* Observe first that any derivation $D^\prime \in \der(\g)$ satisfying $D^\prime (I \otimes \mfr{k}) \subseteq (I \otimes \mfr{k})$ integrates to a unique $1$-parameter group $t \mapsto e^{t D^\prime}$ of automorphisms on $\g$ that leave the ideal $I \otimes \mfr{k} \subseteq \g$ invariant. Indeed, this follows from fact that the corresponding statement is true for the finite-dimensional Lie algebra $\der(\g_n)$ for every $n \in \N$, where we use that $\g$ is the projective limit $\g = \varprojlim_n \g_n$. In particular this applies to the $\T$-action $e^{tD}$ on $\g$, which therefore leaves $I \otimes \mfr{k}$ invariant. It thus induces a continuous $\T$-action on $V^\ast \otimes \mfr{k} \cong (I \otimes \mfr{k})/(I^2 \otimes \mfr{k})$, which integrates the linear operator $-\mc{L}_{{\bm{v}_{\mrm{l}}}} \otimes 1 + 1\otimes \ad_{\sigma_0}$ on $V^\ast \otimes \mfr{k}$. This implies that ${\bm{v}_{\mrm{l}}} \in \gl(V)$ and $\ad_{\sigma_0} \in \der(\mfr{k})$ integrate to continuous $\T = \R/2\pi\Z$-actions on $V$ and $\mfr{k}$, respectively. As $\T$ is compact it follows in particular that ${\bm{v}_{\mrm{l}}}$ is semisimple and that $\Spec({\bm{v}_{\mrm{l}}}) \cup \Spec(\ad_{\sigma_0}) \subseteq 2\pi i\Z$. By lem: splitting_off_constants we know that there is some degree-1 MC-element $\sigma^\prime \in L_I^{\sigma_0}$ such that $\sigma = {\sigma_0} + \sigma^\prime$. By the previous item, it follows that we may assume that $\sigma^\prime$ satisfies $-\mc{L}_{{\bm{v}_{\mrm{l}}}}\sigma^\prime + [{\sigma_0}, \sigma^\prime] = 0$, by acting with gauge transformations in $L_I^{\sigma_0}$ if necessary. Let $n \in \N$. The $\T$-action on $\g_n = R_n \otimes \mfr{k}$ must be unitarizable because $\T$ is compact, so that its generator $D_n := -\mc{L}_{{\bm{v}_{\mrm{l}}}} + [j^n\sigma, \--] \in \der(\g_n)$ must be semisimple. Notice further that $-\mc{L}_{\bm{v}_{\mrm{l}}} + [{\sigma_0}, \--]$ is semisimple on $\g_n$ whereas $[j^n\sigma^\prime, \--]$ is nilpotent. Since $-\mc{L}_{\bm{v}_{\mrm{l}}}\sigma^\prime + [{\sigma_0}, \sigma^\prime] = 0$, the operators $-\mc{L}_{\bm{v}_{\mrm{l}}} + [{\sigma_0}, \--]$ and $[j^n\sigma^\prime, \--]$ on $\g_n$ commute. Thus $D_n = \big(-\mc{L}_{\bm{v}_{\mrm{l}}} + [{\sigma_0}, \--]\big) + [j^n\sigma^\prime, \--]$ is the Jordan decomposition of $D_n$. As $D_n$ is semisimple, this implies that $[j^n\sigma^\prime, \--]=0$. Thus $j^n\sigma^\prime \in \mfr{Z}(\g_n)$, where $\mfr{Z}(\g_n)$ denotes the center of $\g_n$. As $\mfr{k}$ is simple, we know that $\mfr{Z}(\g_n) = P^n(V)\otimes \mfr{k} \subseteq \g_n$. Thus $\sigma^\prime \in I^{n-1}\otimes \mfr{k}$ for every $n \in \N$, where $I^0 := R$. As $\bigcap_{n\in \N}(I^{n-1}\otimes \mfr{k}) = \{0\}$, it follows that $\sigma^\prime = 0$. Hence $\sigma = {\sigma_0} \in \mfr{t}$.

§ PROJECTIVE UNITARY G.P.E. REPRESENTATIONS OF $(R \OTIMES \MFR{K}) \RTIMES_D \P$

Having obtained the normal form results thm: normalformverticaltwistss and thm: normalformverticaltwistoned, 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, $R = \R\llbracket V^\ast \rrbracket := \prod_{n=0}^\infty P^k(V)$ 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 $\g = R \otimes \mfr{k}$, where $\mfr{k}$ is a compact simple Lie algebra and $\p$ is a finite-dimensional real Lie algebra acting on $\g$ by the homomorphism $D : \p \to \der(\g)$, which splits into a horizontal and a vertical part according to $D(p) = -\mc{L}_{\bm{v}(p)} + \ad_{\sigma(p)}$, where $\bm{v} : \p \to \X_I^{\op}$ is a homomorphism and where the linear map $\sigma : \p \to \g$ satisfies the Maurer Cartan equation (<ref>). Let $P$ and $K$ be the $1$-connected Lie groups integrating $\p$ and $K$, respectively. For $n \in \N$ write $G_n := J^n_0(V; K)$, $G_n^\sharp := J^n_0(V; K) \rtimes P$ and $\g_n^\sharp := \g_n \rtimes_D \p$. Define further $G := J^\infty_0(V; K) := \varprojlim_n G_{n}$, $G^\sharp :=G \rtimes_\alpha P$ and $\g^\sharp := \g \rtimes_D \p \cong \varprojlim_n \g_n^\sharp$.

In the following, we are interested in understanding the extent to which the linearization $\bm{v}_{\mrm{l}}$ of $\bm{v}$ and the values of $\sigma(p)$ at the origin already determine properties of the class of representations which are of g.p.e. at a given cone $\mfr{C} \subseteq \p$. To describe the main results, we first have to introduce some more notation.

Define $\sigma_0 := \ev_0 \circ \sigma : \p \to \mfr{k}$ and let ${\bm{v}_{\mrm{l}}} = j^1 \bm{v} : \p \to \gl(V)$ be the linearization of $\bm{v}$ at the origin. For $p \in \p$, the vector fields $\bm{v}(p)$ splits as $\bm{v}(p) = \vl{p} + \vho{p}$ for some formal vector field $\vho{p} \in \X_{I^2}$ vanishing up to first order at the origin. Let $\vl{p} = \vls{p} + \vln{p}$ be the Jordan decomposition of $\vl{p}$ over $\C$. Let $V_{\mrm{c}}^\C(p)$ denote the span in $V_\C$ of all generalized eigenspaces of $\vl{p}$ corresponding to eigenvalues with zero real part. Set $V_{\mrm{c}}(p) := V_{\mrm{c}}^\C(p) \cap V$. If $\mfr{C} \subseteq \p$ is a subset, define $V_{\mrm{c}}(\mfr{C}) := \bigcap_{p \in \mfr{C}}V_{\mrm{c}}(p)$. We call $V_{\mrm{c}}(\mfr{C})$ the center subspace associated to $\mfr{C}$, in analogy with the center manifold of a fixed point of a dynamical system. Let $V_{\mrm{c}}(\mfr{C})^\perp\subseteq V^\ast$ denote the annihilator of $V_{\mrm{c}}(\mfr{C})$ in $V^\ast$. For any $p \in \p$, let $\Sigma_p \subseteq \C$ denote the additive subsemigroup of $\C$ generated by $\Spec(\vl{p})$. Recall from def: qpe that for any continuous projective unitary representation $\olpi$ of $\g\rtimes_D \p$, the set $\mfr{C}(\olpi)$ consists of all points $p \in \p$ for which $\olpi$ 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:

Notice that $\Spec(\ad_{\sigma_0(p)}) = \set{\alpha(\sigma_0(p)) \st \alpha \in \Delta} \cup \{0\}$ is a finite subset of $i\R$. In particular, the condition $\Spec(\vl{p}) \cap \Spec(\ad_{\sigma_0(p)}) = \emptyset$ is satisfied if ${\bm{v}_{\mrm{l}}}$ has no purely imaginary eigenvalues.

This is complemented by the following results, which in particular give sufficient conditions for $\restr{\olpi}{\g}$ to factor through $\mfr{k}$, because $\R\llbracket V_{\mrm{c}}(\mfr{C})^\ast \rrbracket \otimes \mfr{k} \cong \mfr{k}$ whenever $V_{\mrm{c}}(\mfr{C}) = \{0\}$.

Let $\mfr{t} \subseteq \mfr{k}$ be a maximal Abelian subalgebra. Let $\olpi$ be a continuous projective unitary representation of $\g^\sharp$. Let $\mfr{C} \subseteq \mfr{C}(\olpi)$. Assume for every $p \in \mfr{C}$ that $\sigma(p) \in R \otimes \mfr{t}$ and $[\vls{p},\, \vho{p}] = 0$.  Then $RV_{\mrm{c}}(\mfr{C})^\perp\otimes \mfr{k} \subseteq \ker \olpi$ and hence $\restr{\olpi}{\g}$ factors through $\R\llbracket V_{\mrm{c}}(\mfr{C})^\ast \rrbracket \otimes \mfr{k}$.

To prove these results, we consider in sec: second_la_cohom the second continuous Lie algebra cohomology $H_{\ct}^2(\g \rtimes_D \p; \R)$ so as to obtain particular representatives $\omega$ of cohomology classes therein. We proceed in sec: fact_fin_jets to show that any irreducible smooth projective unitary representation $G \rtimes_\alpha P$ factors through the finite-dimensional $G_n \rtimes P$ for some $n \in \N$. This gives us access to techniques that are available for finite-dimensional Lie groups, and in particular to cor: pe_vanishing_ideal, which leads to thm: fact_pe_case. In sec: case_p_R and sec: restr_mom_gen_case we study the kernel of the quadratic form $\xi \mapsto \omega(D(p)\xi, \xi)$. Recalling from cor: qpe_kernel_special_case that

\[ [D(p)\eta, \eta] = 0 \implies \bigg(\omega(D(p)\eta, \eta) = 0 \iff \olpi(D(p)\eta) = 0 \bigg), \qquad \forall \eta \in \g,	\]

this leads to an ideal in $\g$ contained in $\ker \olpi$, and to the proof of thm: fact_result_spectral_condition and thm: fact_result_nform_condition. These results are supplemented in sec: restr_mom_set_semisimple by a consideration of the special case where $\p$ is a simple non-compact Lie algebra, in which case thm: normalformverticaltwistss is available. This leads to the following:

Assume that $\p$ is non-compact and simple. Let $\olpi$ be a continuous projective unitary representation of $\g \rtimes_D\p$. Write $\mfr{C} := \mfr{C}(\olpi) \subseteq \p$. Then $\restr{\olpi}{\g}$ factors through $\R\llbracket V_{\mrm{c}}(\mfr{C})^\ast \rrbracket \otimes \mfr{k}$.

§.§ The Second Lie Algebra Cohomology $H_{\ct}^2(\g \rtimes_D \p, \R)$.

We next determine suitable representatives of classes in the second Lie algebra cohomology $H^2_{\ct}(\g \rtimes_D \p, \R)$, which classifies the continuous $\R$-central extensions of $\g \rtimes \p =: \g^\sharp$ up to equivalence. As an intermediate step we first consider $H^2_{\ct}(\g, \R)$, which is completely understood.

Define $\Omega_R^k := R \otimes \bigwedge^k V^\ast$, equipped with projective limit topology obtained from $\Omega_R^k = \varprojlim_n \Omega_{R_n}^k$, where $\Omega_{R_n}^k := R_n \otimes \bigwedge^k V^\ast$. This makes $\Omega_R^k$ into a Fréchet space. In particular $\Omega_R^0 = R$. Since $J_0^\infty(\Omega^k(V)) \cong \Omega_R^k$, we can define a continuous differential $d : \Omega_R^k \to \Omega_R^{k+1}$ by $dj^\infty_0\alpha := j^\infty_0d\alpha \in J^\infty_0(\Omega^{n+1}(V)) \cong \Omega_R^{k+1}$, which is indeed well-defined. Choosing a basis $(e_\mu)_{\mu=1}^d$ of $V$ with dual basis $(d x_\mu)_{\mu=1}^d$ of $V^\ast$, the above differential $d$ is on $R$ given by $df = \sum_\mu (\partial_\mu f) \otimes dx_\mu$ for $f \in R$.

Let $E$ be a topological $R$-module and let $D : R \to E$ be a continuous derivation. Then there exists a unique continuous $R$-linear map $\overline{D} : \Omega^1_R \to E$ such that $D = \overline{D} \circ d$.

