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Symplectic reduction along a submanifold

Published online by Cambridge University Press:  18 October 2022

Peter Crooks
Affiliation:
Department of Mathematics and Statistics, Utah State University, 3900 Old Main Hill, Logan, UT 84322, USA peter.crooks@usu.edu
Maxence Mayrand
Affiliation:
Département de mathématiques, Université de Sherbrooke, 2500 Bd de l'Université, Sherbrooke, QC, J1K 2R1, Canada maxence.mayrand@usherbrooke.ca
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Abstract

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden–Weinstein–Meyer reduction, Mikami–Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg–Kazhdan construction of Moore–Tachikawa varieties in topological quantum field theory. A key feature of our construction is a concrete and systematic association of a Hamiltonian $G$-space $\mathfrak {M}_{G, S}$ to each pair $(G,S)$, where $G$ is any Lie group and $S\subseteq \mathrm {Lie}(G)^{*}$ is any submanifold satisfying certain non-degeneracy conditions. The spaces $\mathfrak {M}_{G, S}$ satisfy a universal property for symplectic reduction which generalizes that of the universal imploded cross-section. Although these Hamiltonian $G$-spaces are explicit and natural from a Lie-theoretic perspective, some of them appear to be new.

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Research Article
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original article is properly cited. Compositio Mathematica is © Foundation Compositio Mathematica.
Copyright
© 2022 The Author(s)

1. Introduction

1.1 Informal context

Noether's approach to symmetries and conserved quantities in classical mechanics naturally gives rise to quotient constructions in symplectic geometry. The most basic of these constructions is Marsden–Weinstein–Meyer reduction [Reference MeyerMey73, Reference Marsden and WeinsteinMW74] for Hamiltonian Lie group actions, a cornerstone of symplectic geometry [Reference Guillemin and SternbergGS82a, Reference Sjamaar and LermanSL91, Reference SjamaarSja95, Reference Meinrenken and SjamaarMS99] with wide-ranging implications for algebraic geometry [Reference Atiyah and BottAB83, Reference KirwanKir84, Reference HitchinHit87, Reference LosevLos09] and geometric representation theory [Reference Kazhdan, Kostant and SternbergKKS78, Reference NakajimaNak98, Reference Etingof and GinzburgEG02, Reference Gan and GinzburgGG04, Reference Bezrukavnikov, Finkelberg and MirkovićBFM05, Reference Braverman, Finkelberg and NakajimaBFN19, Reference BălibanuBăl21]. This construction applies to a Hamiltonian $G$-space, i.e. a symplectic manifold $M$ equipped with a Hamiltonian action of a Lie group $G$ and a moment map $\mu :M\longrightarrow \mathfrak {g}^{*}$, where $\mathfrak {g}$ is the Lie algebra of $G$. Each element $\xi \in \mathfrak {g}^{*}$ has a $G$-stabilizer $G_{\xi }\subseteq G$ and determines a topological quotient

(1.1)\begin{equation} M \mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{\xi}} G := \mu^{-1}(\xi) / G_\xi, \end{equation}

called the Marsden–Weinstein–Meyer reduction of $M$ by $G$ at level $\xi$. The space $M \mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{\xi }} G$ is a symplectic manifold under certain hypotheses.

Motivated by recent advances in topological quantum field theory [Reference Moore and TachikawaMT12, Reference Ginzburg and KazhdanGK], we introduce and study a generalization of Marsden–Weinstein–Meyer reduction. Our starting point is the work of Mikami–Weinstein [Reference Mikami and WeinsteinMW88], which defines the notion of a Hamiltonian action of a symplectic groupoid $\mathcal {G} \rightrightarrows X$ on a symplectic manifold $M$ with moment map $\mu : M \longrightarrow X$. These authors proceed to define the symplectic reduction $M \mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{x}} \mathcal {G} := \mu ^{-1}(x) / \mathcal {G}_x$ of $M$ by $\mathcal {G}$ at level $x \in X$, where $\mathcal {G}_x$ is the isotropy group at $x$. Our construction upgrades the level $x \in X$ and subgroup $\mathcal {G}_{x} \subseteq \mathcal {G}$ to a submanifold $S \subseteq X$ and subgroupoid $\mathcal {G}_S \subseteq \mathcal {G}$, respectively, in such a way that the quotient $M \mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S}} \mathcal {G} := \mu ^{-1}(S) / \mathcal {G}_S$ is a symplectic manifold.

The equivalence between usual Hamiltonian $G$-spaces and Hamiltonian spaces for the cotangent groupoid $T^{*}G\rightrightarrows \mathfrak {g}^{*}$ (see [Reference Mikami and WeinsteinMW88]) then allows us to generalize (1.1) by replacing the level $\xi \in \mathfrak {g}^{*}$ with a submanifold $S\subseteq \mathfrak {g}^{*}$. In other words, given a Hamiltonian $G$-space $M$ with moment map $\mu : M \longrightarrow \mathfrak {g}^{*}$, we obtain a notion of symplectic reduction

(1.2)\begin{equation} M \mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S}} G := \mu^{-1}(S) / \mathcal{G}_S \end{equation}

of $M$ by $G$ at level $S \subseteq \mathfrak {g}^{*}$, where $\mathcal {G}_S$ is a certain subgroupoid of $T^{*}G$ determined by $S$. As with (1.1), the space (1.2) is a symplectic manifold under reasonable hypotheses. As $\mathcal {G}_S$ replaces the stabilizer subgroup $G_\xi$ in (1.1), we call it the ‘stabilizer subgroupoid’ of $S$.

To be somewhat more precise, let us address the class of submanifolds $S\subseteq X$ for which our construction can be performed; these are the so-called pre-Poisson submanifolds of $X$, as defined and studied by Cattaneo and Zambon [Reference Cattaneo and ZambonCZ07, Reference Cattaneo and ZambonCZ09]. Poisson transversals and Poisson submanifolds are automatically pre-Poisson, and there is a reasonable sense in which generic submanifolds of $X$ are pre-Poisson. At the same time, each pre-Poisson submanifold $S\subseteq X$ canonically determines a Lie subalgebroid $L_S \longrightarrow S$ of the Lie algebroid of $\mathcal {G}$. The notion of a stabilizer subgroupoid for $S$ is then transparent; it refers to any isotropic immersed Lie subgroupoid $\mathcal {H}\rightrightarrows S$ of $\mathcal {G}$ integrating $L_S$. One has a non-empty, discrete family of stabilizer subgroupoids for $S$, exactly one of which is source-connected and source-simply-connected (ssc).

Our generalization of Marsden–Weinstein–Meyer reduction amounts to the topological space

\begin{align*} M \mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S,\mathcal{H}}}\mathcal{G}:= \mu^{-1}(S) / \mathcal{H} \end{align*}

being a symplectic manifold under reasonable hypotheses, where $\mu :M\longrightarrow X$ is the moment map. Any two source-connected choices of $\mathcal {H}\rightrightarrows S$ yield the same quotient $M \mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S,\mathcal {H}}}\mathcal {G}$, allowing us to define

(1.3)\begin{equation} M\mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S}}\mathcal{G}:= M \mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S,\mathcal{H}}}\mathcal{G}\quad\text{for any source-connected }\mathcal{H}\rightrightarrows S. \end{equation}

We use the nomenclature symplectic reduction along a submanifold for the construction described in these last two sentences. In very general terms, the following are some features that it enjoys. The reader is referred to § 1.2 for more precise descriptions of these features.

Categories

Our construction has analogs in the categories of complex analytic spaces, complex algebraic varieties, and derived Artin stacks, all of which are developed and proved in this paper.

Relation to other constructions

Several well-studied quotient constructions in symplectic geometry can be realized as special cases of symplectic reduction along a submanifold. Such constructions include Marsden–Weinstein–Meyer reduction, Mikami–Weinstein reduction for Hamiltonian symplectic groupoid actions [Reference Mikami and WeinsteinMW88], the pre-images of Poisson transversals under moment maps [Reference BielawskiBie97, Reference Frejlich and MarcutFM17, Reference Crooks and RöserCR20, Reference Crooks and van PruijssenCP21], and the Ginzburg–Kazhdan presentation [Reference Ginzburg and KazhdanGK] of Moore–Tachikawa varieties [Reference Moore and TachikawaMT12]. We also find that symplectic cutting [Reference LermanLer95, Reference Lerman, Meinrenken, Tolman and WoodwardLMTW98, Reference WeitsmanWei01, Reference Martens and ThaddeusMT12, Reference Fisher and RayanFR16] and symplectic implosion [Reference Guillemin, Jeffrey and SjamaarGJS02, Reference Dancer, Kirwan and SwannDKS13, Reference Dancer, Kirwan and RöserDKR16, Reference SafronovSaf17] may be described more simply as reduction along a polyhedral set and reduction along a closed Weyl chamber, respectively. Our work also has an interpretation in shifted symplectic geometry [Reference Pantev, Toën, Vaquié and VezzosiPTVV13] as a derived intersection of two Lagrangians.

