected Lie groups integrating p and K, respectively. For n ∈ N write G n := J n 0 (V ; K), G ♯ n := J n 0 (V ; K) ⋊ P and g ♯ n := g n ⋊ D p. Define further G : In the following, we are interested in understanding the extent to which the linearization v l of v and the values of σ(p) at the origin already determine properties of the class of representations which are of g.p.e. at a given cone C ⊆ p. To describe the main results, we first have to introduce some more notation.
Define σ 0 := ev 0 •σ : p → k and let v l = j 1 v : p → gl(V ) be the linearization of v at the origin. For p ∈ p, the vector fields v(p) splits as v(p) = v l (p) + v ho (p) for some formal vector field v ho (p) ∈ X I 2 vanishing up to first order at the origin. Let v l (p) = v l (p) s + v l (p) n be the Jordan decomposition of v l (p) over C. Let V C c (p) denote the span in V C of all generalized eigenspaces of v l (p) corresponding to eigenvalues with zero real part.
. We call V c (C) the 'center subspace associated to C', in analogy with the center manifold of a fixed point of a dynamical system. Let V c (C) ⊥ ⊆ V * denote the annihilator of V c (C) in V * . For any p ∈ p, let Σ p ⊆ C denote the additive subsemigroup of C generated by Spec(v l (p)). Recall from Definition 4.1 that for any continuous projective unitary representation π of g ⋊ D p, the set C(π) consists of all points p ∈ p for which π is of generalized positive energy at p.
Let us describe the main results of this section. In the context of positive energy representations, we have: Theorem 8.1. Let ρ be a smooth projective unitary representation of G ⋊ α P which is of positive energy at p ∈ p. Assume that Spec(ad σ0(p) ) ∩ Spec(v l (p)) = ∅. Then ρ factors through Remark 8.2. Notice that Spec(ad σ0(p) ) = { α(σ 0 (p)) : α ∈ ∆ } ∪ {0} is a finite subset of iR. In particular, the condition Spec(v l (p)) ∩ Spec(ad σ0(p) ) = ∅ is satisfied if v l has no purely imaginary eigenvalues. This is complemented by the following results, which in particular give sufficient conditions for π| g to factor through k, because R V c (C) * ⊗ k ∼ = k whenever V c (C) = {0}.
Theorem 8.4. Let t ⊆ k be a maximal Abelian subalgebra. Let π be a continuous projective unitary representation of g ♯ . Let C ⊆ C(π). Assume for every p ∈ C that σ(p) ∈ R ⊗ t and [v l (p) s , v ho (p)] = 0. Then RV c (C) ⊥ ⊗ k ⊆ ker π and hence π| g factors through R V c (C) * ⊗ k.
To prove these results, we consider in Section 8.1 the second continuous Lie algebra cohomology H 2 ct (g⋊ D p; R) so as to obtain particular representatives ω of cohomology classes therein. We proceed in Section 8.2 to show that any irreducible smooth projective unitary representation G ⋊ α P factors through the finite-dimensional G n ⋊ P for some n ∈ N. This gives us access to techniques that are available for finite-dimensional Lie groups, and in particular to Corollary 3.6, which leads to Theorem 8.1. In Section 8.3 and Section 8.4 we study the kernel of the quadratic form ξ → ω(D(p)ξ, ξ). Recalling from Corollary 4.6 that this leads to an ideal in g contained in ker π, and to the proof of Theorem 8.3 and Theorem 8.4. These results are supplemented in Section 8.5 by a consideration of the special case where p is a simple non-compact Lie algebra, in which case Theorem 7.12 is available. This leads to the following: Theorem 8.5. Assume that p is non-compact and simple. Let π be a continuous projective unitary representation of g ⋊ D p. Write C := C(π) ⊆ p. Then π| g factors through R V c (C) * ⊗ k.
Theorem 8.6. Assume that p is non-compact and simple. Suppose that v l defines a non-trivial irreducible p-representation on V . Let π be a continuous projective unitary representation of g ⋊ D p. Let C ⊆ C(π) be a P -invariant convex cone. Either C is pointed or π| g factors through k.

The Second Lie Algebra Cohomology
We next determine suitable representatives of classes in the second Lie algebra cohomology H 2 ct (g ⋊ D p, R), which classifies the continuous R-central extensions of g ⋊ p =: g ♯ up to equivalence. As an intermediate step we first consider H 2 ct (g, R), which is completely understood.