Let $\R[V^\ast]$ denote the ring of polynomial functions on $V$. As $\R[V^\ast] \subseteq R$, $E$ is also a $\R[V^\ast]$-module and $\restr{D}{\R[V^\ast]} : \R[V^\ast] \to E$ is a derivation. Using the universal property of the Kähler differential forms $\Omega^1_{\R[V^\ast]} \cong \R[V^\ast] \otimes V^\ast$, there is a unique $\R[V^\ast]$-linear map $\overline{D} : \Omega^1_{\R[V^\ast]} \to E$ such that $\restr{\overline{D} \circ d}{\R[V^\ast]} = \restr{D}{\R[V^\ast]}$. As $\Omega^1_{\R[V^\ast]}$ is dense in $\Omega_R^1$, it remains to extend $\overline{D}$ continuously to the Fréchet space $\Omega_R^1$. Let $\alpha \in \Omega^1_R$ and let $(\alpha_n)_{n\in \N}$ be a sequence in $\Omega^1_{\R[V^\ast]}$ s.t. $\alpha_n \to \alpha$ in $\Omega^1_R$. We show that $D\alpha := \lim_{n \to \infty}\overline{D} \alpha_n$ exists and is independent of the approximating sequence $(\alpha_n)$. Choose a basis $(dx_\mu)_{\mu=1}^d$ of $V^\ast$. Write $\alpha_n = \sum_{\mu=1}^d f_\mu^{(n)} dx_\mu$ and $\alpha = \sum_{\mu=1}^d f_\mu dx_\mu$ for some unique $f_\mu^{(n)} \in \R[V^\ast]$ and $f_\mu \in R$. Then $f_\mu^{(n)} \to f_\mu$ in $R$ for every $\mu$ and hence $\lim_{n \to \infty} \overline{D}\alpha_n  = \sum_{\mu=1}^d \lim_{n \to \infty} f_\mu^{(n)} Dx_\mu = \sum_{\mu=1}^d f_\mu Dx_\mu$, which is independent of the approximating sequence $(\alpha_n)$. It follows that $\overline{D}$ extends to a continuous $R$-linear map $D : \Omega^1_R \to E$, which satisfies $\overline{D} \circ d = D$ by construction. It is unique with these properties because its restriction to the dense subspace $\R[V^\ast]\subseteq R$ is so.

lem: one_forms_universal entails that $d : R \to \Omega_R^1$ is the universal differential module $R$ in the category of complete locally convex $\R$-modules, in the sense of <cit.>

Any class $[\omega] \in H_{\ct}^2(\g, \R)$ has a unique representative of the form $\omega(\xi, \eta) = \lambda(\kappa(\xi, d\eta))$, where $\lambda \in {\Omega^1_R}^\ast$ is closed and continuous functional on $\Omega^1_R$ and $\kappa$ is the Killing form on $\mfr{k}$. (Closed meaning that $\lambda(dR) = 0$.) Conversely, any such $\lambda$ 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 $(\Omega^1_R / dR)$.

This is a special case of <cit.>, seeing as $\g = R \otimes \mfr{k}$, where $R$ is a unital, associative and commutative Fréchet algebra and $\mfr{k}$ is a simple Lie algebra.

Let $\omega$ be an extension of the 2-cocycle $\lambda(\kappa(\xi, d\eta))$ on $\g$ to a $2$-cocycle on $\g^\sharp = \g \rtimes \p$. Then $\lambda$ is $\p$-invariant in the sense that $\lambda(\mc{L}_{\bm{v}(p)} \Omega^1_R) = 0$ for every $p \in \p$.

Take $\xi = f \otimes X$ and $\eta = g \otimes X$ for $f,g \in R$ and $0\neq X \in \mfr{k}$. Notice that $[\xi, \eta] = 0$ in $R \otimes\mfr{k}$. Using the cocycle identity, this implies

\begin{align*}
\omega(D(p)\xi, \eta) + \omega(\xi, D(p)\eta) &= \omega(D(p), [\xi, \eta]) = 0\\
\omega([\sigma(p), \xi], \eta) + \omega(\xi, [\sigma(p), \eta]) &= \omega(\sigma(p), [\xi, \eta]) = 0.
\end{align*}

Using $D(p)\xi = -\mc{L}_{\bm{v}(p)}\xi + [\sigma(p), \xi]$ it follows that $0 =\omega(\mc{L}_{\bm{v}(p)} \xi, \eta) + \omega(\xi, \mc{L}_{\bm{v}(p)}\eta) = \lambda(\mc{L}_{\bm{v}(p)}fdg)\kappa(X, X)$. As $\kappa(X, X) \neq 0$ and $R dR = \Omega^1_R$, this shows the claim.

Let $\omega : \g^\sharp \times \g^\sharp \to \R$ be a continuous $2$-cocycle on $\g^\sharp = \g \rtimes_D \p$. Then there exists $n \in \N$ and a $2$-cocycle $\omega_n$ on $\g_n^\sharp$ such that $\omega(\xi, \eta) = \omega_n(j^n\xi, j^n\eta)$ for all $\xi, \eta \in \g^\sharp$.

Let $\omega : \g^\sharp \times \g^\sharp \to \R$ be a continuous $2$-cocycle on $\g^\sharp$. Choose norms $\norm{\--}_{n}$ on the finite-dimensional Lie algebras $\g_n^\sharp$ s.t. the quotient maps $j^n : \g_m^\sharp \to \g_n^\sharp$ are contractive for any $n,m \in \N$ with $n \leq m$. The topology on $\g^\sharp = \varprojlim \g_n^\sharp$ is specified by the seminorms $\xi \mapsto \norm{j^n\xi}_n$ for $n \in \N$ and $\xi \in \g^\sharp$. As $\omega$ is continuous and the maps $\g_m^\sharp \to \g_n^\sharp$ are contractive for $n \leq m$, there exist $n \in \N$ such that $|\omega(\xi, \eta)| \leq \norm{j^n\xi}_n \norm{j^n\eta}_n$ for all $\xi, \eta \in \g^\sharp$ (e.g. using <cit.>). As $j^n : \g^\sharp \to \g_n^\sharp$ is surjective, it follows that $\omega(\xi, \eta) = \omega_n(j^n \xi, j^n \eta)$ for a unique $2$-cocycle $\omega_n$ on $\g_n^\sharp$.

§.§ Factorization Through Finite Jets

In the context of smooth projective unitary representations $\overline{\rho}$ of the Lie group $G^\sharp$, it is no loss of generality to consider the case where $\overline{\rho}$ factors through the finite-dimensional Lie group $G_n^\sharp$ for some $n \in \N$:

Let $\overline{\rho}$ be a smooth projective unitary representation of $G^\sharp$ with lift $\rho : \circled{G} \to \U(\mcH_\rho)$ for some central $\T$-extension $\circled{G}$ of $G^\sharp$. Then $\rho$ decomposes as a (possibly uncountable) direct sum $\rho = \bigoplus_{i \in \mc{I}}\rho_i$ s.t. for every $i \in \mc{I}$ there exists $n \in \N$ s.t. the projective unitary representations $\overline{\rho}_i$ associated to $\rho_i$ factors through $G_n^\sharp$.  In particular, if $\overline{\rho}$ is irreducible then it factors through $G_n^\sharp$ for some $n \in \N$.

Write $N_{m}^\sharp := \ker\bigg(j^{m} : G^\sharp \to G_{m}^\sharp\bigg)$ and $\n_{m}^\sharp := \ker\bigg(j^{m} : \g^\sharp \to \g_{m}^\sharp\bigg)$ for any $m \in \N_{\geq 0}$, so that $G_m^\sharp \cong G^\sharp/N_m^\sharp$ for any $m \in \N_{\geq 0}$. Notice that $N_m^\sharp \subseteq N_{n}^\sharp$ whenever $n\leq m$. Since $\overline{\rho}$ is a smooth projective representation, it follows from <cit.> that $\circled{G}$ is a Lie group. It is moreover regular by <cit.>, because both $G^\sharp$ and $\T$ are so. Let $\circled{\g} := \Lie(\circled{G})$. Then $\circled{\g}$ is a central $\R$-extension of $\g^\sharp$ in the category of locally convex Lie algebras. Let the continuous $2$-cocycle $\omega : \g^\sharp \times \g^\sharp \to \R$ represent the corresponding class in $H^2_{\ct}(\g^\sharp, \R)$. By lem: lambda_factors_through_finite_jets, there is some $n \in \N$ such that for all $\xi \in \g^\sharp$ we have $j^{n}\xi = 0 \implies \omega(\xi, \eta) = 0$ for all $\eta \in \g^\sharp$. Let $\circled{N}_n$ be the closed normal subgroup of $\circled{G}$ covering $N_n^\sharp$ and let $\circled{\n}_n$ be its Lie algebra. Then $\circled{N}_n$ is a central $\T$-extension of $N_n^\sharp$ integrating $\circled{\n}_n$. Since $\restr{\omega}{\n_n^\sharp \times \n_n^\sharp} = 0$, the central $\R$-extension $\circled{\n}_n$ is trivial. Hence $\circled{\n}_n \cong \R \oplus \n_n^\sharp$ as central $\R$-extensions of $\n_n^\sharp$. As $N_n^\sharp$ is regular and $1$-connected, it follows from <cit.> that there is a commutative diagram

\[
\begin{tikzcd}
{\R} \arrow{r} \arrow{d}{e^{2\pi i \, \bcdot}} & {\R \times N_n^\sharp} \arrow{r} \arrow{d}{\widetilde{\phi}} & {N_n^\sharp} \arrow{d}{\id} \\
{\T} \arrow{r}           & {\circled{N}_n} \arrow{r}           & {N_n^\sharp}
\end{tikzcd}
\]

of locally convex regular Lie groups. Observe that $\widetilde{\phi}$ is surjective and that $\ker \widetilde{\phi} = \Z$. Thus $\circled{N}_n \cong \T \times N_n^\sharp$ as central $\T$-extension of $N_n^\sharp$. Let $\phi : \T \times N_n^\sharp \to \circled{N}$ realize the isomorphism. For any integer $m \geq n$, let $\mc{N}_m := \phi(\{1\} \times N_m^\sharp) \subseteq \circled{N}_n \subseteq \circled{G}$, which is a closed normal subgroup of $\circled{G}$ isomorphic to and covering $N_m^\sharp$. Then $\mc{N} := \{\mc{N}_m\}_{m \geq n}$ is a filter basis of (decreasing) closed normal subgroups of $\circled{G}$ satisfying $\varinjlim \mc{N} = \{1\}$, in the sense that for any $1$-neighborhood $U$ of $\circled{G}$ there exists $m \geq n$ such that $\mc{N}_m \subseteq U$. Indeed, since $G^\sharp = \varprojlim_m G_m^\sharp$ carries the projective limit topology and $\circled{G}$ is a locally trivial principal $\T$-bundle over $G^\sharp$ <cit.>, it follows that any $1$-neighborhood $U \subseteq \circled{G}$ contains $\phi(I \times {N}_m^\sharp)$ for large enough $m$ and some open $1$-neighborhood $I \subseteq \T$. It now follows from <cit.> that $\rho$ decomposes as a possibly uncountable direct sum $\rho\cong \bigoplus_{i \in \mc{I}} \rho_i$ such that for every $i \in \mc{I}$ there exists some $m \geq n$ with $\rho_i(\mc{N}_m) = \{1\}$, which implies that $\overline{\rho}_i(N_m^\sharp) = \{1\}$.

thm: always_factor_through_fin_jet gives us access to techniques that are available for finite-dimensional Lie groups, and in particular to cor: pe_vanishing_ideal. This can be used to prove thm: fact_pe_case.

To prove thm: fact_pe_case, it suffices by thm: always_factor_through_fin_jet to consider the case where $\overline{\rho}$ factors through the finite-dimensional Lie group $G_k^\sharp$ for some $k \in \N$, which we thus assume. Write $\mfr{a}$ for the ideal in $\g_k^\sharp$ generated by $p \in \p$. Let $\mfr{a}_n \subseteq \mfr{a}$ denote the maximal nilpotent ideal in $\mfr{a}$. According to cor: pe_vanishing_ideal we have $[\mfr{a}, [\mfr{a}_n, \mfr{a}_n]] \subseteq \ker d\overline{\rho}$. Recall that $I_k := I/I^{k+1}$.

Suppose that $V^\ast \otimes \mfr{k} \subseteq j^1 \mfr{a}_n$. Then $I_k \otimes \mfr{k} \subseteq \mfr{a}_n$.

By assumption $V^\ast \otimes \mfr{k} \subseteq \mfr{a}_n + I_k^2 \otimes \mfr{k}$. As $\mfr{k}$ is perfect it follows that

\[I_k^{l+1}\otimes \mfr{k} = [V^\ast \otimes \mfr{k}, I_k^l \otimes \mfr{k}] \subseteq \mfr{a}_n + [I_k^2 \otimes \mfr{k}, I_k^l \otimes \mfr{k}] = \mfr{a}_n + I_k^{l+2}\otimes \mfr{k}, \qquad \forall l \in \N.\]

Thus it follows by induction that $V^\ast \otimes \mfr{k} \subseteq \mfr{a}_n + I_k^{l+1}\otimes \mfr{k}$ for all $l \in \N$. As $\bigcap_l (\mfr{a}_n + I_k^{l+1}\otimes \mfr{k}) = \mfr{a}_n$, it follows that $V^\ast \otimes \mfr{k} \subseteq \mfr{a}_n$ and hence $I_k \otimes \mfr{k} \subseteq \mfr{a}_n$.

We may assume that $\overline{\rho}$ factors through $G_k^\sharp$ for some $k \in \N$. It suffices to show that $d\overline{\rho}$ factors through $\g_2^\sharp$ and that the image of $-\mc{L}_{\vl{p}} + \ad_{\sigma_0(p)}$ in $P^2(V) \otimes \mfr{k} \subseteq \g_2$ is contained in $\ker d\overline{\rho}$. By cor: pe_vanishing_ideal we know that $\big[\mfr{a}, [\mfr{a}_n, \mfr{a}_n]\big] \subseteq \ker d\overline{\rho}$. Moreover $I_k^3 \otimes \mfr{k} = [I_k \otimes \mfr{k}, [I_k \otimes \mfr{k}, I_k \otimes \mfr{k}]]$, because $\mfr{k}$ is perfect. To see that $d\overline{\rho}$ factors through $\g_2^\sharp$ it thus suffices to show that $I_k \otimes \mfr{k} \subseteq \mfr{a}_n$. By lem: computation_for_an it is further sufficient to show that $V^\ast \otimes \mfr{k} \subseteq j^1(\mfr{a}_n)$. Write $D_1(p) := -\mc{L}_{\vl{p}} + [\sigma_0(p), \--]$. Notice that $j^1(D(p)\xi) = D_1(p) \xi$ for $\xi \in V^\ast \otimes \mfr{k}$. The assumption $\Spec(\vl{p}) \cap \Spec(\ad_{\sigma_0(p)}) = \emptyset$ implies that $D_1(p)$ is invertible on $V^\ast \otimes \mfr{k}$. Thus if $\eta \in V^\ast \otimes \mfr{k}$ is arbitrary, there exists $\xi \in V^\ast \otimes \mfr{k}$ such that $\eta = D_1(p) \xi$. Then $\eta = D_1(p) \xi = j^1(D(p) \xi) = j^1([p,\xi]) \in j^1(\mfr{a}_n)$. Thus $V^\ast \otimes \mfr{k} \subseteq j^1(\mfr{a}_n)$. We obtain that $I_k \otimes \mfr{k} \subseteq \mfr{a}_n$ and $I_k^3 \otimes \mfr{k} \subseteq \ker d\overline{\rho}$, so $d\overline{\rho}$ factors through $\g_2^\sharp$. We may thus assume that $k=2$. We then obtain

\[D_1(p)(P^2(V)\otimes \mfr{k}) = D(p)(I_k^2 \otimes \mfr{k}) = [p, [I_k \otimes \mfr{k}, I_k \otimes \mfr{k}]] \subseteq [\mfr{a}, [\mfr{a}_n, \mfr{a}_n] \subseteq \ker d\overline{\rho}.\qedhere\]

§.§ The Case Where $\p = \R$

We proceed with the study of projective unitary representations $\olpi$ of $\g^\sharp = \g \rtimes_D \p$ which are of generalized positive energy. We first specialize to the case where $\p = \R$, aiming to consider its consequences for the general case afterwards.