Universal reduced spaces

Our construction yields a very simple, systematic, Lie-theoretic technique for producing Hamiltonian Lie group spaces. One begins with a Lie group $G$ and pre-Poisson submanifold $S\subseteq \mathfrak {g}^{*}$, e.g. $S$ could be any $G$-invariant submanifold or any Poisson transversal. If the pair $(G,S)$ satisfies some non-degeneracy conditions, we show that it determines a universal Hamiltonian $G$-space $\mathfrak {M}_{G, S}$; the precise meaning of ‘universal’ is given in § 1.2.

We realize several Hamiltonian $G$-spaces in this way, including products of coadjoint orbits, open Moore–Tachikawa varieties [Reference BielawskiBie21, Reference Ginzburg and KazhdanGK], universal imploded cross-sections [Reference Guillemin, Jeffrey and SjamaarGJS02], and some spaces that appear to be new. We also show $\mathfrak {M}_{G,S}$ to be a symplectic groupoid when $S\subseteq \mathfrak {g}^{*}$ is $G$-invariant and appropriate non-degeneracy conditions are imposed.

Remark 1.1 Some follow-up work (by Bălibanu and Mayrand) will address a few generalizations of our construction in the smooth category; it will concern Hamiltonian actions of quasi-symplectic groupoids on Dirac manifolds, as well as quotients by more general subgroupoids.

1.2 Statement of main results

We now give precise formulations of our main results.

1.2.1 The smooth category

Let $(X,\sigma )$ be a Poisson manifold with Poisson bivector field $\sigma :T^{*}X\longrightarrow TX$. A submanifold $S\subseteq X$ is called pre-Poisson if $\sigma ^{-1}(TS)\cap TS^{\circ }$ has constant rank over $S$, where $TS^{\circ }\subseteq T^{*}X$ is the annihilator of $TS\subseteq TX$. Any symplectic groupoid $\mathcal {G}\rightrightarrows X$ has the property that $X$ is Poisson, and we may therefore consider a pre-Poisson submanifold $S\subseteq X$. One then has a Lie subalgebroid

\begin{align*} L_S:=\sigma^{-1}(TS)\cap TS^{\circ} \end{align*}

of the Lie algebroid of $\mathcal {G}\rightrightarrows X$. We use the term stabilizer subgroupoid of $S$ for any isotropic Lie subgroupoidFootnote 1 $\mathcal {H}\rightrightarrows S$ of $\mathcal {G}\rightrightarrows X$ having $L_S$ as its Lie algebroid.

On the other hand, Mikami and Weinstein [Reference Mikami and WeinsteinMW88] define what it means for $\mathcal {G}\rightrightarrows X$ to act in a Hamiltonian fashion on a symplectic manifold $(M,\omega )$. Any stabilizer subgroupoid $\mathcal {H}\rightrightarrows S$ then acts on $N:= \mu ^{-1}(S)$, where $\mu :M\longrightarrow X$ is the moment map realizing the Hamiltonian action of $\mathcal {G}\rightrightarrows X$ on $M$. We call the quotient topological space

\begin{align*} M\mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S,\mathcal{H}}}\mathcal{G}:= N/\mathcal{H} \end{align*}

the symplectic reduction of $M$ by $\mathcal {G}$ along $S$ with respect to $\mathcal {H}$. This space features in the following result.

Theorem A (Smooth category)

Suppose that a symplectic groupoid $\mathcal {G}\rightrightarrows X$ acts on a symplectic manifold $(M,\omega )$ in a Hamiltonian fashion with moment map $\mu :M\longrightarrow X$. Let $\mathcal {H}\rightrightarrows S$ be a stabilizer subgroupoid of a pre-Poisson submanifold $S\subseteq X$ and set $N:= \mu ^{-1}(S)$.

  1. (i) Assume that $S$ intersects $\mu$ cleanly and that $M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S,\mathcal {H}}}\mathcal {G} := N / \mathcal {H}$ has a smooth manifold structure for which the quotient map $\pi :N\longrightarrow M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S,\mathcal {H}}}\mathcal {G}$ is a submersion. The manifold $M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S,\mathcal {H}}}\mathcal {G}$ then carries a unique symplectic form $\bar {\omega }$ for which

    \begin{align*} \pi^{*}\bar{\omega}=i^{*}\omega, \end{align*}
    where $i:N\longrightarrow M$ is the inclusion map.
  2. (ii) If $\mathcal {H}$ acts freely on $N$, then $S$ is transverse to $\mu$. The hypotheses of part (i) are therefore satisfied if $\mathcal {H}$ acts freely and properly on $N$.

  3. (iii) Let $\sigma :T^{*}X\longrightarrow TX$ denote the Poisson bivector field on $X$. Assume that the hypotheses of part (i) are satisfied and that $S/\mathcal {H}$ has a smooth manifold structure for which the quotient map $\rho :S\longrightarrow S/\mathcal {H}$ is a submersion. The manifold $S/\mathcal {H}$ then has a unique Poisson structure satisfying the following condition: if $x\in S$, $f,g$ are smooth functions on $S/\mathcal {H}$ defined near $\rho (x)$, and $F,G$ are smooth functions on $X$ defined near $x$ with $dF(\sigma (TS^{\circ }))=0=dG(\sigma (TS^{\circ }))$, then

    \begin{align*} \rho^{*}\{f,g\}_{\mathcal{S}/\mathcal{H}}=\{F,G\}_X\vert_S. \end{align*}
    Furthermore, the moment map $\mu :M\longrightarrow X$ descends to a Poisson map $M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S,\mathcal {H}}}\mathcal {G}\longrightarrow S/\mathcal {H}$.

This result appears in the main text as Theorems 2.14, 2.20, and 2.24.

Let us suppose that $S=\{x\}$ is a singleton and that $\mathcal {H}$ is the isotropy group at $x$. In this case, $M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S,\mathcal {H}}}\mathcal {G}$ is precisely the Mikami–Weinstein reduction of $M$ at level $x$ (see [Reference Mikami and WeinsteinMW88]). In other words, Mikami–Weinstein reduction is a special case of Theorem A. The same is therefore true of Marsden–Weinstein–Meyer reduction, as it is a special case of Mikami–Weinstein reduction.

It is also illuminating to apply Theorem A in the case of a Poisson transversal $S\subseteq X$. The trivial groupoid $\mathcal {H}\rightrightarrows S$ is then a stabilizer subgroupoid of $S$ in $\mathcal {G}$, and one obtains the symplectic submanifold $M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S,\mathcal {H}}}\mathcal {G}=\mu ^{-1}(S)\subseteq M$.

1.2.2 The complex analytic and algebraic categories

Theorem A and its consequences have natural counterparts in the category of complex analytic spaces. To this end, call a complex analytic space $(X,\mathcal {O}_X)$ Poisson if $\mathcal {O}_X$ is a sheaf of complex Poisson algebras. Now assume that $X$ is a holomorphic Poisson manifold and denote its Poisson bivector field by $\sigma :T^{*}X\longrightarrow TX$. We call a complex submanifold $S\subseteq X$ a pre-Poisson complex submanifold if $\sigma ^{-1}(TS)\cap TS^{\circ }$ has constant rank over $S$. The base space of a holomorphic symplectic groupoid $\mathcal {G}\rightrightarrows X$ is a holomorphic Poisson manifold, in which context we may consider a pre-Poisson complex submanifold $S\subseteq X$. In analogy with § 1.2.1, we may define what it means for $\mathcal {H}\rightrightarrows S$ to be a holomorphic stabilizer subgroupoid of $S$. Hamiltonian actions are also defined analogously, i.e. one has an analogous notion of $\mathcal {G}\rightrightarrows X$ acting on a holomorphic symplectic manifold $M$ in a Hamiltonian fashion with moment map $\mu :M\longrightarrow X$. Let us assume that $N:= \mu ^{-1}(S)$ is reduced. Let us also consider a complex analytic quotient $\pi :N\longrightarrow Q$ of $N$ by $\mathcal {H}$; by this, we mean that $Q$ is a complex analytic space, that $\pi$ is holomorphic, and that the canonical map $\mathcal {O}_{Q}\longrightarrow (\pi _*\mathcal {O}_N)^{\mathcal {H}}$ is an isomorphism. We refer to $Q$ as the symplectic reduction of $M$ by $\mathcal {G}$ along $S$ with respect to $\mathcal {H}$ and $\pi$ and write

\begin{align*} M\mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S,\mathcal{H},\pi}}\mathcal{G}:= Q. \end{align*}

These considerations yield the following counterpart of Theorem A.

Theorem B (Complex analytic category)

Let a holomorphic symplectic groupoid $\mathcal {G}\rightrightarrows X$ act on a holomorphic symplectic manifold $(M,\omega )$ in a Hamiltonian fashion with moment map $\mu :M\longrightarrow X$. Suppose that $\mathcal {H}\rightrightarrows S$ is a holomorphic stabilizer subgroupoid of a pre-Poisson complex submanifold $S\subseteq X$ and that $N:= \mu ^{-1}(S)$ is reduced. Let us also suppose that $\pi :N\longrightarrow Q$ is a complex analytic quotient of $N$ by $\mathcal {H}$.

  1. (i) If $p\in N$ and $f\in \mathcal {O}_{Q,\pi (p)}$, then there exists $F\in \mathcal {O}_{M,p}$ satisfying $\pi ^{*}f=F\vert _N$ and $dF(TN^{\omega })=0$, where $TN^{\omega }$ is the annihilator of $TN$ with respect to $\omega$.

