text:
\section*{Introduction}
When a group $G$ acts on a Poisson manifold $M$ by Poisson automorphisms, the action is called Hamiltonian if there exists a moment map
\[\Phi:M\longrightarrow \mathfrak{g}^*\]
whose target is the dual of the Lie algebra of $G$. According to the Marsden--Weinstein Theorem \cite{mar.wei:74}, we may use this moment map to ``reduce'' the symmetries of the manifold $M$ at any point $\xi$ of $\mathfrak{g}^*$ which is a regular value by taking the quotient
\[\Phi^{-1}(\xi)/G_\xi\]
of the fiber above $\xi$ by the action of the stabilizer $G_\xi$. If this quotient is a manifold, it inherits a Poisson structure from $M$.

More generally, we can replace the space $\mathfrak{g}^*$ by an arbitrary Poisson manifold $(X,\pi)$. The graph $L$ of the induced map
\[\pi^\#:T^*X\longrightarrow TX\]
is a Lagrangian subbundle of $TX\oplus T^*X$ and has the natural structure of a Lie algebroid. Any Poisson map
\[\Phi:M\longrightarrow X\]
from a Poisson manifold $M$ induces a Lie algebroid action of $L$ on $M$, and for any point $x$ of $X$ the isotropy Lie algebra $L_{x}$ acts on the fiber above $x$. If $x$ is a regular value and the reduced space
\[\Phi^{-1}(x)/L_x\]
is a manifold, reduction equips it with a natural Poisson structure.

When the space $(X,L_X)$ is instead Dirac, the correct analogue of a moment map is a \emph{Dirac realization} or \emph{strong Dirac map}, a notion introduced by Bursztyn and Crainic \cite{bur.cra:05, bur.cra:09}. A Dirac realization is a forward-Dirac map
\[\Phi:M\longrightarrow X\]
from a Dirac manifold $(M,L_M)$ which satisfies an additional nondegeneracy condition. Such a map induces an action of the Lie algebroid $L_X$ on $M$, and at any point $x$ of $X$ which is a regular value we can form the reduced space
\[\Phi^{-1}(x)/L_{X,x}.\]
If it is a manifold, the Dirac pullback of $L_M$ to the fiber $\Phi^{-1}(x)$ descends to a Poisson structure on this reduced space.

We develop a further generalization of this reduction procedure by introducing the notion of a \emph{generalized reduction level} of $X$. This is a triple $(S,\gamma,\varphi)$ consisting of
\begin{itemize}[topsep=0pt]
\item a submanifold $S$ of $X$,
\item a $2$-form $\gamma\in\Omega^2(S)$, and
\item a Dirac map $\varphi:S\longrightarrow Z$ of constant rank to a Dirac manifold $Z$,
\end{itemize}
which satisfy certain compatibility conditions. Each reduction level is equipped with a \emph{stabilizer subalgebroid}
\[A_{S,\gamma,\varphi}\subset L_X,\]
and we prove the following main theorem, as Theorem \ref{main}.

\begin{theorem*}
Let $\Phi:M\longrightarrow X$ be a Dirac realization, and suppose that $(S,\gamma,\varphi)$ is a generalized reduction level of $X$ which intersects $\Phi$ cleanly. If the quotient
\[\Phi^{-1}(S)/A_{S,\gamma,\varphi}\]
is a manifold, it has a natural Dirac structure, and the map $\varphi$ descends to a Dirac realization
\[\overline{\varphi}:\Phi^{-1}(S)/A_{S,\gamma,\varphi}\longrightarrow Z.\]
\end{theorem*}

\noindent In other words, the symmetries induced on $M$ by the action of $L_X$ are reduced to $L_Z$-symmetries on the quotient space $\Phi^{-1}(S)/A_{S,\gamma,\varphi}$. In the special case when $Z$ is a point, all symmetries are removed and the resulting Dirac structure is Poisson.

This theorem recovers several familiar constructions in Poisson and quasi-Poisson geometry. Among them are Marsden--Weinstein reduction at a point and along a coadjoint orbit \cite{mar.wei:74}, parabolic reduction for the action of a complex reductive group, and the reduction of quasi-Hamiltonian and quasi-Poisson manifolds introduced by Alexeev, Malkin, and Meinrenken \cite{ale.mal.mei:98} and by Alexeev, Kosmann-Schwarzbach, and Meinrenken \cite{ale.kos.mei:02}. When $M$ is Hamiltonian or quasi-Hamiltonian for the action of a compact Lie group, this theorem also recovers symplectic implosion as defined by Guillemin, Jeffrey, and Sjamaar \cite{gui.jef.sja:02}, as well as its quasi-Hamiltonian counterpart introduced by Hurtubise, Jeffrey, and Sjamaar \cite{hur.jef.sja:06}. Moreover, in the case when $M$ is a symplectic manifold, our results can be specialized to the procedure of symplectic reduction along a submanifold which was developed by the second named author together with Crooks \cite{cro.may:21}.