As $\p = \R$, we may as well identify $\bm{v}$ with $\bm{v}(1) \in \X_I$, $D$ with $D(1)$ and $\sigma$ with $\sigma(1) \in \g$. Recall that the derivation $D$ is given by $D = -\mc{L}_{\bm{v}} + [\sigma, \--]$. Write $\bm{v}  = \bm{v}_{\mrm{l}} + \bm{v}_{\mrm{ho}}$, where $\bm{v}_{\mrm{l}} := j^1 \bm{v} \in \gl(V)$ is the linearization of $\bm{v}$ at $0 \in V$ and where $\bm{v}_{\mrm{ho}} \in \X_{I^2}$. Let $\bm{v}_{\mrm{l}} = \bm{v}_{\mrm{l,s}} + \bm{v}_{\mrm{l,n}}$ denote the Jordan decomposition of $\bm{v}_{\mrm{l}}$ over $\C$. Write $V_{\mrm{c}}^\C$ for the span of the eigenspaces of $\bm{v}_{\mrm{l}}$ whose corresponding eigenvalue has zero real part. Set $V_{\mrm{c}} := V_{\mrm{c}}^\C \cap V$. Write $\bm{d} := (0,1) \in \g^\sharp = \g \rtimes_D \R$. Let $V_{\mrm{c}}^\perp \subseteq V^\ast$ denote the annihilator of $V_{\mrm{c}}$ in $V^\ast$, so $V_{\mrm{c}}^\perp \cong (V/V_{\mrm{c}})^\ast$.

Let $\mfr{t} \subseteq \mfr{k}$ be a maximal Abelian subalgebra. Assume that $\sigma \in R \otimes \mfr{t} \subseteq \g$ and $[\bm{v}_{\mrm{l,s}}, \bm{v}_{\mrm{ho}}] = 0$ in $\X_I$. Let $\olpi$ be a continuous projective unitary representation of $\g^\sharp$ on the pre-Hilbert space $\mc{D}$. Assume that $\olpi$ is of g.p.e. at $\bm{d} \in \g \rtimes_D \R\bm{d}$. Then $RV_{\mrm{c}}^\perp \subseteq \ker \olpi$. Consequently, $\restr{\olpi}{\g}$ factors through $\R\llbracket V_{\mrm{c}}^\ast \rrbracket \otimes \mfr{k}$. In particular, if $V_{\mrm{c}} = \{0\}$ then $\restr{\olpi}{\g}$ factors through $\mfr{k}$.

By acting with formal diffeomorphisms if necessary, one may by thm: poincare_dulac always bring $\bm{v}$ into a normal form, in the sense that $[\bm{v}_{\mrm{l,s}},\bm{v}_{\mrm{ho}}] = 0$ in $\X_I$. Moreover, thm: normalformverticaltwistoned provides sufficient conditions guaranteeing that $\sigma$ is gauge equivalent to some element in $R\otimes \mfr{t}$.

§.§.§ Proof of thm: general_factorization_result

Let $\omega$ be a continuous $2$-cocycle on $\g^\sharp$ that represents the class in $H^{2}_{\ct}(\g^\sharp, \R)$ corresponding to the central $\R$-extension of $\g^\sharp$ obtained from $\olpi$ by pulling back $\mfr{u}(\mcD) \to \pu(\mcD)$ along $\olpi$. In view of prop: characterization_of_cts_second_cohom and lem: lambda_p_invariant, we may and do assume that $\omega$ satisfies $\omega(\xi, \eta) = \lambda(\kappa(\xi, d\eta))$ for any $\xi, \eta \in \g$, where $\lambda : \Omega_R \to \R$ is continuous, $\p$-invariant and closed. We write $fX$ instead of $f \otimes X$ for $f \in R_\C$ and $X \in \mfr{k}_\C$. Let $\Delta\subseteq i\mfr{t}^\ast$ denote the set of roots of $\mfr{k}$. Finally, write $\h := \mfr{t}_\C \subseteq \mfr{k}_\C$. Recall from cor: qpe_kernel_special_case that

\begin{equation}\label{eq: qpe_kernel_special_case_repeat}
[D\eta, \eta] = 0 \implies \bigg(\omega(D\eta, \eta) = 0 \iff \olpi(D\eta) = 0 \bigg), \qquad \forall \eta \in \g.
\end{equation}

Moreover, $\omega(D\eta, \eta) \geq 0$ whenever $[D\eta, \eta] = 0$. In the present setting, this yields:

Fix $f \in R$. Then $\olpi(R\mc{L}_{\bm{v}}f \otimes \mfr{k}) = \{0\} \iff \lambda(fd\mc{L}_{\bm{v}}f) = 0$.

For any $H \in \mfr{t}$, observe that $DfH = -\mc{L}_{\bm{v}}fH$ because $\sigma \in R \otimes \mfr{t}$, so $[DfH, fH] = -[\mc{L}_{\bm{v}}fH, fH] = 0$. Using (<ref>) we obtain that

\begin{equation}\label{eq: kernel_rerp_1temp}
\kappa(H,H)\lambda(\mc{L}_{\bm{v}}(f)df) = 0 \iff \olpi(\mc{L}_{\bm{v}}fH) = 0, \qquad \forall H \in \mfr{t}.
\end{equation}

Assume that $\olpi(R\mc{L}_{\bm{v}}f \otimes \mfr{k}) = \{0\}$. Then $\olpi(\mc{L}_{\bm{v}}fH) = 0$ for any $H \in \mfr{t}$, so $\lambda(\mc{L}_{\bm{v}}(f)df) = 0$ by (<ref>). Conversely, suppose that $\lambda(\mc{L}_{\bm{v}}(f)df) = 0$. Then $\olpi(\mc{L}_{\bm{v}}fH) = 0$ for all $H \in \mfr{t}$, by (<ref>). Taking the commutator with $\olpi(gX_\alpha)$, where $g \in R$, $\alpha \in \Delta$ is a root and $X_\alpha \in (\mfr{k}_\C)_\alpha$ is a corresponding root vector, it follows that

\begin{equation}\label{eq: lem_kernel_intermediate_1}
\olpi(g\mc{L}_{\bm{v}}f X_\alpha) = 0 \qquad \forall X_\alpha \in (\mfr{k}_{\C})_{\alpha}, \; g \in R.
\end{equation}

Take $X_{\alpha} \in (\mfr{k}_\C)_{\alpha}$ and $Y_{-\alpha} \in (\mfr{k}_\C)_{-\alpha}$. Write $H_\alpha = [X_\alpha, Y_{-\alpha}]$. By taking commutators with $\olpi(1\otimes Y_{-\alpha})$ in eq: lem_kernel_intermediate_1 we find that $\olpi(g\mc{L}_{\bm{v}}f H_{\alpha}) = 0$. As $\h = \sum_{\alpha} [(\mfr{k}_{\C})_{\alpha}, (\mfr{k}_{\C})_{-\alpha}]$, this shows by linearity together with eq: lem_kernel_intermediate_1 and the root space decomposition that $\olpi(R\mc{L}_{\bm{v}}f~\otimes~\mfr{k})~=~\{0\}$.

Define the quadratic form $q(f) := \lambda(\mc{L}_{\bm{v}}(f)df) = -\lambda(fd\mc{L}_{\bm{v}}f)$ on $R$. Let $\mc{N} := \ker q$ denote its kernel. By prop: kernel_of_repr_and_quadr_form, $\mcN$ generates an ideal $J \otimes \mfr{k}$ on which $\olpi$ vanishes, where $J := R\mc{L}_{\bm{v}}\mcN$.

Set $J := R \mc{L}_{\bm{v}}\mcN$. Then $J\otimes \mfr{k} \subseteq \ker(\olpi)$.

Together with the fact that $\lambda$ vanishes on exact forms and is $\mc{L}_{\bm{v}}$-invariant, this puts severe restrictions on the representation $\olpi$ and leads to thm: general_factorization_result. Let us also remark the following:

The bilinear form $\beta_q(f,g) := \lambda(\mc{L}_{\bm{v}}(f)dg)$ on $R$ associated to $q$ is symmetric, the quadratic form $q$ is positive semi-definite and $\mc{N} = \set{f \in R \st \beta_q(f,g) = 0 \;\; \forall g \in R}$.

As $\lambda$ is closed and $\mc{L}_{\bm{v}}$-invariant, it follows that $\beta$ is symmetric. To see that $q$ is positive semi-definite, let $f \in R$ and $0 \neq H \in \mfr{t}$. Write $\eta := fH$ and notice that $[D\eta, \eta] = 0$. By cor: qpe_kernel_special_case we know that $-\kappa(H,H)\lambda(\mc{L}_{\bm{v}}(f)df) = \omega(D\eta, \eta) \geq 0$. As $\kappa$ is negative definite on $\mfr{k}$ we obtain that $q$ is positive semi-definite. It follows that $|\beta_q(f,g)|^2 \leq q(f)q(g)$, which implies $\mc{N} = \set{f \in R \st \beta_q(f,g) = 0 \qquad \forall g \in R}$.

The following observation is also noteworthy, although it will not be used:

$\mc{N} \subseteq R$ is a subalgebra.

Let $f,g \in \mc{N}$. Then using the Leibniz rule and prop: kernel_of_repr_and_quadr_form we obtain

\[\olpi(R\mc{L}_{\bm{v}}(fg)\otimes \mfr{k}) \subseteq \olpi(fR\mc{L}_{\bm{v}}g\otimes \mfr{k}) +  \olpi(gR\mc{L}_{\bm{v}}f\otimes \mfr{k}) \subseteq \{0\},\]

Applying prop: kernel_of_repr_and_quadr_form once more, we conclude that $fg \in \mc{N}$.

$\lambda \circ \mc{L}_{\bm{v}_{\mrm{l,s}}} = 0$.

As $\lambda : \Omega^1_R \to \R$ is continuous, it factors through the finite-dimensional space $\Omega^1_{R_k} = R_k \otimes V^\ast$ for some $k \in \N$. Notice that both $\mc{L}_{\bm{v}_{\mrm{l,n}}}$ and $\mc{L}_{\bm{v}_{\mrm{ho}}}$ are nilpotent on $\Omega^1_{R_k} \otimes_\R \C$, whereas $\mc{L}_{\bm{v}_{\mrm{l,s}}}$ is semisimple on it. Also $[\mc{L}_{\bm{v}_{\mrm{l,s}}}, \mc{L}_{\bm{v}_{\mrm{l,n}}} + \mc{L}_{\bm{v}_{\mrm{ho}}}] = 0$ because $[\bm{v}_{\mrm{l,s}}, \bm{v}_{\mrm{ho}}] = [\bm{v}_{\mrm{l,s}}, \bm{v}_{\mrm{l,n}}] = 0$. Thus $\mc{L}_{\bm{v}} = \mc{L}_{\bm{v}_{\mrm{l,s}}} + \big(\mc{L}_{\bm{v}_{\mrm{l,n}}} + \mc{L}_{\bm{v}_{\mrm{ho}}}\big)$ is the Jordan decomposition of $\mc{L}_{\bm{v}}$ acting on $\Omega^1_{R_k} \otimes_\R \C$. Thus $\mrm{Im}(\mc{L}_{\bm{v}_{\mrm{l,s}}}) \subseteq \mrm{Im}(\mc{L}_{\bm{v}})$ when $\mc{L}_{\bm{v}_{\mrm{l,s}}}$ and $\mc{L}_{\bm{v}}$ are considered as operators on $\Omega^1_{R_k} \otimes_\R \C$. As $\lambda$ is $\mc{L}_{\bm{v}}$-invariant, we know $\lambda \circ \mc{L}_{\bm{v}} = 0$. Thus $\lambda \circ \mc{L}_{\bm{v}_{\mrm{l,s}}} = 0$.

In particular, $\lambda$ vanishes on the eigenspaces in $\Omega^1_{R_\C}$ of $\mc{L}_{\bm{v}_{\mrm{l,s}}}$ corresponding to non-zero eigenvalues. We introduce some more notation. Let $E_\C$ denote the span of all eigenspaces in $R_\C$ of $\mc{L}_{\bm{v}_{\mrm{l,s}}}$ corresponding to eigenvalues with non-zero real part. Define $E := E_\C \cap R$ and $E^n := E \cap I^n$.

$E \subseteq \mcN$.