  2. (ii) The complex analytic space $Q$ has a unique Poisson structure with the following property: if $p\in N$, $f,g\in \mathcal {O}_{Q,\pi (p)}$, and $F,G\in \mathcal {O}_{M,p}$ satisfy $\pi ^{*}f=F\vert _N$, $\pi ^{*}g=G\vert _N$, and $dF(TN^{\omega })=0=dG(TN^{\omega })$, then

    \begin{align*} \pi^{*}\{f,g\}_Q=\{F,G\}_M\vert_N. \end{align*}
  3. (iii) Assume that there exists $p\in N$ such that $d\pi _p$ is surjective, $\pi ^{-1}(\pi (p))$ is an $\mathcal {H}$-orbit, and $\pi (p)$ is a smooth point of $Q$. The Poisson structure in part (ii) is then non-degenerate on an open dense subset of the smooth locus of $Q$ containing $\pi (p)$.

  4. (iv) Suppose that $S$ intersects $\mu$ cleanly, and that the topological quotient $Q = N/\mathcal {H}$ has a complex manifold structure such that $\pi : N \longrightarrow Q$ is a holomorphic submersion. The Poisson structure on $Q$ is then induced by a holomorphic symplectic form $\bar {\omega }$ satisfying

    \begin{align*} \pi^{*}\bar{\omega} = i^{*}\omega, \end{align*}
    where $i : N \longrightarrow M$ is the inclusion map.
  5. (v) If $\mathcal {H}$ acts freely on $N$, then $S$ is transverse to $\mu$. The hypotheses of part (iv) are therefore satisfied if $\mathcal {H}$ acts freely and properly on $N$.

  6. (vi) Let $\sigma :T^{*}X\longrightarrow TX$ denote the Poisson bivector field on $X$. If $\rho : S \longrightarrow R$ is a complex analytic quotient of $S$ by $\mathcal {H}$, then there is a unique holomorphic Poisson structure on $R$ such that

    \begin{align*} \rho^{*}\{f, g\}_R = \{F, G\}_X\vert_S \end{align*}
    for all $x \in S$, $f, g \in \mathcal {O}_{R, \rho (x)}$, and $F, G \in \mathcal {O}_{X, x}$ satisfying $\rho ^{*}f = F\vert _S$, $\rho ^{*}g = G\vert _S$, and $dF(\sigma (TS^{\circ }))=0=dG(\sigma (TS^{\circ }))$.
  7. (vii) Assume that the hypothesis of part (vi) holds. If $\mu : M \longrightarrow X$ descends to a holomorphic map $\bar {\mu } : Q \longrightarrow R$, then $\bar {\mu }$ is Poisson with respect to the Poisson structures in parts (ii) and (vi).

This result appears in the main text as Theorem 3.10, Propositions 3.14, 3.15, and Theorem 3.16.

Theorem B has incarnations in complex algebraic geometry. The setup is entirely analogous to that outlined in the paragraph preceding Theorem B; one simply replaces each complex analytic notion with its algebro-geometric counterpart, e.g. holomorphic symplectic groupoids with algebraic symplectic groupoids, holomorphic Poisson manifolds with smooth Poisson varieties, and complex analytic quotients with algebraic quotients.Footnote 2 The algebro-geometric definition of $M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S,\mathcal {H},\pi }}\mathcal {G}$ is then analogous to that given in the complex analytic setting.

Theorem C (Complex algebraic category)

Let an algebraic symplectic groupoid $\mathcal {G}\rightrightarrows X$ act on a symplectic variety $(M,\omega )$ in a Hamiltonian fashion with moment map $\mu :M\longrightarrow X$. Suppose that $\mathcal {H}\rightrightarrows S$ is a stabilizer subgroupoid of a pre-Poisson subvariety $S\subseteq X$ and that $N:= \mu ^{-1}(S)$ is reduced. Suppose also that $\pi :N\longrightarrow Q$ is an algebraic quotient of $N$ by $\mathcal {H}$. Let $\mathcal {O}_M^{{\rm an}}$ denote the structure sheaf of $M$ as a complex analytic space.

  1. (i) If $p\in N$ and $f\in \mathcal {O}_{Q,\pi (p)}$, then there exists $F$ in $\mathcal {O}_{M,p}^{{\rm an}}$ satisfying $\pi ^{*}f=F\vert _N$ and ${dF(TN^{\omega })=0}$.

  2. (ii) The variety $Q$ has a unique algebraic Poisson structure with the following property: if $p\in N$, $f,g\in \mathcal {O}_{Q,\pi (p)}$, and $F,G\in \mathcal {O}_{M,p}^{{\rm an}}$ satisfy $\pi ^{*}f=F\vert _N$, $\pi ^{*}g=G\vert _N$, and $dF(TN^{\omega })=0=dG(TN^{\omega })$, then

    \begin{align*} \pi^{*}\{f,g\}_Q=\{F,G\}_M\vert_N. \end{align*}
  3. (iii) Assume that there exists $p \in N$ such that $d\pi _p$ is surjective, $\pi ^{-1}(\pi (p))$ is an $\mathcal {H}$-orbit, and $\pi (p)$ is a smooth point of $Q$. The Poisson structure in part (ii) is then non-degenerate on a Zariski-open subset of the smooth locus of $Q$ containing $\pi (p)$.

Theorem C appears in the main text as Theorem 5.3.

1.2.3 Symplectic reduction by a Lie group along a submanifold

Our results have implications for classical Hamiltonian $G$-spaces in both the smooth and complex analytic categories, where $G$ is any real or complex Lie group with Lie algebra $\mathfrak {g}$. We develop the implications for smooth Hamiltonian $G$-spaces in parallel with those for holomorphic Hamiltonian $G$-spaces, allowing context to resolve any ambiguities that this may create.

Recall the identification of Hamiltonian $G$-spaces with Hamiltonian spaces for the cotangent groupoid $T^{*}G\rightrightarrows \mathfrak {g}^{*}$ described in [Reference Mikami and WeinsteinMW88] and mentioned in § 1.1. This gives rise to the notation

\begin{align*} M\mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S,\mathcal{H}}}G:= M\mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S,\mathcal{H}}}T^{*}G \end{align*}

for the reduction of a Hamiltonian $G$-space $M$ along a pre-Poisson submanifold $S\subseteq \mathfrak {g}^{*}$ with respect to a stabilizer subgroupoid $\mathcal {H}\rightrightarrows S$ in $T^{*}G$. As per (1.3) and the discussion preceding it, we may write

\begin{align*} M\mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S}} G:= M\mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S,\mathcal{H}}} G\quad\text{for any source-connected }\mathcal{H}\rightrightarrows S. \end{align*}

On the other hand, we call the pair $(S,\mathcal {H})$ a clean reduction datum if it satisfies the hypotheses of Theorems A(i) or B(iv). Note that $M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S,\mathcal {H}}}G$ is a symplectic manifold in this case.

One may specialize this discussion to the Hamiltonian $G$-space $M=T^{*}G$, where $G$ acts on $T^{*}G$ by right translations. Let us begin by considering

\begin{align*} \mathfrak{M}_{G,S,\mathcal{H}}:= T^{*}G\mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S,\mathcal{H}}}G\quad\text{and}\quad\mathfrak{M}_{G,S}:= \mathfrak{M}_{G,S,\mathcal{H}}\quad\text{for any source-connected }\mathcal{H}\rightrightarrows S. \end{align*}

Note that $\mathfrak {M}_{G,S,\mathcal {H}}$ is a symplectic manifold if $(S,\mathcal {H})$ is a clean reduction datum for the right translation action of $G$ on $T^{*}G$. In this case, the left translation action descends to a Hamiltonian $G$-action on $\mathfrak {M}_{G,S,\mathcal {H}}$, i.e. $\mathfrak {M}_{G,S,\mathcal {H}}$ is a Hamiltonian $G$-space. Part (i) of the following result justifies our calling $\mathfrak {M}_{G,S,\mathcal {H}}$ the universal reduced space associated to $(G,S,\mathcal {H})$.

Theorem D (Reduction by a Lie group)

Let $S\subseteq \mathfrak {g}^{*}$ be a pre-Poisson submanifold.