Define Ω k R := R ⊗ k V * , equipped with projective limit topology obtained from Ω k As Ω 1 R[V * ] is dense in Ω 1 R , it remains to extend D continuously to the Fréchet space Ω 1 R . Let α ∈ Ω 1 R and let (α n ) n∈N be a sequence in Ω 1 R[V * ] s.t. α n → α in Ω 1 R . We show that Dα := lim n→∞ Dα n exists and is independent of the approximating sequence (α n ). Choose a basis ( which is independent of the approximating sequence (α n ). It follows that D extends to a continuous R-linear map D : Ω 1 R → E, which satisfies D • d = D by construction. It is unique with these properties because its restriction to the dense subspace R[V * ] ⊆ R is so.
Remark 8.8. Lemma 8.7 entails that d : R → Ω 1 R is the universal differential module R in the category of complete locally convex R-modules, in the sense of [Mai02, Thm. 6] Proposition 8.9. Any class [ω] ∈ H 2 ct (g, R) has a unique representative of the form ω(ξ, η) = λ(κ(ξ, dη)), where λ ∈ Ω 1 R * is closed and continuous functional on Ω 1 R and κ is the Killing form on k. (Closed meaning that λ(dR) = 0.) Conversely, any such λ defines a 2-cocycle representing some non-zero class in H 2 ct (g, R). Consequently, the center of the universal central extension of g is (Ω 1 R /dR). Proof. This is a special case of [Mai02, Theorem 16], seeing as g = R ⊗ k, where R is a unital, associative and commutative Fréchet algebra and k is a simple Lie algebra.
Lemma 8.11. Let ω : g ♯ × g ♯ → R be a continuous 2-cocycle on g ♯ = g ⋊ D p. Then there exists n ∈ N and a 2-cocycle ω n on g ♯ n such that ω(ξ, η) = ω n (j n ξ, j n η) for all ξ, η ∈ g ♯ . Proof. Let ω : g ♯ × g ♯ → R be a continuous 2-cocycle on g ♯ . Choose norms − n on the finite-dimensional Lie algebras g ♯ n s.t. the quotient maps j n : g ♯ m → g ♯ n are contractive for any n, m ∈ N with n ≤ m. The topology on g ♯ = lim ← − g ♯ n is specified by the seminorms ξ → j n ξ n for n ∈ N and ξ ∈ g ♯ . As ω is continuous and the maps g ♯ m → g ♯ n are contractive for n ≤ m, there exist n ∈ N such that |ω(ξ, η)| ≤ j n ξ n j n η n for all ξ, η ∈ g ♯ (e.g. using [Tre67,Prop. 43.1 and Prop. 43.4]). As j n : g ♯ → g ♯ n is surjective, it follows that ω(ξ, η) = ω n (j n ξ, j n η) for a unique 2-cocycle ω n on g ♯ n .

Factorization Through Finite Jets
In the context of smooth projective unitary representations ρ of the Lie group G ♯ , it is no loss of generality to consider the case where ρ factors through the finite-dimensional Lie group G ♯ n for some n ∈ N: Theorem 8.12. Let ρ be a smooth projective unitary representation of G ♯ with lift ρ :  g is a central R-extension of g ♯ in the category of locally convex Lie algebras. Let the continuous 2-cocycle ω : g ♯ × g ♯ → R represent the corresponding class in H 2 ct (g ♯ , R). By Lemma 8.11, there is some n ∈ N such that for all ξ ∈ g ♯ we have j n ξ = 0 =⇒ ω(ξ, η) = 0 for all η ∈ g ♯ . Let 2] that ρ decomposes as a possibly uncountable direct sum ρ ∼ = i∈I ρ i such that for every i ∈ I there exists some m ≥ n with ρ i (N m ) = {1}, which implies that ρ i (N ♯ m ) = {1}. Theorem 8.12 gives us access to techniques that are available for finite-dimensional Lie groups, and in particular to Corollary 3.6. This can be used to prove Theorem 8.1.