Let $\mu \in \Spec(\mc{L}_{\bm{v}_{\mrm{l,s}}})$ with $\mrm{Re}(\mu) \neq 0$. Set $E_\mu := \ker(\mc{L}_{\bm{v}_{\mrm{l,s}}} - \mu I) \subseteq R_\C$. Suppose first that $\mu \in \R$. If $\psi \in E_\mu \cap R$ then because $\mc{L}_{\bm{v}}$ leaves the eigenspaces of $\mc{L}_{\bm{v}_{\mrm{l,s}}}$ invariant, the 1-form $\psi d \mc{L}_{\bm{v}}\psi$ is an eigenvector of $\mc{L}_{\bm{v}_{\mrm{l,s}}}$ with non-zero eigenvalue $2\mu$. By lem: lambda_invariant_under_linear_ss_part it follows that $q(\psi) = 0$ and hence $\psi \in \mc{N}$. Thus $E_\mu \subseteq \mc{N}$. Next, suppose that $\mu$ is not real. Then also $\overline{\mu}$ is an eigenvalue of $\mc{L}_{\bm{v}_{\mrm{l,s}}}$. Write $W_\C := E_\mu \oplus E_{\overline{\mu}}$ and $W := W_\C\cap R$. Take $\psi \in W$ arbitrary. Then $\psi = \eta + \overline{\eta}$ for some $\eta \in E_\mu$ (and hence $\overline{\eta} \in E_{\overline{\mu}}$). As $\mu + \overline{\mu} = 2\mrm{Re}(\mu) \neq 0$ and $\mc{L}_{\bm{v}}$ leaves the eigenspaces of $\mc{L}_{\bm{v}_{\mrm{l,s}}}$ invariant, each of the 1-forms $\eta d\mc{L}_{\bm{v}}\eta,\; \eta d\mc{L}_{\bm{v}}\overline{\eta}, \; \overline{\eta} d\mc{L}_{\bm{v}}\eta$ and $\overline{\eta} d\mc{L}_{\bm{v}}\overline{\eta}$ are eigenvectors of $\mc{L}_{\bm{v}_{\mrm{l,s}}}$ with non-zero eigenvalue. Using lem: lambda_invariant_under_linear_ss_part it follows that $q(\psi) = 0$ and hence $\psi \in \mc{N}$. Thus $W \subseteq \mc{N}$. As $\mc{N}$ is a linear subspace, we have shown $E \subseteq \mcN$.

$RE \subseteq R\mc{L}_{\bm{v}}E$.

Write $J := R\mc{L}_{\bm{v}}E$. As $J$ is an ideal in $R$ it suffices to show $E \subseteq J$. We claim that $E^n \subseteq J + E^{n+1}$ for every $n \in \N_{\geq 0}$. Indeed, take $\psi \in E^n$. As $\mc{L}_{\bm{v}_{\mrm{l}}}$ is invertible on $E^n$ (which is true because $\mc{L}_{\bm{v}_{\mrm{l}}}$ is invertible on every finite-dimensional and $\mc{L}_{\bm{v}_{\mrm{l}}}$-invariant subspace $E \cap P^k(V) \subseteq E$), there exists some $\eta \in E^n$ s.t. $\mc{L}_{\bm{v}_{\mrm{l}}} \eta = \psi$. Observe that $\mc{L}_{\bm{v}_{\mrm{ho}}} E \subseteq E$ because $[\mc{L}_{\bm{v}_{\mrm{l,s}}}, \mc{L}_{\bm{v}_{\mrm{ho}}}] = 0$. Also $\mc{L}_{\bm{v}_{\mrm{ho}}} I^n \subseteq I^{n+1}$, since ${\bm{v}_{\mrm{ho}}} \in \X_{I^2}$. Thus $\mc{L}_{\bm{v}_{\mrm{ho}}} E^n \subseteq E^{n+1}$. In particular $\mc{L}_{\bm{v}_{\mrm{ho}}} \eta \in E^{n+1}$. Then $\psi =  \mc{L}_{\bm{v}_{\mrm{l}}} \eta =  \mc{L}_{\bm{v}}\eta - \mc{L}_{\bm{v}_{\mrm{ho}}}\eta \in J + E^{n+1}$, as required. By induction it follows that $E = E^0 \subseteq J + E^{n}$ for every $n \in \N$. As $\bigcap_{n \in \N}(J + E^{n}) = J$, this implies $E \subseteq J$.

Using lem: center_subspace_in_kernel we obtain $E \subseteq \mcN$. By cor: vanishing_ideal_from_full_quadratic_form, this implies $J\otimes \mfr{k} \subseteq \ker \olpi$, where $J = R\mc{L}_{\bm{v}}E$. By lem: char_of_ideal_when_lin_part_non_deg, we know $RE \subseteq J$. Notice that $E \cap V^\ast = V_{\mrm{c}}^\perp$, so in particular $R V_{\mrm{c}}^\perp \subseteq J$. Thus $RV_{\mrm{c}}^\perp \otimes \mfr{k} \subseteq \ker \olpi$. Notice that $R/(R V_{\mrm{c}}^\perp) \cong \R \llbracket V_{\mrm{c}}^\ast \rrbracket$, because $V_{\mrm{c}}^\ast = V^\ast/V_{\mrm{c}}^\perp$. We conclude that $\olpi$ factors through the quotient $(R \otimes \mfr{k})/(R V_{\mrm{c}}^\perp \otimes \mfr{k}) \cong (\R\llbracket V_{\mrm{c}}^\ast \rrbracket \otimes \mfr{k})$.

§.§ 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 sec: restr_proj_reps.

Define $\sigma_0 := \ev_0 \circ \sigma : \p \to \mfr{k}$ and let ${\bm{v}_{\mrm{l}}} = j^1 \bm{v} : \p \to \gl(V)$ be the linearization of $\bm{v}$ at $0 \in V$. For $p \in \p$, the vector fields $\bm{v}(p)$ splits as $\bm{v}(p) = \vl{p} + \vho{p}$ for some formal vector field $\vho{p} \in \X_{I^2}$ vanishing up to first order at the origin. Let $\vl{p} = \vls{p} + \vln{p}$ be the Jordan decomposition of $\vl{p}$ over $\C$. Let $V_{\mrm{c}}^\C(p)$ denote the span in $V_\C$ of all generalized eigenspaces of $\vl{p}$ corresponding to eigenvalues with zero real part. Set $V_{\mrm{c}}(p) := V_{\mrm{c}}^\C(p) \cap V$. If $\mfr{C} \subseteq \p$ is a subset, define $V_{\mrm{c}}(\mfr{C}) := \bigcap_{p \in \mfr{C}} V_{\mrm{c}}(p)$. Let $V_{\mrm{c}}(\mfr{C})^\perp \subseteq V^\ast$ denote the annihilator of $V_{\mrm{c}}(\mfr{C})$ in $V^\ast$. Let $\Sigma_p \subseteq \C$ denote the additive subsemigroup of $\C$ generated by $\Spec(\vl{p})$. For any continuous projective unitary representation $\olpi$ of $\g\rtimes_D \p$, the set $\mfr{C}(\olpi)$ consists of all points $p \in \p$ for which $\olpi$ is of generalized positive  energy at $p$.

We use thm: general_factorization_result combined with suitable normal form results to prove thm: fact_result_spectral_condition and thm: fact_result_nform_condition.

Let $W$ be a finite-dimensional real vector space and let $W_i \subseteq W$ be a collection of linear subspaces, where $i \in \mc{I}$ for some indexing set $\mc{I}$. Then $\big(\bigcap_{i \in \mc{I}} W_i\big)^\perp = \Span_{i \in \mc{I}}W_i^\perp$.

Notice first that $\bigcap_{i \in \mc{I}} W_i^\perp  = \big[\Span_{i \in \mc{I}} W_i\big]^\perp$. Applying this observation to the subspaces $W_i^\perp\subseteq W^\ast$, we obtain that $\bigcap_{i \in \mc{I}} W_i = \bigcap_{i \in \mc{I}} (W_i^\perp)^\perp = \big[\Span_{i \in \mc{I}} W_i^\perp\big]^\perp$, where we also used that $(W_i^{\perp})^{\perp} \cong W_i$ for any $i\in \mc{I}$. Taking annihilators, this implies $\big(\bigcap_{i \in \mc{I}} W_i\big)^\perp = \Span_{i \in \mc{I}}W_i^\perp$.

Let $p \in \mfr{C}$. By thm: poincare_dulac, there is a formal vector field $\who{p} \in \X_{I^2}$ satisfying $[\vls{p}, \who{p}] = 0$ s.t. $\bm{v}(p)$ is formally equivalent to $\bm{w}(p) := \vl{p} + \who{p}$. If $h \in \Aut(R)\subseteq \Aut(\g)$ is a formal diffeomorphism s.t. $\bm{w}(p) = h.\bm{v}(p)$, then $h$ leaves the constant part $\sigma_0(p)$ of $\sigma(p)$ fixed, so $\ev_0(h.\sigma(p)) = \ev_0\sigma(p) = \sigma_0(p)$. Thus, we may assume that $[\vls{p}, \vho{p}] = 0$ and $\Spec(\ad_{\sigma_0(p)}) \cap \Sigma_p = \emptyset$. By acting with gauge transformations, we may by thm: normalformverticaltwistoned further assume that $\sigma \in R \otimes \mfr{t}$, where $\mfr{t}$ is a maximal torus containing $\sigma_0(p)$. By thm: general_factorization_result, it follows that $RV_{\mrm{c}}(p)^\perp\subseteq \ker \olpi$. The above holds for all $p \in \mfr{C}$, so $\Span_{p \in \mfr{C}}RV_{\mrm{c}}(p)^\perp \subseteq \ker \olpi$. By lem: ann_of_intersection we know $\Span_{p \in \mfr{C}}\big(V_{\mrm{c}}(p)^\perp\big) =  V_{\mrm{c}}(\mfr{C})^\perp$, so that $R/(\Span_{p \in \mfr{C}}RV_{\mrm{c}}(p)^\perp) \cong \R\llbracket V_{\mrm{c}}(\mfr{C})^\ast \rrbracket$.

By thm: general_factorization_result it follows that $\Span_{p \in \mfr{C}}RV_{\mrm{c}}(p)^\perp = RV_{\mrm{c}}(\mfr{C})^\perp \subseteq \ker \olpi$.

§.§ 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 $\bm{v}$ and $\sigma$ are available (see thm: herman_linearization and thm: normalformverticaltwistss). In particular, $\bm{v} : \p \to \X_I^{\op}$ is always formally equivalent to its linearization $\bm{v}_{\mrm{l}}$ at $0 \in V$. Similarly, by thm: normalformverticaltwistss, the vertical twist $\sigma: \p \to \g$ is gauge-equivalent to some Lie algebra homomorphism $\sigma_0 : \p \to \mfr{k}$. In particular, if $\p$ is not compact then we may and do assume that $\sigma =  0$ by acting with gauge transformations if necessary, for in that case there are no homomorphisms $\p \to \mfr{k}$ (because $\mfr{k}$ is compact, see lem: no_homs_noncompact_to_compact). Combined with thm: fact_result_nform_condition we immediately obtain thm: factorization_result_semisimple below, where $V_{\mrm{c}}(\mfr{C}) := \bigcap_{p \in \mfr{C}}V_{\mrm{c}}(p)$.

Let $\p = \mfr{k}_0 \oplus \p_0$ be a Cartan decomposition of $\p$, so that $\mfr{k}_0$ and $\p_0$ are the $+1$ and $-1$ eigenspaces of a Cartan-involution $\theta$, respectively <cit.>. Let $\mfr{a}_0 \subseteq \p_0$ be a maximal Abelian subalgebra of $\p_0$. According to the Iwasawa decomposition <cit.>, $\p$ decomposes as $\p \cong \mfr{k}_0 \oplus \mfr{a}_0 \oplus \n_0$, where $\n_0 \subseteq \p$ is nilpotent. For $p \in \p$ we write $p = p_e + p_h + p_n$ for the corresponding decomposition of $p$, where $p_e \in \mfr{k}_0, p_h \in \mfr{a}_0$ and $p_n \in \n_0$. Then $\Spec(\ad_{p_e}) \subseteq i\R$, $\Spec(\ad_{p_h}) \subseteq \R$ and $\ad_{p_n}$ is nilpotent <cit.>. Moreover, $\mfr{a}_0$ is contained in a Cartan subalgebra of $\p_0$ <cit.>.

Suppose that $\p$ is simple and that the $\p$-representation $\bm{v}_{\mrm{l}}$ on $V$ is non-trivial and irreducible. Let $\mfr{C}\subseteq \p$ be an $\Ad_P$-invariant convex cone and let $V_{\mrm{c}}(\mfr{C}) := \bigcap_{p \in \mfr{C}}V_{\mrm{c}}(p)$. Assume that $\mfr{C}$ contains some non-zero $p_h \in \mfr{a}_0$. Then $V_{\mrm{c}}(\mfr{C}) = \{0\}$.

Notice first that as $P$ is $1$-connected, the $\p$-action $\bm{v}_{\mrm{l}} : \p \to \gl(V)$ integrates to a continuous representation of $P$ on $V$. As $\mfr{C}$ is $\Ad_P$-invariant, the subspace $V_{\mrm{c}}(\mfr{C})$ is $P$-invariant. Thus either $V_{\mrm{c}}(\mfr{C}) = \{0\}$ or $V_{\mrm{c}}(\mfr{C}) = V$, so it suffices to show $V_{\mrm{c}}(\mfr{C}) \neq V$. By assumption $p_h \neq 0$. In view of Cartan's unitary trick, see e.g. <cit.>, the image of elements in $\mfr{a}_0$ in any finite-dimensional representation are semisimple and have real spectrum. Thus $\Spec(\vl{p_h}) \subseteq \R$. As $\p$ is simple and $\bm{v}_{\mrm{l}}$ is a non-trivial $\p$-representation by assumption, it follows that $\bm{v}_{\mrm{l}}$ is injective. As $\vl{p_h}\in \gl(V)$ is semisimple, there exists $0\neq v \in V$ s.t. $\vl{p_h}v = \mu v$ for some $0 \neq \mu \in \R$. Thus $0\neq v \notin V_{\mrm{c}}(\mfr{C})$. Hence $V_{\mrm{c}}(\mfr{C}) \neq V$ and so $V_{\mrm{c}}(\mfr{C}) = \{0\}$.

Notice that if $\p$ is simple and $\mcC$ is a closed $\Ad_P$-invariant convex cone which is not pointed, then $\mcC \cap -\mcC = \p$ and hence $\mcC = \p$.