  1. (i) Suppose that $(S,\mathcal {H})$ is a clean reduction datum for both a Hamiltonian $G$-space $M$ and the right translation action of $G$ on $T^{*}G$. We then have a canonical symplectomorphism

    \begin{align*} M \mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S, \mathcal{H}}} G \cong (M \times \mathfrak{M}_{G, S, \mathcal{H}}^{-}) \mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{0}} G, \end{align*}
    where $\mathfrak {M}_{G, S, \mathcal {H}}^{-}$ is $\mathfrak {M}_{G, S, \mathcal {H}}$ with the negated symplectic form, $G$ acts diagonally on $M \times \mathfrak {M}_{G, S, \mathcal {H}}^{-}$, and $\,\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{0}}$ denotes Marsden–Weinstein–Meyer reduction at level $0$.
  2. (ii) Suppose that the Lie subalgebroid $L_S := \sigma ^{-1}(TS) \cap TS^{\circ } \subseteq T^{*}\mathfrak {g}^{*}$ is contained in the kernel of the Kirillov–Kostant–Souriau Poisson structure $\sigma :T^{*}\mathfrak {g}^{*}\longrightarrow T\mathfrak {g}^{*}$. The subspace

    \begin{align*} \mathfrak{h}_{\xi}:= (T_{\xi}S)^{\circ}\cap\mathfrak{g}_{\xi} \end{align*}
    is then a Lie subalgebra of $\mathfrak {g}$ for all $\xi \in S$, where $(T_{\xi }S)^{\circ }\subseteq \mathfrak {g}$ is the annihilator of $T_{\xi }S\subseteq \mathfrak {g}^{*}$ and $\mathfrak {g}_{\xi }\subseteq \mathfrak {g}$ is the centralizer of $\xi$. Furthermore, we have
    \begin{align*} L_S=\{(x,\xi)\in \mathfrak{g}\times S:x\in\mathfrak{h}_{\xi}\}. \end{align*}
  3. (iii) Retain the hypothesis of part (ii). Let $H_{\xi }\subseteq G$ be the connected Lie subgroup with Lie algebra $\mathfrak {h}_{\xi }$ for all $\xi \in S$, and suppose that

    \begin{align*} \mathcal{H}:=\{(g,\xi)\in G\times S:g\in H_{\xi}\} \end{align*}
    is closed in $G\times S$. The subset $\mathcal {H}$ is then a stabilizer subgroupoid of $S$ in $T^{*}G\rightrightarrows \mathfrak {g}^{*}$ if one uses the left trivialization to identify $T^{*}G$ with $G\times \mathfrak {g}^{*}$.
  4. (iv) Suppose that $S$ is a $G$-invariant submanifold of $\mathfrak {g}^{*}$. The hypothesis of part (ii) is then satisfied. Let us also assume that the hypotheses of part (iii) are satisfied. Then $(S,\mathcal {H})$ is a clean reduction datum for the right translation action of $G$ on $T^{*}G$, and the symplectic manifold $\mathfrak {M}_{G,S}$ inherits the structure of a symplectic groupoid integrating $S$.

Theorem D appears in the main text as Proposition 4.10 and Theorems 4.11, 4.17, and 4.21.

We proceed to obtain the following specific examples of universal reduced spaces.

Theorem E (Specific examples of universal reduced spaces)

Let $G$ be a connected complex semisimple Lie group with Lie algebra $\mathfrak {g}$.

  1. (i) If $G$ is of adjoint type, $\mathcal {S}\subseteq \mathfrak {g}^{*}$ is a principal Slodowy slice, and $\Delta _n\mathcal {S}\subseteq (\mathfrak {g}^{*})^{n}$ is the diagonally embedded copy of $\mathcal {S}$, then $\mathfrak {M}_{G^{n},\Delta _n\mathcal {S}}$ is the $n$th open Moore–Tachikawa variety. Its affinization is the $n$th Moore–Tachikawa variety constructed by Ginzburg–Kazhdan, and the Poisson structure on this variety can be recovered from Theorem C.

  2. (ii) If $(\mathfrak {g}^{*}_{\mathrm {irr}})^{\circ }$ is the set of subregular semisimple elements in $\mathfrak {g}^{*}$, then

    \begin{align*} \mathfrak{M}_{G,(\mathfrak{g}^{*}_{\mathrm{irr}})^{\circ}}=\bigsqcup_{\xi\in (\mathfrak{g}^{*}_{\mathrm{irr}})^{\circ}}G/[G_{\xi},G_{\xi}], \end{align*}
    where $G_{\xi }$ is the $G$-stabilizer of $\xi \in \mathfrak {g}^{*}$. Furthermore, this space has the structure of a holomorphic symplectic groupoid integrating $(\mathfrak {g}^{*}_{\mathrm {irr}})^{\circ }$. Its symplectic form, Hamiltonian $G$-action, and groupoid structure are described in Example 4.36.

Theorem E appears in the main text as Example 4.36 and Theorem 5.10.

A few brief comments are warranted. We first note that the stabilizer subgroupoid used to construct $\mathfrak {M}_{G^{n},\Delta _n\mathcal {S}}$ is a genuine group scheme, i.e. a group scheme whose fibers are not all isomorphic to one another. This contrasts with the more familiar construction of reduced spaces as quotients by Lie group actions. The Ginzburg–Kazhdan construction of Moore–Tachikawa varieties thereby inspired much of the setup in this paper. In particular, this construction was the main motivation for our work.

Our second comment is that Theorem E(ii) holds in greater generality; one can reduce along any decomposition class $\mathcal {D}\subseteq \mathfrak {g}^{*}$ (see [Reference Borho and KraftBK79]) and obtain an explicit description of the Hamiltonian $G$-space $\mathfrak {M}_{G,\mathcal {D}}$. The Hamiltonian $G$-spaces $\mathfrak {M}_{G,\mathcal {D}}$ do not seem to appear in the literature.

1.2.4 Examples of symplectic reduction along a submanifold

In addition to Marsden– Weinstein–Meyer reduction, Mikami–Weinstein reduction, and the pre-images of Poisson transversals under moment maps, we realize each of the following as special cases of symplectic reduction along a submanifold. Retain the notation used in § 1.2.3 and identify $T^{*}G$ with $G\times \mathfrak {g}^{*}$ via the left trivialization as appropriate. Write $G_{\xi }$ for the $G$-stabilizer of $\xi \in \mathfrak {g}^{*}$.

Theorem F (General examples)

Let $G$ be a Lie group with Lie algebra $\mathfrak {g}$. Suppose that $G$ acts on a symplectic manifold $M$ in a Hamiltonian fashion and with moment map $\mu :M\longrightarrow \mathfrak {g}^{*}$.

  1. (i) If $\mathcal {O}\subseteq \mathfrak {g}^{*}$ is a coadjoint orbit, $\mathcal {H}=\{(g,\xi )\in G\times \mathcal {O}:g\in G_{\xi }\}$, $(\mathcal {O},\mathcal {H})$ is a clean reduction datum, and $\mathcal {O}^{-}$ is $\mathcal {O}$ with the negated symplectic form, then $M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{\mathcal {O},\mathcal {H}}}G=\mu ^{-1}(\mathcal {O})/G\times \mathcal {O}^{-}$.

  2. (ii) If $G$ is compact and connected and $\mathfrak {t}_+^{*}\subseteq \mathfrak {g}^{*}$ is a closed fundamental Weyl chamber, then $M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{\mathfrak {t}_+^{*}}}G$ is the imploded cross-section of $M$.

  3. (iii) If $G=T$ is a compact torus and $P\subseteq \mathfrak {g}^{*}=\mathfrak {t}^{*}$ is a polyhedral set, then $M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{P}}T$ is the symplectic cut of $M$ with respect to $P$.

Although $\mathfrak {t}_+^{*}$ and $P$ are not submanifolds, each is stratified into pre-Poisson submanifolds. This allows one to form each of $M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{\mathfrak {t}_+^{*}}}G$ and $M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{P}}T$ on a stratum-by-stratum basis. We refer the reader to Remark 2.17 for further details.

Theorem F appears in the main text as Propositions 4.26, 4.29, and 4.32.

1.2.5 Shifted symplectic interpretation

The notion of symplectic reduction along a submanifold can be interpreted as a construction in shifted symplectic geometry [Reference Pantev, Toën, Vaquié and VezzosiPTVV13] similar to that for Marsden–Weinstein–Meyer reduction [Reference Calaque, Anel and CatrenCal20, § 2.1.2] and quasi-Hamiltonian reduction [Reference SafronovSaf16]. More precisely, we obtain it as a derived intersection of two Lagrangian morphisms [Reference Pantev, Toën, Vaquié and VezzosiPTVV13, Theorem 2.9] in the $1$-shifted symplectic stack $[X/\mathcal {G}]$ associated to a symplectic groupoid $\mathcal {G} \rightrightarrows X$. One consequence is that most of the assumptions of Theorem C can be dropped at the expense of obtaining a derived stack $[M \mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S, \mathcal {H}}} \mathcal {G}]$ endowed with a 0-shifted symplectic structure. This picture also explains the definition of a stabilizer subgroupoid, as shown by part (i) of the following theorem.

Theorem G (Shifted symplectic interpretation)

Let $\mathcal {G} \rightrightarrows X$ be an algebraic symplectic groupoid.

  1. (i) An algebraic subgroupoid $\mathcal {H} \rightrightarrows S$ of $\mathcal {G} \rightrightarrows X$ is a stabilizer subgroupoid of a pre-Poisson subvariety if and only if the zero $2$-form on $S$ is a Lagrangian structure on the morphism of quotient stacks $[S/\mathcal {H}] \longrightarrow [X/\mathcal {G}]$.

  2. (ii) Suppose that $\mathcal {G} \rightrightarrows X$ acts on a symplectic variety $M$ in a Hamiltonian fashion with moment map $\mu : M \longrightarrow X$, and let $\mathcal {H} \rightrightarrows S$ be a stabilizer subgroupoid of a pre-Poisson subvariety $S$ in $\mathcal {G} \rightrightarrows X$. We then have two Lagrangian morphisms $[S/\mathcal {H}] \longrightarrow [X/\mathcal {G}]$ and $[\mu ] : [M/\mathcal {G}] \longrightarrow [X/\mathcal {G}]$ on the $1$-shifted symplectic stack $[X/\mathcal {G}]$, inducing a $0$-shifted symplectic structure on the derived fiber product

    \begin{align*} [M\mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S, \mathcal{H}}} \mathcal{G}] := [S/\mathcal{H}] \times_{[X/\mathcal{G}]}^{h} [M/\mathcal{G}]. \end{align*}

This result appears in the main text as Proposition 6.1 and Theorem 6.2.