Assume that π(RL v f ⊗ k) = {0}. Then π(L v f H) = 0 for any H ∈ t, so λ(L v (f )df ) = 0 by (20). Conversely, suppose that λ(L v (f )df ) = 0. Then π(L v f H) = 0 for all H ∈ t, by (20). Taking the commutator with π(gX α ), where g ∈ R, α ∈ ∆ is a root and X α ∈ (k C ) α is a corresponding root vector, it follows that Together with the fact that λ vanishes on exact forms and is L v -invariant, this puts severe restrictions on the representation π and leads to Theorem 8.14. Let us also remark the following: Proof. As λ is closed and L v -invariant, it follows that β is symmetric. To see that q is positive semi-definite, let f ∈ R and 0 = H ∈ t. Write η := f H and notice that [Dη, η] = 0. By Corollary 4.6 we know that As κ is negative definite on k we obtain that q is positive semi-definite.
The following observation is also noteworthy, although it will not be used: Proof. Let f, g ∈ N . Then using the Leibniz rule and Proposition 8.16 we obtain Applying Proposition 8.16 once more, we conclude that f g ∈ N .
Proof. As λ : Ω 1 R → R is continuous, it factors through the finite-dimensional space Ω 1 R k = R k ⊗ V * for some k ∈ N. Notice that both L v l,n and L v ho are nilpotent on Ω 1 In particular, λ vanishes on the eigenspaces in Ω 1 R C of L v l,s corresponding to non-zero eigenvalues. We introduce some more notation. Let E C denote the span of all eigenspaces in R C of L v l,s corresponding to eigenvalues with non-zero real part. Define E := E C ∩ R and E n := E ∩ I n .
Proof. Let µ ∈ Spec(L v l,s ) with Re(µ) = 0. Set E µ := ker(L v l,s − µI) ⊆ R C . Suppose first that µ ∈ R. If ψ ∈ E µ ∩ R then because L v leaves the eigenspaces of L v l,s invariant, the 1-form ψdL v ψ is an eigenvector of L v l,s with non-zero eigenvalue 2µ. By Lemma 8.20 it follows that q(ψ) = 0 and hence ψ ∈ N . Thus E µ ⊆ N . Next, suppose that µ is not real. Then also µ is an eigenvalue of L v l,s . Write W C := E µ ⊕E µ and W := W C ∩R. Take ψ ∈ W arbitrary. Then ψ = η + η for some η ∈ E µ (and hence η ∈ E µ ). As µ + µ = 2Re(µ) = 0 and L v leaves the eigenspaces of L v l,s invariant, each of the 1-forms ηdL v η, ηdL v η, ηdL v η and ηdL v η are eigenvectors of L v l,s with non-zero eigenvalue. Using Lemma 8.20 it follows that q(ψ) = 0 and hence ψ ∈ N . Thus W ⊆ N . As N is a linear subspace, we have shown E ⊆ N .
Proof. Write J := RL v E. As J is an ideal in R it suffices to show E ⊆ J. We claim that E n ⊆ J + E n+1 for every n ∈ N ≥0 . Indeed, take ψ ∈ E n . As L v l is invertible on E n (which is true because L v l is invertible on every finite-dimensional and L v l -invariant subspace E ∩ P k (V ) ⊆ E), there exists some η ∈ E n s.t.
, as required. By induction it follows that E = E 0 ⊆ J + E n for every n ∈ N. As n∈N (J + E n ) = J, this implies E ⊆ J.
Proof of Theorem 8.14: Using Lemma 8.21 we obtain E ⊆ N . By Corollary 8.17, this implies J ⊗ k ⊆ ker π, where J = RL v E. By Lemma 8.22, we know RE ⊆ J.
We conclude that π factors through the quotient

The Case of General p
Let us return to the case where P is a 1-connected finite-dimensional Lie group with Lie algebra p. Let us recall some of the notation introduced earlier in Section 8.
Define σ 0 := ev 0 •σ : p → k and let v l = j 1 v : p → gl(V ) be the linearization of v at 0 ∈ V . For p ∈ p, the vector fields v(p) splits as v(p) = v l (p) + v ho (p) for some formal vector field v ho (p) ∈ X I 2 vanishing up to first order at the origin. Let v l (p) = v l (p) s + v l (p) n be the Jordan decomposition of v l (p) over C. Let V C c (p) denote the span in V C of all generalized eigenspaces of v l (p) corresponding to eigenvalues with zero real part. Set V c (p) : Let Σ p ⊆ C denote the additive subsemigroup of C generated by Spec(v l (p)). For any continuous projective unitary representation π of g ⋊ D p, the set C(π) consists of all points p ∈ p for which π is of generalized positive energy at p.