The edge $\mfr{e} := \overline{\mfr{C}} \cap -\overline{\mfr{C}}$ of the closure $\overline{\mfr{C}}$ of $\mfr{C}$ is an ideal in $\p$. Assume that $\mfr{C}$ is not pointed. Then neither is $\overline{\mfr{C}}$. As $\p$ is simple, it follows that $\mfr{e} = \p$ and hence $\overline{\mfr{C}} = \p$. Thus $\mfr{C}$ is a dense convex cone in the finite-dimensional real vector space $\p$, which implies that $\mfr{C} = \p$. As $\p$ is non-compact, it contains some hyperbolic element. Thus, so does $\mfr{C}$. By prop: cone_must_not_contain_hyperbolics it follows that $V_{\mrm{c}}(\mfr{C}) = \{0\}$ and hence thm: factorization_result_semisimple implies that $\olpi$ factors through $\mfr{k}$.

Thus if $\mc{C}$ 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 $\mcC \subseteq \p$ other than those which factor through $\mfr{k}$. On the other hand, we know by <cit.> 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 $\dim(\z(\mfr{k}_0)) = 1$, where $\p = \mfr{k}_0 \oplus \p_0$ is a Cartan decomposition of $\p$ and where $\mfr{k}_0$ 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.

Suppose that $P$ is a non-compact simple connected Lie group. If $(\sigma, \mcH_\sigma)$ is a unitary $P$-representation that is norm-continuous, then $\sigma$ is trivial.

As $\p$ is simple, $d\sigma$ is either injective or trivial. Assume that $d\sigma$ is not trivial. Let $\p = \mfr{k}_0 \oplus \mfr{a}_0 \oplus \n_0$ be the Iwasawa decomposition of $\p$. Take $x \in \mfr{a}_0$. Then $\ad_x$ is semisimple and $\Spec(\ad_x) \subseteq \R$. As $\sigma$ is unitary and $d\sigma$ is injective, $z \mapsto \norm{d\sigma(z)}_{\B(\mcH)}$ defines a $P$-invariant norm on $\p$. With respect to this norm, $e^{t \ad_x}$ is an isometry on $\p$ for every $t \in \R$. As $\ad_x$ is semisimple, it follows that $\Spec(\ad_x) \subseteq i\R$. So $\Spec(\ad_x) \subseteq \R \cap i\R = \{0\}$ and hence $\ad_x = 0$. Since $\p$ has trivial center it follows that $x = 0$. So $\mfr{a}_0 = \{0\}$ and hence $\p$ is compact. But $P$ is non-compact by assumption. So $d\sigma$ must be trivial. As $P$ is connected, it follows that $\sigma$ is trivial.

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 $\overline{\rho}$ be a continuous projective unitary representation of $G$ which is of positive energy at $\mcC := \p$. Then $\restr{\overline{\rho}}{G}$ factors through $K$.

By thm: always_factor_through_fin_jet it suffices to consider the case where $\overline{\rho}$ factors through $G_k$ for some $k\in \N$. From Whitehead's Second Lemma, <cit.>, we know that $H^2(\p, \R) = \{0\}$. Using in addition that $P$ is simply connected, it follows that $\restr{\overline{\rho}}{P}$ lifts to a continuous unitary representation $\sigma : P \to \U(\mcH_\rho)$ of $P$, so that $\overline{\rho}(p) = [\sigma(p)]$ in $\PU(\mcH_\rho)$ for all $p \in P$. By lem: full_cone_then_norm_continuous, the fact that $\restr{\overline{\rho}}{P}$ is of p.e. at $\mcC = \p$ implies that $\sigma$ is norm-continuous. It follows from lem: non_compact_norm_continuous_trivial that $\sigma$ is trivial. Thus $\overline{\rho}(\alpha_p(g)) = \overline{\rho}(g)$ for all $g \in G$ and $p \in P$. It follows that $d\overline{\rho}$ 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 $I\otimes \mfr{k}$. Thus $I \otimes \mfr{k} \subseteq \ker d\overline{\rho}$. This implies that $\restr{\overline{\rho}}{G_k}$ factors through $K$.

The following provides a simple example of a projective p.e. representation $\overline{\rho}$ of $G_1 \rtimes P$ s.t. $\restr{\overline{\rho}}{G_1}$ does not factor through $K$.

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 $\g = R \otimes \mfr{k}$ is given by $D(p) = -\mc{L}_{\bm{v}(p)}$. In this case the generator $p_0$ of rotations generates the unique (up to a sign) non-trivial closed, pointed, $P$-invariant convex cone $\mcC$ in $\p$. Explicitly, $\bm{v}(p_0) = y\partial_x - x\partial_y$. Let us construct a non-trivial continuous projective unitary representation of $G_1 \rtimes P \cong (V^\ast \otimes \mfr{k}) \rtimes (K \times P)$ that is of p.e. at the cone $\mcC \subseteq \p$. Write $W := V^\ast \otimes \mfr{k}$.

We begin by specifying a suitable $2$-cocycle on $V^\ast \otimes \mfr{k} \subseteq \g_1$. Notice that $\big(\bigwedge^{2} V\big)^\p \cong \R$ is one-dimensional. Let $0\neq \lambda \in \big(\bigwedge^{2}V\big)^\p$ and consider it as a $\p$-invariant bilinear map $V^\ast \times V^\ast \to \R$. To be consistent with prop: characterization_of_cts_second_cohom, let us write $\lambda(fdg)$ instead of $\lambda(f,g)$ for $f,g \in V^\ast$. Let $x,y \in V^\ast$ be the usual basis of $V^\ast$. Then $\lambda$ is fully specified by the number $\lambda(ydx)$. If $\lambda(ydx) > 0$, then the quadratic form $q(v) := \lambda(\mc{L}_{\bm{v}(p_0)}v dv)$ is positive-definite, because $q(ax+by) = (a^2+b^2)\lambda(ydx)$ for $a,b \in \R$. Let $\omega$ be the unique symplectic form on $W$ satisfying $\omega(vX, wY) := \lambda(vdw)\kappa(X,Y)$ for $X,Y \in \mfr{k}$ and $v,w \in V^\ast$. Then $\omega(D(p_0)\xi, \xi) \geq 0$ for every $\xi \in W$ (recalling that $\kappa$ is negative definite). Let $\mrm{H}(W, \omega)$ be the corresponding Heisenberg group. Let $L_\pm$ be the $\pm i$-eigenspaces in $W_\C$ of the complex structure $\mc{J} := D(p_0)$ on $W_\C$, so that $W_\C = L_- \oplus L_+$. The $\p$-invariance of $\lambda$ ensures that $\mc{J}^\ast \omega = \omega$. Indeed, extend $\omega$ $\C$-bilinearly to $W_\C$. As $\lambda$ is $\p$-invariant, it follows that $\omega(\mc{J}\xi, \eta) + \omega(\xi, \mc{J}\eta) = 0$ for all $\xi, \eta \in W_\C$, which implies that $L_\pm \subseteq W_\C$ are $\mc{J}$-invariant Lagrangian subspaces for $\omega$. Then $\mc{J}^\ast \omega = \omega$ follows from $\mc{J}^\ast \omega(w_+, w_-) = \omega(iw_+, -iw_-) = \omega(w_+, w_-)$ for $w_\pm \in L_\pm$. Notice further that $\omega(\mc{J}\xi, \xi) \geq 0$ holds for all $\xi \in W$, by construction. Equip $L_+$ with the positive definite hermitian form defined by $\langle v, w\rangle_{L_+} := -2i \omega(\overline{v}, w)$ for $v,w \in L_+$. For each $n \in \N$, equip the symmetric algebra $S^n(L_+)$ with the inner product satisfying

\[ \langle v_1\cdots v_n, w_1\cdots w_n\rangle := \sum_{\sigma \in S_n}\prod_{k=1}^n \langle v_{\sigma_k}, w_k\rangle_{L_+}, \qquad v_k, w_k \in L_+.\]

Let $\F := \overline{S^\bullet(L_+)}$ be the Hilbert space completion, where $S^\bullet(L_+) = \bigoplus_{n=0}^\infty S^n(L_+)$. The metaplectic representation $\rho$ of $\mrm{H}(W, \omega) \rtimes \Mp(W, \omega)$, with $\rho(z) = zI$ on the central $\T$ component, can be realized on the Fock space $\F$, where $\Mp(W, \omega)$ denotes the metaplectic group <cit.>. Notice that $\SL(2,\R) \hookrightarrow \Sp(W, \omega)$ because $\lambda$ is $\p$-invariant. By pulling back the metaplectic representation we obtain a continuous unitary representation of $\mrm{H}(W, \omega) \rtimes P$ which is of p.e. at $\mc{C}$ and does not factor through $K$.

§ FROM GERMS TO JETS

Let $\mc{K} \to M$ be locally trivial bundle of Lie groups with typical fiber a finite-dimensional Lie group $G$ with Lie algebra $\g$. Write $\mfr{K} \to M$ for the corresponding Lie algebra bundle. The following justifies the claim made in sec: introduction that any continuous projective unitary representation of $\Gamma_{\mrm{c}}(\mc{K})$ which factors through the germs at a point $a \in M$ actually factors through the $\infty$-jets $J^\infty_a(\mc{K})$ at $a \in M$. The group $\Gamma_{\mrm{c}}(\mc{K})$ is a locally exponential Lie group modeled on the LF-Lie algebra $\Gamma_{\mrm{c}}(\mfr{K})$ <cit.>.

Let $U \subseteq \R^d$ be an open neighborhood of the origin. Let $C^\infty_{\mrm{flat}}(U)$ denote the kernel of the $\infty$-jet projection

\[j^\infty_0 : C_c^\infty(U) \to J_0^\infty(C^\infty_{\mrm{c}}(U)) \cong \R \llbracket x_1, \cdots, x_d \rrbracket\]

at $0 \in U$. In the following we show the known fact that the closure $C^\infty_{\mrm{c}}(U\setminus \{0\})$ in $C^\infty(U)$ is $C^\infty_{\mrm{flat}}(U)$. As a consequence, we deduce that if a continuous projective unitary representation of the Lie algebra $\Gamma_{\mrm{c}}(\mfr{K})$ factors through the germs at a point $a \in M$, then it factors through the $\infty$-jets $J^\infty_a(\mfr{K})$ at $a \in M$. In turn, this implies a group level-analogue.

If $K \subset U$ is a compact set, we write $C^\infty_K(U)$ for the subspace of $C^\infty(U)$ consisting of functions on $U$ with support in $K$. Then $C^\infty_K(U)$ is the projective limit $C^\infty_K(U) = \varprojlim_n C_K^n(U)$ of the Banach spaces $C_K^n(U)$, which we equip with the norm $\norm{f}_{C_K^n(U)} := \sup_{|k|\leq n} \norm{D^k f}_{C_K(U)}$, where the supremum runs over all multi-indices $k \in \N_{\geq 0}^d$ with $|k| \leq n$. Then $C^\infty_{\mrm{c}}(U) := \varinjlim C^\infty_K(U)$ is the corresponding locally convex inductive limit. See e.g. <cit.> for a description of this topology. For $r > 0$, write $B_r := \set{x \in \R^d \st \norm{x} \leq r}$ for the closed ball centered at $0 \in \R^d$ with radius $r$.

The closure of $C^\infty_{\mrm{c}}(U\setminus\{0\})$ in $C_c^\infty(U)$ is $C^\infty_{\mrm{flat}}(U)$.

As $C^\infty_{\mrm{flat}}(U) \subseteq C_c^\infty(U)$ is closed and $C^\infty_{\mrm{c}}(U\setminus\{0\}) \subseteq C^\infty_{\mrm{flat}}(U)$, it follows that $\overline{C^\infty_{\mrm{c}}(U\setminus\{0\})} \subseteq C^\infty_{\mrm{flat}}(U)$. It remains to show the reverse inclusion. Let $f \in C^\infty_{\mrm{flat}}(U)\subseteq C_c^\infty(U)$. We show $f \in \overline{C^\infty_{\mrm{c}}(U\setminus\{0\})}$. Let $K_0 \subseteq M$ be a relatively compact open subset such that $\supp f \subseteq K_0$. Set $K := \overline{K_0}$. We may assume that $0 \in K_0$, for otherwise $f \in C^\infty_{\mrm{c}}(U\setminus\{0\})$ and we are done. By <cit.>, we can find constants $C_k > 0$ for $k \in \N^d_{\geq 0}$, depending only on $k$, such that for any $0 < r < 1$ with $B_{2r} \subseteq K_0$, there exists a smooth function $\psi_r \in C^\infty(\R^d)$ s.t. $\psi_r \geq 0$, $\restr{\psi_r}{B_r} = 0$, $\restr{\psi_r}{(\R^d\setminus B_{2r})} = 1$ and $\sup_{x \in \R^d}|D^k\psi_r(x)| \leq C_k r^{-|k|}$ for every $k \in \N^{d}_{\geq 0}$. In particular $f \psi_r \in C^\infty_{\mrm{c}}(U\setminus\{0\})$ and $\supp f \psi_r \subseteq K$. Moreover, observe that $\supp (1-\psi_r) \subseteq B_{2r}$ and $\norm{(1-\psi_r)}_{C^n_{B_{2r}}(\R^d)} \lesssim r^{-n}$ for some constant depending on $n \in \N_{\geq 0}$, where we used that $0<r<1$. On the other hand, suppose that $\alpha \in \N_{\geq 0}^d$ is a multi-index. Since $j^\infty_0(D^\alpha f) = 0$, it follows from Taylor's Theorem that $\norm{D^\alpha f}_{C(B_{2r})} \lesssim r^{l}$ for arbitrary $l \in \N_{\geq 0}$, with a constant depending on $f$, $\alpha$ and $l$ but not on $r$. Thus $\norm{f}_{C^n(B_{2r})} \lesssim r^{l}$ for arbitrary $n,l \in \N_{\geq 0}$. In particular $\norm{f}_{C^n(B_{2r})} \lesssim r^{n+1}$. Combining the previous observations, we obtain that

\[\norm{f - f \psi_r}_{C^n(K)} = \norm{f(1-\psi_r)}_{C^n(K)} = \norm{f(1-\psi_r)}_{C^n(B_{2r})} \lesssim \norm{f}_{C^n(B_{2r})} \norm{(1-\psi_r)}_{C^n(B_{2r})} \lesssim r,\]

the constants depending only on $f$ and $n$ but not on $r$. This shows that $f \psi_r \to f$ in $C^\infty_K(U)$ as $r\to 0$. Thus $f \psi_r \to f$ in $C^\infty_{\mrm{c}}(U)$. Since $\psi_r f \in C^\infty_{\mrm{c}}(U\setminus\{0\})$ for every $r$, we conclude that $f \in \overline{C^\infty_{\mrm{c}}(U\setminus\{0\})}$.