1.3 Organization

Our paper is organized as follows. Sections 2, 3, 4, 5, and 6 are based around the discussions, proofs, and implications of Theorems A, B, D, C, and G, respectively. Theorem E is divided between §§ 4 and 5, whereas Theorem F appears in § 4. Each of the sections in this paper begins with an outline of its content and structure. Section 6 is followed by a list of recurring notation in our paper.

2. Main construction: smooth version

The present section is devoted to the discussion, proof, and implications of Theorem A. We begin with an overview of the Lie groupoid-theoretic preliminaries in § 2.1. This leads to treatment of pre-Poisson submanifolds and stabilizer subgroupoids in § 2.2. The proofs of parts (i), (ii), and (iii) in Theorem A then appear in §§ 2.3, 2.4, and 2.5, respectively. We conclude with § 2.6, which briefly outlines some special cases of Theorem A.

2.1 Preliminaries

We begin by assembling the concepts and conventions needed to define and study symplectic reduction along a submanifold.

2.1.1 Actions of Lie groupoids

Let $\mathcal {G} \rightrightarrows X$ be a Lie groupoid. We denote the source and target maps by $\mathsf {s}:\mathcal {G}\longrightarrow X$ and $\mathsf {t}:\mathcal {G}\longrightarrow X$, respectively. Our convention is that, given $g, h \in \mathcal {G}$, the product $gh$ is defined if and only if $\mathsf {t}(g) = \mathsf {s}(h)$. The base $X$ is identified as a submanifold of $\mathcal {G}$ via the identity bisection $1 : X \hookrightarrow \mathcal {G}$. We say that $\mathcal {G} \rightrightarrows X$ is an ssc Lie groupoid if every fiber of $\mathsf {s}$ is connected and simply-connected. The Lie algebroid of $\mathcal {G} \rightrightarrows X$ shall be denoted $\operatorname {Lie}(\mathcal {G})$ and realized as the vector bundle $(\ker d\mathsf {t})\vert _X \longrightarrow X$ with anchor map $d\mathsf {s} : (\ker d\mathsf {t})\vert _X \longrightarrow TX$.

Now consider a smooth manifold $M$, a smooth map $\mu :M\longrightarrow X$, and the fibered product

\begin{align*} \mathcal{G}_ {{\mathsf{t}}}{\times}_\mu M := \{(g, p) \in \mathcal{G} \times M : \mathsf{t}(g) = \mu(p)\}. \end{align*}

Recall that an action of $\mathcal {G} \rightrightarrows X$ on $M$ with moment map $\mu$ is a smooth map

\begin{align*} \mathcal{G}_{\mathsf{t}}{\times}_\mu M \longrightarrow M, \quad (g, p) \longmapsto g \cdot p, \end{align*}

such that $\mu (g \cdot p) = \mathsf {s}(g)$, $(g \cdot h) \cdot p = g \cdot (h \cdot p)$, and $1_{\mu (p)} \cdot p = p$ for all $p \in M$ and $g, h \in \mathcal {G}$ for which these expressions are defined. The $\mathcal {G}$-orbit of a point $p \in M$ is the set

\begin{align*} \mathcal{G} \cdot p := \{ g \cdot p : (g, p) \in \mathcal{G} _{\mathsf{t}}{\times}_\mu M \}, \end{align*}

an immersed submanifold of $M$. The quotient of $M$ by $\mathcal {G}$, denoted $M / \mathcal {G}$, is the space of orbits with the quotient topology. We say that the action is free if $g \cdot p = p$ for $(g, p) \in \mathcal {G}_{\mathsf {t}}{\times }_\mu M$ implies $g = 1_{\mu (p)}$, and proper if the map

(2.1)\begin{equation} \mathcal{G}_{\mathsf{t}}{\times}_\mu M \longrightarrow M \times M,\quad (g, p) \longmapsto (p, g\cdot p) \end{equation}

is proper. As for Lie group actions, if the action is free and proper, or more generally the image of (2.1) is a closed embedded submanifold, then the quotient $M / \mathcal {G}$ is a topological manifold with a unique smooth structure such that the quotient map $M \longrightarrow M / \mathcal {G}$ is a smooth submersion [Reference MackenzieMac05, Theorem 1.6.20].

2.1.2 Hamiltonian systems

Let us recall the notion of a Hamiltonian action of a symplectic groupoid on a symplectic manifold, following Mikami–Weinstein [Reference Mikami and WeinsteinMW88].

A Poisson bivector field on a smooth manifold $P$ is a bundle map $\sigma : T^{*}P \longrightarrow TP$ such that the bilinear map $\{f, g\} := \sigma (df)g$ on smooth functions $f, g \in C^{\infty }(P)$ is a Lie bracket. In this case, we call the pair $(P, \sigma )$ a Poisson manifold. For a smooth function $f$ on $P$, we denote by $X_f$ the Hamiltonian vector field $X_f := \sigma (df)$. We denote by $P^{-}$ the Poisson manifold with the opposite Poisson structure $-\sigma$. We say that a submanifold $Q \subseteq P$ is coisotropic if $\sigma (TQ^{\circ }) \subseteq TQ$, where $TQ^{\circ }$ is the annihilator of $TQ$ in $T^{*}P$.

A symplectic groupoid [Reference Costé, Dazord and WeinsteinCDW87, Reference WeinsteinWei87] is a Lie groupoid $\mathcal {G}\rightrightarrows X$ together with a symplectic form $\Omega$ on $\mathcal {G}$, such that the graph

\begin{align*} \Gamma_\mathcal{G} := \{(g, h, gh) : (g, h) \in \mathcal{G}_{\mathsf{t}}{\times}_\mathsf{s} \mathcal{G}\} \end{align*}

of multiplication is Lagrangian in $\mathcal {G} \times \mathcal {G} \times \mathcal {G}^{-}$. In this case, there is a canonical vector bundle isomorphism $\operatorname {Lie}(\mathcal {G}) \cong T^{*}X$ given by

(2.2)\begin{equation} \ker d\mathsf{t}_x \longrightarrow T^{*}_xX, \quad v \longmapsto \Omega(v)\vert_{T_xX} \end{equation}

for all $x \in X$, where $\Omega$ is viewed as a bundle map $T\mathcal {G} \longrightarrow T^{*}\mathcal {G}$. The anchor map $\sigma : \operatorname {Lie}(\mathcal {G}) \cong T^{*}X \longrightarrow TX$ is then a Poisson bivector field on $X$ such that the source and target maps are Poisson and anti-Poisson, respectively.

Definition 2.1 (Mikami–Weinstein [Reference Mikami and WeinsteinMW88])

An action of a symplectic groupoid $\mathcal {G}\rightrightarrows X$ on a Poisson manifold $M$ with moment map $\mu : M \longrightarrow X$ is Hamiltonian if its graph

\begin{align*} \Gamma := \{(g, p, q) \in \mathcal{G} \times M \times M : \mathsf{t}(g) = \mu(p) \text{ and } q = g \cdot p \} \end{align*}

is coisotropic in $\mathcal {G} \times M \times M^{-}$. If the Poisson structure on $M$ is induced by a symplectic form $\omega$, we call $((M, \omega ), \mathcal {G} \rightrightarrows X, \mu )$ a Hamiltonian system.

Remark 2.2 The moment map $\mu : M \longrightarrow X$ of a Hamiltonian system is automatically Poisson [Reference Mikami and WeinsteinMW88, Theorem 3.8]. Conversely, any Poisson map $\mu : M \longrightarrow X$ from a symplectic manifold $M$ to a Poisson manifold $X$ uniquely determines an action of the symplectic local groupoid integrating $X$ (see [Reference Costé, Dazord and WeinsteinCDW87, Chapitre III, Théorème 1.1]).

Remark 2.3 Each Lie group $G$ with Lie algebra $\mathfrak {g}$ has an associated symplectic groupoid $T^{*}G\rightrightarrows \mathfrak {g}^{*}$. It has the property that Hamiltonian systems $((M, \omega ), T^{*}G \rightrightarrows \mathfrak {g}^{*}, \mu )$ are equivalent to the more familiar Hamiltonian $G$-spaces. We refer the reader to § 4.1 for further details.

2.2 Pre-Poisson submanifolds and stabilizer subgroupoids

For a submanifold $S$ of a Poisson manifold $(X, \sigma )$, consider the subset of $T^{*}X$ given by

(2.3)\begin{equation} L_S := \sigma^{-1}(TS) \cap TS^{\circ}, \end{equation}

where $TS^{\circ }$ is the annihilator of $TS$ in $T^{*}X$. Our reduction procedure is based on the following class of submanifolds.

Definition 2.4 (Cattaneo–Zambon [Reference Cattaneo and ZambonCZ07] and [Reference Cattaneo and ZambonCZ09, Definition 2.2])

A submanifold $S$ of a Poisson manifold $(X, \sigma )$ is called pre-Poisson if (2.3) has constant rank over $S$.