We use Theorem 8.14 combined with suitable normal form results to prove Theorem 8.3 and Theorem 8.4. Lemma 8.23. Let W be a finite-dimensional real vector space and let W i ⊆ W be a collection of linear subspaces, where i ∈ I for some indexing set I. Then Proof. Notice first that i∈I W ⊥ i = Span i∈I W i ⊥ . Applying this observation to the subspaces W ⊥ i ⊆ W * , we obtain that i∈I W i = i∈I (W ⊥ i ) ⊥ = Span i∈I W ⊥ i ⊥ , where we also used that (W ⊥ i ) ⊥ ∼ = W i for any i ∈ I. Taking annihilators, this implies i∈I W i ⊥ = Span i∈I W ⊥ i .
Theorem 8.4. Let t ⊆ k be a maximal Abelian subalgebra. Let π be a continuous projective unitary representation of g ♯ . Let C ⊆ C(π). Assume for every p ∈ C that σ(p) ∈ R ⊗ t and [v l (p) s , v ho (p)] = 0. Then RV c (C) ⊥ ⊗ k ⊆ ker π and hence π| g factors through R V c (C) * ⊗ k.

The Case Where p is Simple.
Let us consider the special case where p is simple. Let P be a 1-connected Lie group with Lie(P ) = p. In this case, suitable normal form theorems for v and σ are available (see Theorem 7.10 and Theorem 7.12).
In particular, v : p → X op I is always formally equivalent to its linearization v l at 0 ∈ V . Similarly, by Theorem 7.12, the vertical twist σ : p → g is gauge-equivalent to some Lie algebra homomorphism σ 0 : p → k. In particular, if p is not compact then we may and do assume that σ = 0 by acting with gauge transformations if necessary, for in that case there are no homomorphisms p → k (because k is compact, see Lemma 7.20). Combined with Theorem 8.4 we immediately obtain Theorem 8.5 below, where V c (C) := p∈C V c (p).
Theorem 8.5. Assume that p is non-compact and simple. Let π be a continuous projective unitary representation of g ⋊ D p. Write C := C(π) ⊆ p. Then π| g factors through R V c (C) * ⊗ k.
Let p = k 0 ⊕ p 0 be a Cartan decomposition of p, so that k 0 and p 0 are the +1 and −1 eigenspaces of a Cartaninvolution θ, respectively [Kna96, Cor. 6.18]. Let a 0 ⊆ p 0 be a maximal Abelian subalgebra of p 0 . According to the Iwasawa decomposition [Kna96,Prop. 6.43], p decomposes as p ∼ = k 0 ⊕a 0 ⊕n 0 , where n 0 ⊆ p is nilpotent. For p ∈ p we write p = p e + p h + p n for the corresponding decomposition of p, where p e ∈ k 0 , p h ∈ a 0 and p n ∈ n 0 . Then Spec(ad pe ) ⊆ iR, Spec(ad p h ) ⊆ R and ad pn is nilpotent [Kna96, Lem. 6.45]. Moreover, a 0 is contained in a Cartan subalgebra of p 0 [Kna96, Cor. 6.47].
Proposition 8.24. Suppose that p is simple and that the p-representation v l on V is non-trivial and irreducible. Let C ⊆ p be an Ad P -invariant convex cone and let V c (C) := p∈C V c (p). Assume that C contains some non-zero p h ∈ a 0 . Then V c (C) = {0}.
Proof. Notice first that as P is 1-connected, the p-action v l : p → gl(V ) integrates to a continuous representation of P on V . As C is Ad P -invariant, the subspace V c (C) is P -invariant. Thus either V c (C) = {0} or V c (C) = V , so it suffices to show V c (C) = V . By assumption p h = 0. In view of Cartan's unitary trick, see e.g. [Kna01, V. Prop. 5.3], the image of elements in a 0 in any finite-dimensional representation are semisimple and have real spectrum. Thus Spec(v l (p h )) ⊆ R. As p is simple and v l is a non-trivial p-representation by assumption, it follows that v l is injective. As v l (p h ) ∈ gl(V ) is semisimple, there exists 0 = v ∈ V s.t. v l (p h )v = µv for some 0 = µ ∈ R. Thus 0 = v / ∈ V c (C). Hence V c (C) = V and so V c (C) = {0}.
Theorem 8.6. Assume that p is non-compact and simple. Suppose that v l defines a non-trivial irreducible p-representation on V . Let π be a continuous projective unitary representation of g ⋊ D p. Let C ⊆ C(π) be a P -invariant convex cone. Either C is pointed or π| g factors through k.