If $a \in M$, define the spaces of smooth section of $\mc{K}$ and $\mfr{K}$ which are flat at $a \in M$:

\begin{align*}
\Gamma_{\mrm{flat}(a)}(\mc{K}) := \ker\bigg(j^\infty_a : \Gamma_{\mrm{c}}(\mc{K}) \to J^\infty_a(\mc{K})\bigg),\qquad
\Gamma_{\mrm{flat}(a)}(\mfr{K}) := \ker\bigg(j^\infty_a : \Gamma_{\mrm{c}}(\mfr{K}) \to J^\infty_a(\mfr{K})\bigg).
\end{align*}

prop: two_tops_on_inf_jets_agree below clarifies the apparent ambiguity in the topology on $J^\infty_a(\mfr{K})$, for which two candidates are available.

Let $a \in M$. The projective limit topology on $J_a^\infty(\mfr{K}):= \varprojlim_kJ^k(\mfr{K})$ coincides with the quotient topology obtained from $J_a^\infty(\mfr{K}) \cong \Gamma_{\mrm{c}}(\mfr{K})/\Gamma_{\mrm{flat}(a)}(\mfr{K})$.

The continuous $k$-jet projections $j_a^k : \Gamma_{\mrm{c}}(\mfr{K}) \to J_a^k(\mfr{K})$ at $a \in M$ all descend to continuous maps $\Gamma_{\mrm{c}}(\mfr{K})/\Gamma_{\mrm{flat}(a)}(\mfr{K}) \to J_a^k(\mfr{K})$. By the universal property of the projective limit, they induce a continuous map $\Phi: \Gamma_{\mrm{c}}(\mfr{K})/\Gamma_{\mrm{flat}(a)}(\mfr{K}) \to J_a^\infty(\mfr{K})$. Using Borel's Lemma <cit.>, it is not hard to check that this map is bijective. It remains to show it is an open map, which follows immediately from the Open Mapping Theorem <cit.> because $\Gamma_{\mrm{c}}(\mfr{K})/\Gamma_{\mrm{flat}(a)}(\mfr{K})$ and $J^\infty_a(\mfr{K})$ are both Fréchet spaces and $\Phi$ is bijective and continuous.

Let $a \in M$.

* The closure of $\Gamma_{\mrm{c}}(M\setminus\{a\}; \mfr{K})$ in $\Gamma_{\mrm{c}}(M; \mfr{K})$ is $\Gamma_{\mrm{flat}(a)}(\mfr{K})$.

* The closure of $\Gamma_{\mrm{c}}(M\setminus\{a\}; \mc{K})$ in $\Gamma_{\mrm{c}}(M; \mc{K})$ is $\Gamma_{\mrm{flat}(a)}(\mc{K})$.

By a partition of unity argument, we may assume that the bundle $\mfr{K} \to M$ is trivial, that $M \subseteq \R^d$ is open neighborhood of $0 \in \R^d$ and that $a = 0$. Then $\Gamma_{\mrm{c}}(M; \mfr{K}) \cong C^\infty_{\mrm{c}}(M; \mfr{k})$. The claim now follows from lem: closure of g_c_is_g_flat. Notice for the second assertion that $\Gamma_{\mrm{flat}(a)}(M; \mc{K})$ is a locally exponential, being an embedded closed Lie subgroup of the locally exponential Lie group $\Gamma_{\mrm{c}}(M; \mc{K})$. The result is then immediate from the previous point.

Let $a \in M$.

* Let $\olpi : \Gamma_{\mrm{c}}(M; \mfr{K}) \to \mc{L}^\dagger(\mcD)$ be a continuous projective unitary representation on the pre-Hilbert space $\mcD$.  Assume that $\olpi$ vanishes on $\Gamma_{\mrm{c}}(M \setminus\{a\}; \mfr{K})$. Then $\olpi$ factors continuously through $J^\infty_a(\mfr{K})$.

* Let $\overline{\rho} : \Gamma_{\mrm{c}}(M; \mc{K}) \to \PU(\mcH)$ be a continuous projective unitary representation of $\Gamma_{\mrm{c}}(M; \mc{K})$. Assume that $\overline{\rho}$ vanishes on $\Gamma_{\mrm{c}}(M \setminus \{a\}; \mc{K})$. Then $\overline{\rho}$ factors through $\Gamma_{\mrm{flat}(a)}(M; \mc{K})$.

For the first point, notice by continuity that $\olpi$ must also vanish on the closure of $\Gamma_{\mrm{c}}(M \setminus\{a\}; \mfr{K})$ in $\Gamma_{\mrm{c}}(M;\mfr{K})$, which by prop: closure_flat equals $\Gamma_{\mrm{flat}(a)}(M;\mfr{K})$. Thus $\Gamma_{\mrm{c}}(M;\mfr{K})$ factors continuously through the quotient space $J^\infty_a(\mfr{K}) \cong \Gamma_{\mrm{c}}(M;\mfr{K})/\Gamma_{\mrm{flat}(a)}(M;\mfr{K})$, where prop: two_tops_on_inf_jets_agree was used. The second point is proven similarly using prop: closure_flat.


# A Comparative Analysis of Microrings Based Incoherent Photonic GEMM
Accelerators

Sairam Sri Vatsavai, Venkata Sai Praneeth Karempudi, Oluwaseun Adewunmi Alo,
and Ishan Thakkar  Department of Electrical and Computer Engineering,
University of Kentucky, Lexington, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

Several analog photonic architectures based on microring resonators (MRRs)
have been proposed to accelerate general matrix-matrix multiplications (GEMMs)
that compose deep neural networks with exceptional throughput and energy
efficiency. To implement GEMM functions, these MRR-based architectures, in
general, manipulate optical signals in five different ways: (i) Splitting
(copying) of multiple optical signals to achieve a certain fan-out, (ii)
Aggregation (multiplexing) of multiple optical signals to achieve a certain
fan-in, (iii) Modulation of optical signals to imprint input values onto
analog signal amplitude, (iv) Weighting of modulated optical signals to
achieve analog input-weight multiplication, (v) Summation of optical signals.
Prior accelerators undertake the first four ways of signal manipulation in an
arbitrary order ignoring the possible impact of the order of these
manipulations on their performance. In this paper, we conduct a detailed
analysis of accelerator organizations with three different orders of these
manipulations: (1) Modulation-Aggregation-Splitting-Weighting (MASW), (2)
Aggregation-Splitting-Modulation-Weighting (ASMW), and (3) Splitting-
Modulation-Weighting-Aggregation (SMWA). We show that these organizations
affect the crosstalk noise and optical signal losses in different magnitudes,
which renders these organizations with different levels of processing
parallelism at the circuit level, and different magnitudes of throughput and
energy-area efficiency at the system level. Our evaluation results for four
CNN models show that the SMWA organization achieves up to 4.4$\times$,
5$\times$, and 5.2$\times$ better throughput, energy efficiency, and area
efficiency, respectively, compared to the ASMW and MASW organizations on
average.

##  I Introduction

Deep Neural Networks (DNNs) achieve high inference accuracy, which has
revolutionized their use in various artificial intelligence tasks, such as
image recognition, language translation, and autonomous driving [1, 2].
However, DNNs are computationally intensive because they are typically
composed of inherently abundant linear functions such as general matrix-matrix
multiplication (GEMM). The need to tackle the rapidly increasing computing
demands of the abundant GEMM functions of DNNs has pushed for highly
customized hardware GEMM accelerators [3, 4]. Among GEMM accelerators in the
literature, silicon-photonic GEMM accelerators have shown great promise to
provide unparalleled parallelism, ultra-low latency, and high energy
efficiency [5, 6, 7, 8, 9]. A silicon-photonic GEMM accelerator typically
consists of multiple Dot Product Units (DPUs) that perform a total of M dot
product operations in parallel of N-size each. Several DPU-based optical GEMM
accelerators have been proposed in prior works. Among them, the Microring
Resonator (MRR)-enabled analog DPU-based accelerators (e.g., [5, 7, 6, 10, 9,
11]) have shown disruptive performance and energy efficiencies, due to the
MRRs’ compact footprint, low dynamic power consumption, and compatibility with
cascaded dense-wavelength-division multiplexing (DWDM).

A typical DPU employs five blocks of optical components to manipulate optical
signals in five different ways. (1) A splitting block for splitting (copying)
N optical signals in M ways to achieve a fan-out of M per optical signal; (2)
An aggregation block for aggregation (multiplexing) of N optical signals per
waveguide to achieve a fan-in of N per waveguide; (3) A modulation block for
modulation of optical signals to imprint input values onto them; (4) A
weighting block for weighting of modulated optical signals to achieve analog
input-weight multiplication; (5) A summation block to perform summation of
optical signals. Prior accelerators arrange these optical signal manipulation
blocks within a DPU in an arbitrary order, resulting in different DPU
organizations. Different DPU organizations incur different severity levels of
various optical crosstalk effects and signal losses, causing different amounts
of optical power penalty across different DPU organizations. This variation in
optical power penalty causes different DPU organizations to achieve different
values of N (fan-in degree/DPE size) and M (fan-out degree/count of parallel
DPEs). This is because the achievable peak values of N and M highly depend on
the available optical power budget in the DPU, which in turn is determined by
the incurred power penalty in the DPU [12]. It can be intuitively hypothesized
that different values of N and M would render different DPU organizations with
different levels of processing parallelism at the circuit level, and different
magnitudes of throughput and energy-area efficiency at the system level.
However, no prior work has tested this hypothesis. We address this shortcoming
in this paper.

Our key contributions in this paper are summarized below.

  * •

We classify the DPU organizations from prior work into three categories,
namely ASMW, MASW, and SMWA;

  * •

We analyze and discuss the impact of different DPU organizations on various
optical crosstalk effects and signal losses;

  * •

We perform a comparative analysis of the impact of different DPU organizations
on the scalability of achievable N and M values at different bit precision
values;

  * •

We implement and evaluate ASWM, MASW, and SMWA organizations at the system-
level with our in-house simulator, and report the performance in terms of
throughput (FPS), energy-efficiency (FPS/W), and area-efficiency (FPS/W/mm2),
for the inferences of four CNNs.

##  II Preliminaries

###  II-A Processing of CNNs on Hardware Accelerators

In CNNs, the major computational requirement arises from convolutional layers.
These layers involve convolution operations that can be converted to General
Matrix-Matrix Multiplication (GEMM) operations using the Toeplitz matrix or
the im2col transformation [4, 13]. As shown in Fig. 1, the input feature map
(Fmap) belonging to a convolution layer is unfolded into the matrix I. The
weight filters of the convolution layer are flattened and stacked to form the
weight matrix (W). The GEMM operation between I and W gives the resultant
output matrix (O). On conventional CPUs/GPUs, GEMM operations are mapped and
executed using basic linear algebra subprograms (BLAS) or Cuda BLAS (cuBLAS)
[14, 15]. However, conventional CPUs/GPUs cannot efficiently meet the
exponentially growing computational demand of modern CNNs. To meet this
demand, both industry and academia have proposed various dedicated GEMM
accelerators [16, 9, 7, 17], tailored to process CNNs with better performance
and energy efficiency.

###  II-B Related Work on Optical GEMM Accelerators

To accelerate CNN inferences with low latency and low energy consumption,
prior work has demonstrated various GEMM accelerators based on photonic
integrated circuits (PICs) (e.g., [5, 9, 7, 18, 19]). Typically, PIC-based
GEMM accelerators consist of multiple dot product units (DPUs), and each DPU
can perform a GEMM operation on multiple constituent dop product elements
(DPEs) as parallel dot product operations among the rows of the matrix I and
the columns of the matrix W. Some accelerators implement digital DPUs (e.g.,
[10, 17, 19]), whereas some others employ analog DPUs (e.g., [5, 9, 7, 12]).
Different DPU implementations employ MRRs (e.g., [5, 7, 9, 20, 19]) or MZIs
(e.g., [18, 21]) or both (e.g., [10], [17]). Among these, the accelerators
based on MRRs-enabled incoherent DPUs achieve better scalability and lower
footprint, because they use PICs that are based on compact MRRs [7], unlike
the coherent DPUs that use PICs based on bulky MZIs. Various state-of-the-art
PIC-based optical GEMM accelerators are well discussed in survey papers [22,
23, 24].

Figure 1: Convolution operation at a convolution layer with two weight filters
and one input feature map (Fmap) having two channels is transformed into a
GEMM operation between input matrix I and weight matrix W.

##  III Organizations of MRR-based GEMM Accelerators

S. S. Vatsavai et al. in [12] categorized the organizations of optical MRR-
based analog DPUs from prior works into two groups: Aggregate-Modulate-
Modulate (AMM) and Modulate-Aggregate-Modulate (MAM) [12]. Here, the term
”aggregate” refers to the aggregation of multiple wavelength channels into a
single photonic waveguide through wavelength division-multiplexing (WDM). The
first ’Modulate’ refers to the modulation of optical wavelength channels with
input values, and the second ’Modulate’ refers to the modulation (weighting)
of input-modulated optical wavelength signals with weight values. This
categorization classifies prior MRR-based DPUs into AMM [7, 25, 9] and MAM [5,
26, 27] classes. However, this categorization does not consider the spectrally
hitless DPU organization proposed in [8]. In this paper, we bridge this gap
and also improve the comprehensibility of classification categories based on
the order of various optical channel manipulation blocks present in MRR-based
GEMM accelerators. The details of these manipulation blocks and their
different organizations are discussed next.