Remark 2.5 In [Reference Cattaneo and ZambonCZ07, Reference Cattaneo and ZambonCZ09], pre-Poisson submanifolds are defined by the condition that $\sigma (TS^{\circ }) + TS$ has constant rank. The two definitions are equivalent because $(\sigma (TS^{\circ }) + TS)^{\circ } = \sigma ^{-1}(TS) \cap TS^{\circ }$.

Note that there is no constraint on the dimension of $S$. In fact, a generic submanifold is pre-Poisson in the following sense.

Proposition 2.6 For a generic point $x \in X$, a generic subspace $V \subseteq T_xX$, and an arbitrary submanifold $S \subseteq X$ with $x \in S$ and $T_xS = V$, $S$ is pre-Poisson in a neighborhood of $x$.

Proof. The term ‘generic’ used in this statement refers to an element of some fixed open dense set. The set of $x \in X$ such that $\sigma$ has constant rank in a neighborhood of $x$ is open and dense, so this reduces to the case of a Poisson vector space $(W, \sigma )$. The proposition then follows from the fact that the set of $k$-dimensional subspaces $V \subseteq W$ such that $V + \sigma (V^{\circ })$ has maximal dimension is open and dense in the Grassmannian of $k$-planes in $W$.

We have the following important result on pre-Poisson submanifolds.

Theorem 2.7 (Cattaneo–Zambon [Reference Cattaneo and ZambonCZ09, Proposition 3.6 and Proposition 7.2])

Let $\mathcal {G} \rightrightarrows X$ be a symplectic groupoid and $S \subseteq X$ a pre-Poisson submanifold.

  1. (i) The subset $L_S$ of $T^{*}X$ given by (2.3) is a Lie subalgebroid of $\operatorname {Lie}(\mathcal {G}) = T^{*}X$.

  2. (ii) The ssc Lie groupoid $\mathcal {G}_S \rightrightarrows S$ with Lie algebroid $L_S$ is isotropic in $\mathcal {G}$.

Part (ii) requires further explanation. First, recall that a Lie subalgebroid of an integrable Lie algebroid is integrable [Reference Moerdijk and MrčunMM02, Proposition 3.4]. It follows from part (i) that $L_S$ integrates to an ssc Lie groupoid $\mathcal {G}_S \rightrightarrows S$ together with an immersion $\mathcal {G}_S \longrightarrow \mathcal {G}$ (see [Reference Moerdijk and MrčunMM02, Proposition 3.5]). The condition that $\mathcal {G}_S$ is isotropic in $\mathcal {G}$ then means that the pullback of the symplectic form on $\mathcal {G}$ to $\mathcal {G}_S$ vanishes.

In this paper, we follow the convention that a Lie subgroupoid of a Lie groupoid $\mathcal {G}$ is a Lie groupoid $\mathcal {H}$ together with a possibly non-injective immersion $\mathcal {H} \longrightarrow \mathcal {G}$ that is also a Lie groupoid homomorphism. The groupoid $\mathcal {G}_S \rightrightarrows S$ in Theorem 2.7 is therefore an isotropic Lie subgroupoid of $\mathcal {G} \rightrightarrows S$. These considerations motivate the following definition.

Definition 2.8 Let $\mathcal {G} \rightrightarrows X$ be a symplectic groupoid and $S \subseteq X$ a pre-Poisson submanifold. The Lie algebroid $L_S$ defined in (2.3) is called the stabilizer subalgebroid of $S$. A stabilizer subgroupoid of $S$ in $\mathcal {G}$ is any (possibly non-source-connected) isotropic Lie subgroupoid $\mathcal {H} \rightrightarrows S$ of $\mathcal {G} \rightrightarrows X$ with Lie algebroid $L_S$. The unique ssc stabilizer subgroupoid of $S$ in $\mathcal {G}$ is denoted $\mathcal {G}_S \rightrightarrows S$.

Remark 2.9 (i) This terminology is justified by the fact that stabilizer subgroupoids generalize stabilizer subgroups in Marsden–Weinstein–Meyer reduction, which we explain in § 4.2.

(ii) By [Reference Cattaneo and ZambonCZ09, Proposition 7.2], any source-connected Lie subgroupoid $\mathcal {H} \rightrightarrows S$ of $\mathcal {G} \rightrightarrows X$ with Lie algebroid $L_S$ is isotropic, and hence a stabilizer subgroupoid. More generally, if $\mathcal {H}$ is source-connected over an open dense subset of $S$, then it is isotropic.

(iii) On the other hand, a non-source-connected Lie subgroupoid $\mathcal {H} \rightrightarrows S$ with Lie algebroid $L_S$ might not be isotropic. For example,Footnote 3 let $X$ be a smooth manifold with the zero Poisson structure $\sigma = 0$. Then $\mathcal {G} = T^{*}X$ with its canonical symplectic form is a symplectic groupoid integrating $X$ (see [Reference Costé, Dazord and WeinsteinCDW87, Ch. II, Example 3.3]). Consider $S = X$, so that $L_S$ is trivial. For any non-vanishing $1$-form $\alpha$ on $X$,

\begin{align*} \mathcal{H} := \{k\alpha_x : x \in X, k \in \mathbb{Z}\} \end{align*}

is a Lie subgroupoid of $\mathcal {G} = T^{*}X$ with Lie algebroid $L_S = 0$. However, the pullback of the canonical symplectic form on $T^{*}X$ by $\alpha : X \longrightarrow T^{*}X$ is $d\alpha$. It follows that $\mathcal {H}$ is isotropic if and only if $\alpha$ is closed.

(iv) One explanation for the isotropy condition is that stabilizer subgroupoids are precisely the Lie subgroupoids of $\mathcal {G}$ for which the map of quotient stacks $[S/\mathcal {H}] \longrightarrow [X/\mathcal {G}]$ has a canonical Lagrangian structure. This is explained in § 6.

Another important result of [Reference Cattaneo and ZambonCZ09] is that pre-Poisson submanifolds can be embedded coisotropically in Poisson transversals, as we now recall.

We begin by recalling that a submanifold $Y$ of a Poisson manifold $(X, \sigma )$ is called a Poisson transversal (or sometimes a cosymplectic submanifold) if $Y$ intersects every symplectic leaf transversally and in a symplectic submanifold of the leaf [Reference Frejlich and MarcutFM17]. Equivalently, $Y$ is Poisson transversal if

\begin{align*} TX\vert_Y = TY \oplus \sigma(TY^{\circ}). \end{align*}

In this case, $Y$ inherits a canonical Poisson structure from $X$ (see [Reference Frejlich and MarcutFM17, Lemma 3]). To describe this Poisson structure as a bivector field $\sigma _Y:T^{*}Y \longrightarrow TY$, note that the restriction of linear functionals defines an isomorphism $\sigma ^{-1}(TY) \longrightarrow T^{*}Y$. One then defines $\sigma _Y$ by $\sigma _Y(\xi ) = \sigma (\hat {\xi })$, where $\hat {\xi }$ is the unique element of $\sigma ^{-1}(TY)$ mapping to $\xi \in T^{*}Y$.

Cattaneo and Zambon [Reference Cattaneo and ZambonCZ09] prove that every pre-Poisson submanifold $S$ of a Poisson submanifold $(X, \sigma )$ can be embedded in a Poisson transversal $Y \subseteq X$ such that $S$ is coisotropic in $Y$ (see [Reference Cattaneo and ZambonCZ09, Theorem 3.3]).

2.3 Symplectic reduction along a submanifold

Let $\mu : M \longrightarrow X$ be a smooth map between smooth manifolds. We say that a submanifold $S \subseteq X$ intersects $\mu$ cleanly if $\mu ^{-1}(S)$ is a submanifold of $M$ satisfying

\begin{align*} T_p \mu^{-1}(S) = d\mu_p^{-1}(T_{\mu(p)}S) \end{align*}

for all $p \in \mu ^{-1}(S)$. Recall that this condition is satisfied if $S$ is transverse to $\mu$, i.e. if

\begin{align*} T_{\mu(p)}S + \operatorname{im} d\mu_p = T_{\mu(p)}X \end{align*}

for all $p \in \mu ^{-1}(S)$.

We are now equipped to begin proving Theorem A from the introduction. The main idea is that for a Hamiltonian system $((M, \omega ), \mathcal {G} \rightrightarrows X, \mu )$, pre-Poisson submanifold $S \subseteq X$, and stabilizer subgroupoid $\mathcal {H} \rightrightarrows S$, the distribution defined by the kernel of the restriction of $\omega$ to $\mu ^{-1}(S)$ coincides with the orbits of the $\mathcal {H}$-action. This result can be stated more generally for Poisson manifolds.

Theorem 2.10 Let $\mathcal {G} \rightrightarrows X$ be a symplectic groupoid acting on a Poisson manifold $(M, \tau )$ in a Hamiltonian way with moment map $\mu : M \longrightarrow X$. Let $S \subseteq X$ be a pre-Poisson submanifold intersecting $\mu$ cleanly, let $\mathcal {H} \rightrightarrows S$ be a stabilizer subgroupoid of $S$ in $\mathcal {G}$, and let $N := \mu ^{-1}(S)$. We then have

\begin{align*} T_p(\mathcal{H} \cdot p) = T_pN \cap \tau(T_pN^{\circ}) \end{align*}

for all $p \in N$.