Remark 8.25. Notice that if p is simple and C is a closed Ad P -invariant convex cone which is not pointed, then C ∩ −C = p and hence C = p.
Proof of Theorem 8.6: The edge e := C ∩ −C of the closure C of C is an ideal in p. Assume that C is not pointed. Then neither is C. As p is simple, it follows that e = p and hence C = p. Thus C is a dense convex cone in the finite-dimensional real vector space p, which implies that C = p. As p is non-compact, it contains some hyperbolic element. Thus, so does C. By Proposition 8.24 it follows that V c (C) = {0} and hence Theorem 8.5 implies that π factors through k.
Thus if C is an Ad P -invariant convex cone which is not pointed, then g admits no continuous projective unitary representations which are of g.p.e. at C ⊆ p other than those which factor through k. On the other hand, we know by [Pan81,Cor. 2.3] that if p is simple, then a non-trivial pointed closed and P -invariant convex cone exists in p if and only if p is of hermitian type, meaning that dim(z(k 0 )) = 1, where p = k 0 ⊕ p 0 is a Cartan decomposition of p and where k 0 is the Lie algebra of a compact Lie group.
Let us shift our attention to positive energy representations, in which case a different argument is available.
Lemma 8.26. Suppose that P is a non-compact simple connected Lie group. If (σ, H σ ) is a unitary Prepresentation that is norm-continuous, then σ is trivial.
Proof. As p is simple, dσ is either injective or trivial. Assume that dσ is not trivial. Let p = k 0 ⊕ a 0 ⊕ n 0 be the Iwasawa decomposition of p. Take x ∈ a 0 . Then ad x is semisimple and Spec(ad x ) ⊆ R. As σ is unitary and dσ is injective, z → dσ(z) B(H) defines a P -invariant norm on p. With respect to this norm, e tadx is an isometry on p for every t ∈ R. As ad x is semisimple, it follows that Spec(ad x ) ⊆ iR. So Spec(ad x ) ⊆ R ∩ iR = {0} and hence ad x = 0. Since p has trivial center it follows that x = 0. So a 0 = {0} and hence p is compact. But P is non-compact by assumption. So dσ must be trivial. As P is connected, it follows that σ is trivial.
Proposition 8.27. Suppose that P is a non-compact 1-connected simple Lie group. Assume that the P -action on V is irreducible and non-trivial. Let ρ be a continuous projective unitary representation of G which is of positive energy at C := p. Then ρ| G factors through K.
Proof. By Theorem 8.12 it suffices to consider the case where ρ factors through G k for some k ∈ N. From Whitehead's Second Lemma, [Jac79, III.9. Lem. 6], we know that H 2 (p, R) = {0}. Using in addition that P is simply connected, it follows that ρ| P lifts to a continuous unitary representation σ : P → U(H ρ ) of P , so that ρ(p) = [σ(p)] in PU(H ρ ) for all p ∈ P . By Lemma 3.7, the fact that ρ| P is of p.e. at C = p implies that σ is norm-continuous. It follows from Lemma 8.26 that σ is trivial. Thus ρ(α p (g)) = ρ(g) for all g ∈ G and p ∈ P . It follows that dρ vanishes on D(p)g. As p acts irreducibly and non-trivially on V , it follows that the ideal in g generated by D(p)g is I ⊗ k. Thus I ⊗ k ⊆ ker dρ. This implies that ρ| G k factors through K.
The following provides a simple example of a projective p.e. representation ρ of G 1 ⋊ P s.t. ρ| G1 does not factor through K.
Example 8.28. Let P = Mp(2, R) be the double-cover of SL(2, R). Let P act on V := R 2 via the defining action of SL(2, R). We consider a trivial vertical twist, so that the p-action on g = R ⊗ k is given by D(p) = −L v(p) . In this case the generator p 0 of rotations generates the unique (up to a sign) non-trivial closed, pointed, P -invariant convex cone C in p. Explicitly, v(p 0 ) = y∂ x − x∂ y . Let us construct a non-trivial continuous projective unitary representation of G 1 ⋊ P ∼ = (V * ⊗ k) ⋊ (K × P ) that is of p.e. at the cone C ⊆ p.
We begin by specifying a suitable 2-cocycle on V * ⊗ k ⊆ g 1 . Notice that 2 V p ∼ = R is one-dimensional.