Figure 2: (a) Common optical signal manipulation blocks found in optical DPUs.
Illustration of common incoherent photonic DPU organizations; (b) AMSW DPU,
(c) MASW DPU, and (d) SMWA DPU.

###  III-A Description of Various Blocks that Maniplate Optical Channels in
MRR-based GEMM Accelerators

Every DPU in an optical GEMM accelerator employs a total of N laser diodes
that generate N optical wavelength channels ($\lambda_{1}-\lambda_{N}$). These
optical wavelength channels are manipulated by five different blocks in the
DPU, as shown in Fig. 2(a). (i) The splitting block ( S) splits the optical
power of the N optical wavelength channels in M copies, with each copy
supporting one DPE. Thus, a total of M DPEs are supported. (ii) Aggregation
block ( A) multiplexes N wavelength channels into a single waveguide within a
DPE to achieve fan-in of N to send N wavelength channels concurrently toward a
balanced photodetector (BPD). (iii) Modulation block M modulates N or
N$\times$M wavelength channels to imprint a sequence of input values on each
of the channels. These input-sequence-imprinted wavelength channels are
referred to as optical signals. Each optical signal will thus be a temporal
train of optical symbols, with each symbol representing a value from the input
sequence. (iv) Weighting block ( W) applies weighting to the input-imprinted
wavelength channels (i.e., optical signals). After weighting, each symbol
carried in an optical signal represents the product of an input value and a
weight value. Therefore, each such weighted optical signal is referred to as
an optical product signal, with each temporal symbol of the signal
representing a product value. (v) Summation block ( $\Sigma$) employs
incoherent superposition at each BPD to perform symbol-wise summation of the N
optical product signals that are sent to the BPD. The BPD generates a
resultant photocurrent signal, each temporal symbol of which provides a dot
product of size N (i.e., summation of N product symbols/values). This
photocurrent signal is fed to a transimpedance amplifier, followed by an
analog-to-digital converter, to obtain the sequence of N-sized dot-product
results in the digital format [12]. The $\Sigma$ block involves a total of M
BPDs corresponding to M DPEs, thereby generating M parallel dot product
results every symbol cycle. In a DPU, generally, the five blocks in the DPU
are arranged in a way so that the $\Sigma$ block comes towards the end of the
input-to-output propagation path of optical channels/signals within the DPU,
and the M block always comes before the W block. However, different DPU
organizations can differ in the way the permutational order of the blocks M,
W, S, and A appear in them. Based on this order of these blocks, we classify
different DPU organizations found in prior works on MRR-based GEMM
accelerators into three categories: ASMW, MASW, and SMWA. Each of these
organizations is explained below.

####  III-A1 ASMW DPU Organization

Fig. 2(b) illustrates a DPU of ASMW organization. First, the A block appears,
wherein N wavelength channels ($\lambda_{1}-\lambda_{N}$) from LDs are
multiplexed into a single waveguide. Then, in the S block, the power of each
wavelength channel is equally split into M waveguides to generate a total of
N$\times$M parallel wavelength channels. These wavelength channels then
encounter the M and W blocks in that order. The M block ( W block) employs one
array of N input MRMs (N weight MRRs) per waveguide. At the output of the W
block, a total of N$\times$M optical product signals emerge (N signals per
waveguide in M parallel waveguides), which are sent to the $\Sigma$ block
containing M parallel BPDs.

####  III-A2 MASW DPU Organization

Fig. 2(c) illustrates a DPU of MASW organization. Here, the N optical
wavelength channels generated by LDs are coupled into N parallel waveguides
(one channel per waveguide). Then, the M block appears wherein each of the N
waveguides couples with one input MRM. These N input MRMs in the M block
generate N optical signals, which are then aggregated into a single waveguide
in the subsequently appearing A block. Then, in the S block, these N optical
signals are split into M waveguides to generate their M copies, with each copy
feeding a DPE. These N$\times$M parallel optical signals then encounter the W
block, which employs an array of N weighting MRRs per waveguide. Consequently,
at the output of the W block, a total of N$\times$M optical product signals
emerge, which are sent to the $\Sigma$ block containing M parallel BPDs.

####  III-A3 SMWA DPU Organization

Fig. 2(d) illustrates a DPU of SMWA organization. In this organization, the S
block appears first which splits the optical power of each of the N wavelength
channels from LDs equally into M separate waveguides by using a total of N
$1\times M$ splitters. Thus, a total of N$\times$M wavelength channels and
N$\times$M waveguides emerge, with each waveguide propagating only a single
wavelength channel. Then, the M and W blocks appear in that order, with both
blocks containing a single MRM-MRR pair coupled to each of the N$\times$M
waveguides. At the output of the W block, a total of N$\times$M optical
product signals emerge (each signal in a dedicated waveguide), which are sent
to the A block. There, a total of M N$\times$1 multiplexers are used to
aggregate the optical product signals in a total of M pairs of aggregation
lanes (each pair propagating N optical product signals). These M pairs of
aggregation lanes feed the $\Sigma$ block containing M parallel BPDs.Table I
reports the classification of prior MRR/MZI-based GEMM accelerators based on
their DPU organization.

TABLE I:  Classification of prior MRR/MZI-based analog accelerators based on
their DPU organizations.

DPU Organization | MRR-based DPU Architectures
---|---
ASMW | Crosslight[9], DEAPCNN [7], Robin [25], RAMM [12]
MASW | Holylight [5], Yang [26], Al-Qadasi[27], PCNNA [28], RMAM [12]
SMWA | Hitless [8], ADEPT [18], Albireo[29]

###  III-B Motivation

The performance achieved by the MRR-based DPUs is largely dependent on three
parameters (1) The maximum achievable value of N (fan-in degree/DPE size).
Often, the achievable value of N for photonic DPUs is less than the dot
product size requirement of GEMM operations corresponding to CNN models [12].
In that case, a DPU breaks the dot product into smaller DPU-compatible chunks
and generates intermediate results known as partial sums (psums). These psums
are later accumulated using electronic reduction networks [30], to generate
the final result. The psum reduction latency and energy consumption are non-
trivial components of overall latency and energy consumption [31]. Therefore,
the value of N plays a crucial role in governing the overall performance of
DPUs. (2) The maximum achievable values of M (fan-out degree/count of parallel
DPEs). The value of M directly decides the parallelism and consequently
achieved throughput by a DPU. (3) The bit precision (B) of input and weight
values. If the supported value of B is less than the precision requirement of
GEMM operations, bit-slicing is applied to input and weight values [32]. Due
to bit-slicing, the overall count of dot product operations increases,
degrading the throughput and energy efficiency [19]. Therefore, the
fundamental driver for achieving high performance from optical DPUslies in
maximizing the values of N, M, and B.

Figure 3: Conceptual breakdown of optical power budget usage and dependency of
DPU size N on supported bit precision B for different values of B={2, 3}-bits
across datarates DR={1, 5} GS/s.

In analog DPUs, a strong trade-off exists among supported values of N, M, and
B [27, 12]. The achievable values of M, N, and B also strongly depend on the
available optical power budget in the DPUs [27, 12]. This is illustrated in
Fig. 3, assuming N=M, which is a common assumption in the literature [27, 12].
For the bit precision B=2, 2B=4 analog optical power levels are required that
consume a large part of the available power budget, and the remaining power
budget is used to support N and power penalty (incurred due to crosstalk
effects and signal losses) in the DPU. As B increases to 3-bits, a larger part
of the power budget is used to support B, and the available power budget to
support N and power penalty further decreases. As a result, the supported
value of N decreases too. Similar impact can be observed when the operating
datarate of DPUs increases (Fig. 3). Low N=M decreases fan-in and fan-out
degrees in the DPU, hampering the achievable throughput and energy efficiency.
No prior work has characterized this impact, which has motivated this work.

##  IV Circuit-Level Comparative Analysis

In this section, we discuss the impact of DPU organization on power penalties
due to various crosstalk effects and optical signal losses.

###  IV-A Impacts on Power Penalty Due to Crosstalk Effects

####  IV-A1 Inter-modulation crosstalk

From Fig. 4(a), inter-modulation crosstalk exists at the M block. Inter-
modulation crosstalk occurs when an MRM unexpectedly modulates a neighboring
wavelength channel instead of its assigned wavelength channel [33, 34].
Therefore, it is possible only if there are multiple wavelength channels
present in the waveguide at a narrow channel spacing when the MRM is
modulating its assigned wavelength channel (Fig. 4(b)). Hence, the necessary
condition for inter-modulation crosstalk to occur is that the M block should
appear after the A block in a DPU organization because, then only, the MRMs
could have accessible neighboring wavelength channels to unexpectedly modulate
them. Therefore, from Table II, inter-modulation crosstalk is present only in
the ASMW DPUs.

Figure 4: (a) Types of losses and power penalties at different optical signal
manipulation blocks of optical DPUs. Illustration of (b) Inter-Modulation
crosstalk at MRM input arrays [33, 34], and (c) Filter crosstalk and signal
truncation at filters [35].

####  IV-A2 Cross-weight penalty

The MRR weight arrays in the W block can exhibit cross-weight penalty [36], as
shown in Fig. 4(a). Due to an insufficient channel gap between the adjacent
optical wavelength channels, a weight MRR could perform undesired weighting on
the neighboring optical wavelength channels leading to cross-weight penalty
[36]. The additional power required to compensate for this crosstalk noise (to
keep the signal-to-noise ratio intact) provides quantification of the cross-
weight penalty [36]. Similar to inter-modulation crosstalk, the necessary
condition for cross-weight penalty to occur is that the W block should appear
after the A block in a DPU organization because, then only, the weight MRRs
could have accessible neighboring wavelength channels to unexpectedly apply
weighting to them. Therefore, from Table II, cross-weight penalty is present
only in the ASMW and MASW DPUs.

####  IV-A3 MRR Filter Penalty

In general, the MRR filter penalty consists of two components [35]: (1)
Optical losses due to signal truncation, and (2) inter-channel crosstalk.
However, the inter-channel crosstalk component manifests only when an MRR
filter is utilized in the demultiplexing configuration to demultiplex a signal
from a waveguide containing multiple signals. Since demultiplexing is not
required in our considered DPU organizations, the inter-channel crosstalk
component remains absent from the MRR filter penalty discussed/analyzed in
this paper. On the other hand, the optical losses due to signal truncation
occur when a modulated optical signal is only partially transmitted through
the MRR filter used as a multiplexer[35]. From Fig. 4(a), this phenomenon
exists at the A block. Each N$\times$1 multiplexer consists of multiple MRR
filters, and when the passband of a filter does not overlap perfectly with the
passband of its corresponding modulated optical signal, it leads to the
truncation of the signal sidelobes. Signal truncation is illustrated in Fig.
4(c) (on the left); when the filter has a high-quality factor (Q), its
passband only partially overlaps with the passband of the modulated optical
signal, resulting in signal truncation. Signal truncation occurs on modulated
optical signals only; it does not occur on unmodulated optical wavelength
channels. This is because unmodulated wavelength channels do that have any
spectral sidelobes. Therefore, the necessary condition for filter penalty to
occur is that the A block should appear after the M block in a DPU
organization. The A block is organized after the M block in both MASW and SMWA
DPUs, therefore signal truncation is present in these organizations (Table
II).

TABLE II: Crosstalks effects present in various DPU organizations. | ASMW | MASW | SMWA
---|---|---|---
Inter-modulation Crosstalk | ✓ | X | X
Cross-Weight Penalty | ✓ | ✓ | X
Signal Truncation at Filters | X | ✓ | ✓

###  IV-B Impacts Due to Optical Signal Losses

####  IV-B1 Through losses

The through losses are the optical power losses experienced by a wavelength
channel as it traverses through MRMs and MRRs that are out-of-resonance to the
wavelength channel in interest but operate on adjacent wavelength channels.
From Fig. 4(a), through losses are present in blocks M, W, and A. The total
amount of through loss experienced by a wavelength channel depends on the
number of devices it traverses before reaching BPD. For instance in Fig. 2(b),
after the splitter, the wavelength $\lambda_{1}$(indicated with the color red)
passes through (N-1) out-of-resonance MRMs and (N-1) out-of-resonance MRRs
before reaching the BPD. The reduction in optical power of $\lambda_{1}$ as it
interacts with these devices is termed as its through loss. The total through
losses can be determined by summing the individual through losses caused by
individual MRRs and MRMs. The through losses vary across DPU organizations as
reported in Table III. For example, from Fig. 2(b), Fig. 2(c), and Fig. 2(d)
the total number out-of-resonance MRMs and MRRs traversed by $\lambda_{1}$ are
2(N-1), N, and 2 in ASMW, MASW, and SMWA DPUs respectively. Therefore,
$\lambda_{1}$ incurs higher through losses in ASMW DPUs than MASW and SMWA DPU
organizations. Similarly, other optical wavelength channels
$\lambda_{2}-\lambda_{N}$ also experience higher through losses in ASMW DPUs.

####  IV-B2 Insertion losses

The insertion losses are the optical power losses encountered by a wavelength
channel while devices such as MRMs, MRRs, and filters operate on it. The total
insertion losses experienced by wavelength channels are approximately the same
across DPU organizations.

####  IV-B3 Waveguide Propagation Losses

Propagation losses are the sum of scattering losses (due to the sidewall
roughness of the waveguide) and absorption losses (due to the material and
free-carrier absorption mechanisms in the waveguide). From Fig.4(a),
propagation losses are present in all the blocks. Propagation losses increase
proportionally with the length of the waveguide. The SMWA DPUs, because of
their spectrally hitless architecture, employ a larger number of longer
waveguides [8], resulting in increased propagation losses compared to ASMW and
MASW DPUs. Additionally, MASW DPUs experience lower propagation losses than
ASMW DPUs. This is because the MRR weight arrays in MASW DPUs share a single
MRM input array, which reduces the overall waveguide length and corresponding
propagation losses.