To prove this theorem, we begin with a few lemmas. Let $((M, \tau ), \mathcal {G} \rightrightarrows X, \mu )$, $\mathcal {H} \rightrightarrows S$, and $N$ be as in Theorem 2.10.

Lemma 2.11 We have

\begin{align*} T_x\mathcal{H} = d\mathsf{s}_x^{-1}(T_xS) \cap d\mathsf{t}_x^{-1}(T_xS) \cap (T_xS)^{\Omega} \end{align*}

for all $x \in S$, where $\Omega$ is the symplectic form on $\mathcal {G}$.

Proof. Let $\varphi : (\ker d\mathsf {t})\vert _X \longrightarrow T^{*}X$ be the isomorphism given by (2.2), so that

\begin{align*} \sigma \circ \varphi = d\mathsf{s} : (\ker d\mathsf{t})\vert_X \longrightarrow TX. \end{align*}

Let $x \in S$. The definition of $\mathcal {H}$ then implies that

\begin{align*} \ker d\mathsf{t}_x \cap T_x\mathcal{H} = \varphi^{-1}(\sigma^{-1}(T_xS) \cap T_xS^{\circ}) = \ker d\mathsf{t}_x \cap d\mathsf{s}^{-1}_x(T_xS) \cap T_xS^{\Omega}. \end{align*}

It follows that

\begin{align*} T_x\mathcal{H} = T_xS \oplus (\ker d\mathsf{t}_x \cap T_x\mathcal{H}) = T_xS \oplus (\ker d\mathsf{t}_x \cap d\mathsf{s}^{-1}_x(T_xS) \cap T_xS^{\Omega}). \end{align*}

However, $T_xS \subseteq T_xS^{\Omega }$ because $X$ is Lagrangian in $\mathcal {G}$, so

\begin{align*} T_x\mathcal{H} = (T_xS \oplus (\ker d\mathsf{t}_x \cap d\mathsf{s}^{-1}_x(T_xS))) \cap T_xS^{\Omega} = d\mathsf{s}_x^{-1}(T_xS) \cap d\mathsf{t}_x^{-1}(T_xS) \cap T_xS^{\Omega}. \end{align*}

Denote the action map by $\psi : \mathcal {G}_{\mathsf {t}}{\times }_\mu M \longrightarrow M$ and the orbit map at a point $p \in M$ over $x := \mu (p) \in X$ by

\begin{align*} \psi_p : \mathcal{G}_x \longrightarrow M, \quad g \longmapsto g \cdot p, \end{align*}

where $\mathcal {G}_x := \mathsf {t}^{-1}(x)$. If $p \in N$, then $T_p(\mathcal {H} \cdot p)$ is the image of $T_x\mathcal {H}_x$ under $d\psi _p$.

Lemma 2.12 For all $f \in C^{\infty }(X)$ and $(g, p) \in \mathcal {G} \times _X M$, we have

\begin{align*} d\psi_p(X_{\mathsf{s}^{*}f}(g)) = X_{\mu^{*}f}(g \cdot p). \end{align*}

Proof. Consider the map

\begin{align*} F : \mathcal{G} \times M \times M^{-} \longrightarrow \mathbb{R}, \quad F(g, p, q) = f(\mathsf{s}(g)) - f(\mu(q)). \end{align*}

The graph $\Gamma$ of the action is coisotropic and $F\vert _\Gamma = 0$, so $X_F$ is tangent to $\Gamma$. However,

\begin{align*} dF = (d(\mathsf{s}^{*}f), 0, -d(\mu^{*}f)), \end{align*}

so $X_F = (X_{\mathsf {s}^{*}f}, 0, X_{\mu ^{*}f})$. Hence,

\begin{align*} (X_{\mathsf{s}^{*}f}(g), 0, X_{\mu^{*}f}(g \cdot p)) = (u, v, d\psi(u, v)) \end{align*}

for some $(u, v) \in T_{(g, p)}(\mathcal {G} \times _X M)$, which implies that $v = 0$ and $u = X_{\mathsf {s}^{*}f}(g)$. We conclude that

\begin{align*} X_{\mu^{*}f}(g \cdot p) = d\psi(u, 0) = d\psi_p(X_{\mathsf{s}^{*}f}(g)). \end{align*}

Lemma 2.13 Let $x \in S$ and $v \in T_x\mathcal {G}$. Then $v \in T_x\mathcal {H}_x$ if and only if $v = X_{\mathsf {s}^{*}f}(x)$ for some $f \in C^{\infty }(X)$ such that $f\vert _S = 0$ and $X_f(x) \in T_xS$.

Proof. By Lemma 2.11, we have

(2.4)\begin{equation} T_x\mathcal{H}_x = \ker d\mathsf{t}_x \cap d\mathsf{s}_x^{-1}(T_xS) \cap (T_xS)^{\Omega}. \end{equation}

Let $v = X_{\mathsf {s}^{*}f}(x)$ be as stated. Then $d\mathsf {t}(v) = 0$ (see [Reference Mikami and WeinsteinMW88, Theorem 1.6(i)]), $d\mathsf {s}(v) = X_f(x) \in T_xS$ because $\mathsf {s}$ is a Poisson map, and

\begin{align*} \Omega(v, T_xS) = df(d\mathsf{s}(T_xS)) = df(T_xS) = 0. \end{align*}

Hence, $v \in T_x\mathcal {H}_x$ by (2.4).

Conversely, let $v \in T_x\mathcal {H}_x$. By (2.4), we have $d\mathsf {t}(v) = 0$, $d\mathsf {s}(v) \in T_xS$ and $\Omega (v)\vert _{T_xX} \in T_xS^{\circ }$. The latter implies that $\Omega (v)\vert _{T_xX} = df_x$ for some $f \in C^{\infty }(X)$ such that $f\vert _S = 0$. Then $\Omega (v) = df \circ d\mathsf {s}_x$ because $T_x\mathcal {G} = T_xX \oplus \ker d\mathsf {s}_x$ and $(\ker d\mathsf {s}_x)^{\Omega } = \ker d\mathsf {t}$. It follows that $v = X_{\mathsf {s}^{*}f}(x)$ and $X_f(x) = d\mathsf {s}(v) \in T_xS$.

We now prove Theorem 2.10.

Proof of Theorem 2.10 To show that

\begin{align*} T_p(\mathcal{H} \cdot p) \subseteq T_pN \cap \tau(T_pN^{\circ}), \end{align*}

let $x := \mu (p)$ and $v \in T_x\mathcal {H}_x$. Note that

\begin{align*} u := d\psi_p(v) \in T_p(\mathcal{H} \cdot p). \end{align*}

Then $u \in T_pN$ because $\mathcal {H}$ acts on $N$. By Lemma 2.13, we have $v = X_{\mathsf {s}^{*}f}(x)$ for some $f \in C^{\infty }(X)$ with $f\vert _S = 0$ and $X_f(x) \in T_xS$, and by Lemma 2.12,

\begin{align*} u = d\psi_p(v) = X_{\mu^{*}f}(p). \end{align*}

However, $d\mu (X_{\mu ^{*}f}(p)) = X_f(x) \in T_xS$ since $\mu$ is a Poisson map; this follows from [Reference Mikami and WeinsteinMW88, Theorem 3.8], as explained in [Reference Bursztyn and CrainicBC05, § 4.2]. Hence, $u \in T_pN$. We also have

\begin{align*} d(\mu^{*}f)(T_pN) = df(d\mu(T_pN)) \subseteq df(T_xS) = 0, \end{align*}

so $u = X_{\mu ^{*}f}(p) \in \tau (T_pN^{\circ })$.

Conversely, let $v \in T_pN \cap \tau (T_pN^{\circ })$. As

\begin{align*} T_pN^{\circ} = (d\mu^{-1}(T_xS))^{\circ} = d\mu^{*}(T_xS^{\circ}), \end{align*}

we have $v = X_{\mu ^{*}f}(p)$ for some $f \in C^{\infty }(X)$ with $f\vert _S = 0$. We also have

\begin{align*} X_f(x) = d\mu(v) \in T_xS. \end{align*}

We thus have $X_{\mathsf {s}^{*}f}(x) \in T_x\mathcal {H}_x$ by Lemma 2.13 and $v = d\psi _p(X_{\mathsf {s}^{*}f}(x))$ by Lemma 2.12, so $v \in T_p(\mathcal {H} \cdot p)$.

The main theorem of this section is the following generalization of Mikami–Weinstein reduction.

Theorem 2.14 (Theorem A(i))

Let $((M, \omega ), \mathcal {G} \rightrightarrows X, \mu )$ be a Hamiltonian system, $S \subseteq X$ a pre-Poisson submanifold, $\mathcal {H} \rightrightarrows S$ a stabilizer subgroupoid of $S$ in $\mathcal {G}$, and $N := \mu ^{-1}(S)$. Suppose that $S$ intersects $\mu$ cleanly and that $N / \mathcal {H}$ has the structure of a smooth manifold for which the quotient map $\pi : N \longrightarrow N / \mathcal {H}$ is a smooth submersion. Then there is a unique symplectic form $\bar {\omega }$ on $N / \mathcal {H}$ satisfying $\pi ^{*}\bar {\omega } = i^{*}\omega$, where $i : N \longrightarrow M$ is the inclusion map.