TABLE III: Optical losses present in various DPU organizations. | ASMW | MASW | SMWA
---|---|---|---
Through Losses | High | Moderate | Low
Propagation Losses | Moderate | Low | High
TABLE IV: Definition and values of various parameters used in Eq. 1, Eq. 2, and Eq. 3 (from [27]) for the scalability analysis.  Parameter | Definition | Value
---|---|---
$P_{Laser}$ | Laser Power Intensity | 10 dBm
$R_{s}$ | PD Responsivity | 1.2 A/W
$R_{L}$ | Load Resistance | 50 $\Omega$
$I_{d}$ | Dark Current | 35 nA
T | Absolute Temperature | 300 K
RIN | Relative Intensity Noise | -140 dB/Hz
$P_{EC-IL}$ |  | Fiber to Chip Coupling
---
Insertion Loss
1.44
$P_{MRR-W-IL}$ |  | Silicon Waveguide
---
Insertion Loss
0.3 dB/mm
$P_{splitter-IL}$ | Splitter Insertion Loss | 0.01 dB
$P_{MRM-IL}$ |  | Optical Microring Modulator
---
Insertion Loss
4 dB
$P_{MRR-IL}$ |  | Optical Microring Resonator
---
Insertion Loss
0.01 dB
$P_{MRM-OBL}$ | Out of Band Loss | 0.01 dB
$P_{Penalty}$ | MASW Network Penalty | 4.8 dB
| ASMW Network Penalty | 5.8 dB
| SMWA Network Penalty | 1.8 dB

###  IV-C Scalability Analysis

To determine the achievable size N for our ASMW, MASW, and SMWA DPU
organizations, we adopt scalability analysis equations (Eq. 1, Eq. 2, and Eq.
3) from [27] and [12]. Table IV reports the definitions of the parameters and
their values used in these equations. The reported $P_{Penalty}$ includes the
summation of inter-modulation crosstalk, cross-weight penalty, filter penalty,
and propagation losses. We consider optimistic values for these parameters,
with inter-modulation crosstalk of $\leq$1dB, cross-weight penalty of
$\leq$3dB, and filter penalty of $\leq$0.5dB. To achieve such optimistic
inter-modulation crosstalk and cross-weight penalties the channel spacing
should be equal to 0.4$\times$FWHM [36]. We considered Free Spectral Range
(FSR=50nm) [27] with FWHM=0.7nm, resulting in channel spacing of
0.25nm(=0.4$\times$0.7). Then, the FSR limited N value is 200(=FSR/0.25). We
consider M=N and first solve Eq. 1 and Eq. 2 for a set of DRs={1, 5, 10} GS/s,
to find a corresponding set of $P_{PD-opt}$. Then, we solve Eq. 3 for the
maximum value of N that achieves $P_{O/p}$ greater than obtained $P_{PD-opt}$
values across the set of DRs. Fig. 5 reports the achievable N of ASMW, MASW,
and SMWA DPUs for different bit-precision levels (B) across various DRs. The
achievable N value defines the feasible number of MRRs per DPE; thus, this N
also defines the maximum size of the dot product that can be generated in our
DPU. As evident from Fig.5, SMWA can support larger N value compared to ASMW
and MASW at all bit-precision levels across different DRs. For instance, SMWA
achieves larger N=83 for 4-bit precision at 1 GS/s, compared to ASMW and MASW,
which achieve N=36 and N=43, respectively. This is because of SMWA’s DPU
architecture, as reported in Table II, SMWA significantly reduces crosstalk-
related power penalty reducing $P_{penalty}$. This enables SMWA to support
larger N compared to ASMW and MASW at the same input laser power.

$B=\frac{1}{6.02}\Bigg{[}20log_{10}(\frac{R\times P_{PD-
opt}}{\beta\sqrt{\frac{DR}{\sqrt{2}}}}-1.76\Bigg{]}$ (1)

$\beta=\sqrt{2q(RP_{PD-opt}+I_{d})+\frac{4kT}{R_{L}}+R^{2}P_{PD-
opt}^{2}RIN}+\sqrt{2qI_{d}+\frac{4kT}{R_{L}}}$ (2)

$P_{O/p}(dBm)=P_{Laser}-P_{SMF-att}-P_{EC-IL}-P_{Si-att}\times N\times
d_{MRR}-P_{MRM-IL}-(N-1)P_{MRM-OBL}-P_{splitter-IL}\times log_{2}(M)-P_{MRR-W-
IL}-(N-1)P_{MRR-W-OBL}-P_{penalty}-10log_{10}(N)$ (3)

Figure 5: Supported DPU size N (=M) for bit precision={1, 2, 3, 4, 5, 6, 7, 8}
bits at data rates (DRs)={1, 5, 10} GS/s, for AMW, MAW, and MWA DPUs.

##  V Evaluation

###  V-A System Level Implementation

Fig. 6 illustrates the general system-level implementation of incoherent
photonic GEMM accelerators. It consists of global memory that stores CNN
parameters and a pre-processing and mapping unit. It has a mesh network of
tiles. Each tile contains 4 DPUs interconnected (via H-tree) with a unified
buffer as well as pooling and activation units. Each DPU consists of multiple
DPEs and each DPE is equipped with a dedicated input and output FIFO buffer
[37] to store intermittent weights, inputs, and psums values. In addition,
each tile also contains a psum reduction network.

Figure 6: System-level overview of Photonic GEMM accelerator. TABLE V: DPU size (N) and DPU Count (#) at 4-bit precision across various DRs for different accelerators architectures. | Datarate
---|---
| 1 GS/s | 5 GS/s | 10 GS/s
DPU | N | # | N | # | N | #
ASMW | 36 | 160 | 17 | 265 | 12 | 291
MASW | 43 | 186 | 21 | 275 | 15 | 295
SMWA | 83 | 50 | 42 | 147 | 30 | 198

[b]  Power(mW) Latency Area($mm^{2}$) Reduction Network 0.050 3.125ns 3.00E-5
Activation Unit 0.52 0.78ns 6.00E-5 IO Interface 140.18 0.78ns 2.44E-2 Pooling
Unit 0.4 3.125ns 2.40E-4 eDRAM 41.1 1.56ns 1.66E-1 Bus 7 5 cycles 9.00E-3
Router 42 2 cycles 1.50E-2 DAC [38] 12.5 0.78ns 2.50E-3 ADC(1 GS/s) [39] 2.55
0.78ns 2E-3 ADC(5 GS/s) [40] 11 0.78ns 21E-3 ADC(10 GS/s) [41] 30 0.78ns
103E-3 EO Tuning 80 $\mu$W/FSR 20ns - TO Tuning 275 mW/FSR 4$\mu$s -

TABLE VI: Accelerator Peripherals and DPU Parameters [12]

###  V-B Simulation Setup

For evaluation, we model system-level implementation of AMSW, MASW, and SMWA
GEMM accelerator architectures using our developed custom, transaction-level,
event-driven Python-based simulator. Using the simulator, we simulated the
inference of four CNN models (with a batch size of 1): GoogleNet[42],
ResNet50[43], MobileNet_V2 [44], and ShuffleNet_V2 [45]. We evaluate the
metrics such as Frames per second (FPS), FPS/W (energy efficiency), and
FPS/W/mm2 (area efficiency). Further, prior optical GEMM accelerators [7, 5,
9] show minimal or no loss in inference accuracy.

We compared AMSW, MASW, and SMWA accelerator architectures for inference of
8-bit integer quantized CNN models. All accelerators are operated for 4-bit
integer precision across data rates 1GS/s, 5GS/s, and 10GS/s, from Fig. 5, for
these parameters SMWA, ASMW, and MASW achieve N reported in Table V,
respectively. We omitted comparison with CMOS-based digital CNN accelerators
as prior photonic accelerators (e.g., [7, 5, 9]) have already outperformed
them. Our evaluation is based on output stationary dataflow. For a fair
comparison, we performed area proportionate analysis, wherein we altered the
DPU count for each photonic CNN accelerator across all of the accelerator’s
DPUs to match with the area of SMWA (N=83) having 50 DPUs. Table V reports the
scaled DPU count of ASMW, MASW, and SMWA across various datarates.

Table VI gives the parameters used for evaluating the overhead of the
peripherals in our evaluated accelerators. We consider each laser diode to
emit input optical power of 10 mW (10 dBm) (Table IV)[7], multiplexer and
splitter parameters are taken from [5].

###  V-C Evaluation Results

Fig. 7(a) shows Normalized FPS results for various accelerators with batch
size=1 at different datarates (DRs), normalized to ASMW for ResNet50 at 10
GS/s. SMWA accelerator outperforms MASW and ASMW accelerators, respectively,
on gmean across four CNN models for all data rates. At 1 GS/s, SMWA achieves
up to 2.5$\times$ and 2.3$\times$ better FPS than ASMW and MASW, respectively.
As DR increases to 5 GS/s and 10 GS/s, SMWA shows better improvements in FPS,
achieving up to 3.9$\times$ and 4.4$\times$ better FPS than ASMW,
respectively. Similarly, SMWA achieves up to 3.6$\times$ and 3.9$\times$
better FPS than MASW at 5 GS/s and 10 GS/s.

These significant improvements in throughput for SMWA are due to two reasons.
First, inter-modulation crosstalk and cross-weight related power penalties are
absent in SMWA (refer Table. II) due to its hitless architecture. This allows
SMWA to support a larger DPU size (N=83), i.e., the size of the dot product
operation N (refer Table V) and the number of parallel dot product operations
M (=N). Consequently, the overall throughput is increased with improved
parallelism. Secondly, larger N generates less number of psums which reduces
the use of a partial sum reduction network. This, in turn, reduces the latency
associated with psum reductions and improves FPS. Among ASMW and MASW, MASW
performs slightly better than ASMW at all datarates. MASW with input array
sharing mitigates inter-modulation crosstalk power penalty at MRM input array
and also incurs lower through losses compared to ASMW (refer Table III), due
to these benefits MASW achieves slightly better N compared to ASMW (refer
Table V). MASW with higher N achieves better parallelism and decreases
reduction latency, resulting in better FPS.

Furthermore, as datarate increases the FPS of each accelerator decreases, at
5GS/s and 10 GS/s, the value of N decreases for all the accelerators (refer
Table V) which results in a higher number of psums. Therefore, the latency
corresponding to psum reduction increases with an increase in datarate and
leads to lower FPS for accelerators. Overall, SMWA architecture with higher N
achieves better throughput than ASMW and MASW architectures.

Figure 7: (a) Normalized FPS (log scale) (b) Normalized FPS/W (log scale) (c)
Normalized FPS/W/mm2 (log scale) for AMW, MAW, and MWA accelerators with input
batch size=1. Results of FPS, FPS/W, FPS/W/mm2 are normalized with respect to
AMW executing ResNet50 at 10 GS/s.

Fig. 7(b) shows FPS/W (log scale) results for ASMW, MASW, and SMWA accelerator
with batch size=1 at different DRs, normalized to ASMW for ResNet50 at 10
GS/s. It is evident that the SMWA accelerator attains better energy efficiency
than the MASW and ASMW accelerators. At 1 GS/s, SMWA gains 1.9$\times$ and
2.5$\times$ better FPS/W against analog MASM and ASMW, respectively, on gmean
across the CNNs. As the datarate increases to 5 GS/s and 10 GS/s, SWMA
achieves 3.17$\times$ and 3.3$\times$ improvements in FPS/W when compared to
MASW. SMWA also exhibits a significant 4.4$\times$ and 5$\times$ improvement
in FPS/W when compared to ASMW at 5 GS/s and 10 GS/s. These energy efficiency
benefits of SMWA are due to the improved throughput and decreased energy
consumption of psum reductions. As discussed earlier, superior N supported by
SMWA improves parallelism which decreases dynamic energy consumption with
improved throughput. In addition, SMWA also requires the least number of psum
reductions and this provides energy savings by reducing the usage of psum
reduction network. At higher datarates of 5 GS/s and 10 GS/s, the N value
decreases, consequently requiring more psum reductions and psum reduction
energy consumption. Furthermore, as datarate increases the accelerator
peripherals like ADCs consume more static power (refer Table VI) which also
decreases the FPS/W achieved by each accelerator. Thus, as datarate increase
the FPS/W decreases for ASMW, MASW, and SMWA accelerators. Overall, SMWA
provides better energy efficiency compared to the other accelerators.

Fig. 7(c) shows the area efficiency values (FPS/W/mm2) for each accelerator
across various CNNs. The area efficiency results look similar to energy
efficiency as we match the area of all the accelerators to SMWA (for the area
proportionate analysis). SMWA gains up to 5.2 $\times$ and 3.4$\times$ better
FPS/W/mm2 against ASMW and MSAW, respectively, on gmean across four CNN models
for all data rates. Overall, the SWMA accelerator achieves the best
throughput, energy efficiency, and area efficiency.

##  VI Summary

In this paper, we introduced a systematic approach for classifying prior
incoherent MRR-based GEMM accelerators into three distinct categories based on
their organization of optical signal manipulation blocks: (1) Modulation-
Aggregation-Splitting-Weighting (MASW), (2) Aggregation-Splitting-Modulation-
Weighting (ASMW), and (3) Splitting-Modulation-Weighting-Aggregation (SMWA).
We performed a comprehensive circuit-level comparative analysis of MASW, ASMW,
and SMWA organizations and identified that each organization incurs different
magnitudes of crosstalk noise and optical signal losses. As a result, our
scalability analysis at the circuit level demonstrated that each organization
achieves different levels of processing parallelism. At the system level, our
evaluation results for four CNN models show that SMWA organization achieves up
to 4.4$\times$, 5$\times$, and 5.2$\times$ better throughput, energy
efficiency, and area-energy efficiency, respectively, compared to ASMW and
MASW organizations on average.

## Acknowledgments

We thank the anonymous reviewers whose valuable feedback helped us improve
this paper. We would also like to acknowledge the National Science Foundation
(NSF) as this research was supported by NSF under grant CNS-2139167.

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