Proof. We have $\ker d\pi _p = T_p(\mathcal {H} \cdot p)$ for all $p \in N$ by assumption. Hence, as in standard symplectic reduction [Reference Marsden and WeinsteinMW74, Reference Mikami and WeinsteinMW88], it suffices to show that the distribution $\ker (i^{*}\omega ) = TN \cap TN^{\omega }$ on $N$ coincides with that induced by the $\mathcal {H}$-orbits. This is precisely the content of Theorem 2.10.

Definition 2.15 Let $((M, \omega ), \mathcal {G} \rightrightarrows X, \mu )$ be a Hamiltonian system. A reduction datum is a pair $(S, \mathcal {H})$, where $S$ is a pre-Poisson submanifold of $X$ and $\mathcal {H}\rightrightarrows S$ is a stabilizer subgroupoid of $S$ in $\mathcal {G}$. The quotient topological space $\mu ^{-1}(S) / \mathcal {H}$ is called the symplectic reduction of $M$ by $\mathcal {G}$ along $S$ with respect to $\mathcal {H}$ and denoted

\begin{align*} M \mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S, \mathcal{H}}} \mathcal{G} := \mu^{-1}(S) / \mathcal{H}. \end{align*}

We say that the reduction datum $(S, \mathcal {H})$ is clean if the assumptions of Theorem 2.14 are satisfied, in which case $M \mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S, \mathcal {H}}} \mathcal {G}$ is a symplectic manifold. If $\mathcal {H}$ is source-connected, we use the simplified notation

\begin{align*} M \mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S}} \mathcal{G} := M \mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S, \mathcal{H}}} \mathcal{G} \end{align*}

and terminology symplectic reduction of $M$ by $\mathcal {G}$ along $S$.

Remark 2.16 The quotient $M \mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S, \mathcal {H}}} \mathcal {G}$ depends on $\mathcal {H}$ only up to its image in $\mathcal {G}$. In particular, $M \mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S, \mathcal {H}}} \mathcal {G} = M \mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S, \mathcal {G}_S}} \mathcal {G}$ for any source-connected stabilizer subgroupoid $\mathcal {H}$ of $S$ in $\mathcal {G}$. This explains and justifies the simplified notation $M \mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S}} \mathcal {G}$.

Remark 2.17 More generally, let $S \subseteq X$ be a stratified space whose strata are pre-Poisson submanifolds $S_i$ of $X$. For a collection of stabilizer subgroupoids $\mathcal {H} = (\mathcal {H}_i \rightrightarrows S_i)_i$, we may define

\begin{align*} M \mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S, \mathcal{H}}} \mathcal{G} := \bigcup_i \mu^{-1}(S_i) / \mathcal{H}_i, \end{align*}

and regard it as a topological quotient of $\mu ^{-1}(S)$. This notion will be useful when discussing symplectic implosion (§ 4.7) and symplectic cutting (§ 4.8).

Remark 2.18 Suppose that we had assumed $\mathcal {H}$ to be source-connected in Theorem 2.14. The theorem could then be viewed as a special case reduction along a pre-symplectic submanifold, i.e. a submanifold $i: N \hookrightarrow M$ such that $i^{*}\omega$ has constant rank. The distribution $\ker (i^{*}\omega ) \subseteq TN$ would then necessarily be integrable, so that the leaf space would symplectic when smooth [Reference Guillemin and SternbergGS84, Theorem 25.2]. The essence of Theorem 2.14 is as follows: if $\mu : M \longrightarrow X$ is a Poisson map, then for any pre-Poisson submanifold $S \subseteq X$, the leaves in $N := \mu ^{-1}(S)$ are explicitly realized as the orbits of a groupoid action.

2.4 A sufficient condition for smoothness

In the context of Marsden–Weinstein–Meyer reduction, one knows that zero is a regular value of the moment map if the Hamiltonian action in question is free. The goal of this section is to generalize this classical fact to our setting.

Let $((M, \omega ), \mathcal {G} \rightrightarrows X, \mu )$ be a Hamiltonian system, let $(S, \mathcal {H})$ be a reduction datum, and let $N := \mu ^{-1}(S)$.

Proposition 2.19 Let $p \in N$, let $x := \mu (p)$, and let $\varphi _p : \mathcal {H}_x \longrightarrow N$ be the orbit map $\varphi _p(h) = h \cdot x$. We have

\begin{align*} T_xS + \operatorname{im} d\mu_p = (\ker(d\varphi_p)_x)^{\circ}, \end{align*}

where $\ker (d\varphi _p)_x$ is viewed as a subspace of $T_x^{*}X$ via the isomorphism (2.2).

Proof. Regarding $\ker (d\varphi _p)_x$ as a subspace of $T_x\mathcal {H}_x \subseteq T_x\mathcal {G}$, the statement can be reformulated as

\begin{align*} T_xS + \operatorname{im} d\mu_p = T_xX \cap (\ker (d\varphi_p)_x)^{\Omega}. \end{align*}

We prove this reformulated version in the following.

Lemma 2.11 tells us that

\[ \ker (d\varphi_p)_x \subseteq T_x\mathcal{H} \subseteq (T_xS)^{\Omega}, \]

so

\[ T_xS \subseteq T_xX \cap (\ker (d\varphi_p)_x)^{\Omega}. \]

To show that $\operatorname {im} d\mu _p \subseteq (\ker (d\varphi _p)_x)^{\Omega }$, let $v \in T_pM$ and $w \in \ker (d\varphi _p)_x$. By Lemma 2.13, $w = X_{\mathsf {s}^{*}f}(x)$ for some $f \in C^{\infty }(X)$ such that $f\vert _S = 0$ and $X_f(x) \in T_xS$. Lemma 2.12 then shows that $X_{\mu ^{*}f}(p) = d\varphi _p(w) = 0$, so $d(\mu ^{*}f)_p = 0$. It follows that

\begin{align*} \Omega(d\mu(v), w) =-df(d\mathsf{s}(d\mu(v))) = -df(d\mu(v)) = 0. \end{align*}

It remains only to show that

\begin{align*} T_xS^{\circ} \cap (\operatorname{im} d\mu_p)^{\circ} \subseteq T_xX^{\circ} + \Omega(\ker (d \varphi_p)_x), \end{align*}

where all annihilators are taken in $T^{*}_x\mathcal {G}$. To this end, let $\xi \in T_xS^{\circ } \cap (\operatorname {im} d\mu _p)^{\circ }$ and write $\xi \vert _{T_xX} = \Omega (v)\vert _{T_xX}$ with $v \in \ker d\mathsf {t}_x$. It then suffices to prove that $v \in T_x\mathcal {H}_x$ and $d\varphi _p(v) = 0$. We begin by verifying that $v \in T_x\mathcal {H}_x$. First note that $v \in \ker d\mathsf {t}_x \cap (T_xS)^{\Omega }$. By Lemma 2.11, showing that $d\mathsf {s}(v) \in T_xS$ would suffice to prove that $v \in T_x\mathcal {H}_x$. As $\mu$ is Poisson, we have

\begin{align*} d\mathsf{s}(v) = \sigma(\xi\vert_{T_xX}) = d\mu(\omega^{-1}(d\mu^{*}(\xi\vert_{T_xX}))), \end{align*}

and the latter is $0$ because $\xi \in (\operatorname {im} d\mu _p)^{\circ }$. It follows that $v \in T_x\mathcal {H}_x$. To show that $d\varphi _p(v) = 0$, first note that Lemma 2.13 implies that $v = X_{\mathsf {s}^{*}f}(x)$ for some $f \in C^{\infty }(X)$ such that $f\vert _S = 0$ and $X_f(x) \in T_xX$. However,

\begin{align*} df_x =\Omega(v)\vert_{T_xX} = \xi\vert_{T_xX} \in (\operatorname{im} d\mu_p)^{\circ}, \end{align*}

so $d(\mu ^{*}f)_p = 0$ and hence $d\varphi _p(v) = X_{\mu ^{*}f}(p) = 0$ by Lemma 2.12.

We then deduce the following sufficient conditions for $M \mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S, \mathcal {H}}} \mathcal {G}$ to be smooth.

Theorem 2.20 (Theorem A(ii))

If $\mathcal {H}$ acts freely on $N$, then $S$ is transverse to $\mu$. Hence, if the action of $\mathcal {H}$ on $N$ is also proper, then $(S, \mathcal {H})$ is a clean reduction datum and $M \mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S, \mathcal {H}}} \mathcal {G}$ is a symplectic manifold.

Proof. This follows from Proposition 2.19 using the fact that $\ker (d\varphi _p)_x = 0$ if $\mathcal {H}$ acts freely.

Recall that the restriction of $\mathcal {G}$ to $S$ is the topological groupoid

\begin{align*} \mathcal{G}\vert_S := \mathsf{s}^{-1}(S) \cap \mathsf{t}^{-1}(S). \end{align*}

The following criterion will be useful for investigating the properness of the action of $\mathcal {H}$ on $N$.

Proposition 2.21 If $\mathcal {G}$ acts properly on $M$ and $\mathcal {H}$ is closed in $\mathcal {G}\vert _S$, then $\mathcal {H}$ acts properly on