Motivated by examples from geometric representation theory, we apply this theorem to give a number of new constructions in quasi-Poisson geometry that rely on the structure of complex semisimple and reductive groups. In particular, using the Steinberg cross-section of a simply-connected semisimple group, we introduce a quasi-Poisson analogue of Whittaker reduction as well as multiplicative versions of the open Moore--Tachikawa varieties \cite{moo.tac:12} constructed recently by Ginzburg and Kazhdan \cite{gin.kaz:21}.

We also develop a procedure for quasi-Poisson reduction relative to the action of a certain class of subgroups which we call \emph{reducible}. Let $G$ be a Lie group equipped with a non-degenerate symmetric pairing on its Lie algebra $\mathfrak{g}$. A subgroup $H$ of $G$ is \emph{reducible} if the annihilator $\mathfrak{h}^\perp$ of its Lie algebra $\mathfrak{h}$ is a Lie subalgebra of $\mathfrak{g}$. In this case we denote the corresponding subgroup of $G$ by $H^\perp$ and we prove the following theorem, as Theorem \ref{qpred}.

\begin{theorem*}
Let $(\mu,\nu):M\longrightarrow K\times G$ be a quasi-Poisson $K\times G$-space. If the quotient
\[\nu^{-1}(H)/H^\perp\]
is a manifold, then it has the natural structure of a quasi-Poisson $K\times \overline{H}$-space, where
\[\overline{H}=H/(H\cap H^\perp).\]
\end{theorem*}

\noindent If, moreover, the Lie algebra $\mathfrak{h}$ satisfies the additional condition $[\mathfrak{h}^\perp, \mathfrak{h}^\perp] \subset \mathfrak{h}$, we prove that we can reduce $M$ by the action of $H$ along a conjugacy class $\O$ of $\overline{H}$. If the resulting quotient
\[M \sll{\O} H \coloneqq \nu^{-1}(\O H^\perp)/ H\]
is a manifold, it is a quasi-Poisson $K$-space.

While these results hold for any Lie group whose Lie algebra carries an invariant inner product, several of its most interesting applications are in the case when $G$ is a complex reductive group. Then we may take $H=P$ to be a parabolic subgroup, so that $H^\perp=U$ is its unipotent radical and $\overline{H}=L$ is the Levi quotient. We obtain quasi-Poisson $K$-structures on the reductions
\[\nu^{-1}(U)/P\quad\text{and}\quad\nu^{-1}(P)/P,\]
as well as a quasi-Poisson $K\times L$-structure on
\[\nu^{-1}(P)/U.\]
In particular, this gives quasi-Poisson structures on a number spaces which are of interest to geometric representation theory, including on a multiplicative analogue of the Grothendieck--Springer resolution. Additionally, we recover several quasi-Hamiltonian structures that were first introduced by Boalch \cite{boa:11} in the setting of moduli spaces.\\

\noindent\textbf{Outline.}
In Section \ref{first} we recall some background on Dirac manifolds and we review the technical conditions required for pushing forward Dirac structures along quotient maps. In Section \ref{second}, we introduce reduction levels and generalized reduction levels, and we prove our main theorem on Dirac reduction. We end this section by giving many familiar examples of reduction procedures in Poisson, quasi-Poisson, and Dirac geometry that fit into this framework.

In Section \ref{third} we show that products of Dirac realizations can be reduced along the diagonal using a fiber product construction. This gives a Dirac interpretation for the reduction of the fusion product of two quasi-Poisson manifolds, as well as for the universality constructions which appear in symplectic and quasi-Hamiltonian implosion.

In Section \ref{qpsec} we apply these results to the quasi-Poisson framework. We develop multiplicative analogues of Whittaker reduction and of the Moore--Tachikawa varieties, as well as a procedure for quasi-Poisson reduction by a subgroup. In Section \ref{fifth}, we give an interpretation of our results in terms of shifted symplectic geometry, and use this to extend the main result to quotients by subgroupoids which are not necessarily source-connected.

\vspace{.1in}
\noindent\textbf{Acknowledgments.}
During the completion of this work, A.B. was partially supported by a National Science Foundation MSPRF under award number DMS--1902921.\\

\section{Dirac geometry}
\label{first}
\subsection{Dirac manifolds}
Let $M$ be a (real or complex) manifold and fix a closed $3$-form $\eta\in\Omega^3(M)$. Consider the Courant algebroid
\[\mathbb{T}M\coloneqq TM\oplus T^*M,\]
equipped with the symmetric pairing
\[\langle(v,\alpha),(w,\beta)\rangle\coloneqq \alpha(w)+\beta(v)\]
and the $\eta$-twisted Dorfman bracket
\[\llbracket(v,\alpha),(w,\beta)\rrbracket_\eta\coloneqq ([v,w], \mathcal{L}_v\beta-\imath_wd\alpha+\imath_{v\wedge w}\eta),\]
where $(v,\alpha),(w,\beta)\in\Gamma(\mathbb{T}M)$. An \emph{$\eta$-twisted Dirac structure} on $M$ \cite{bur.cra.wei.zhu:04, sev.wei:01} is a vector subbundle $L\subset \mathbb{T}M$ such that
\begin{itemize}[topsep=1.5pt]
\item $L$ is Lagrangian relative to the pairing $\langle\cdot,\cdot\rangle$, and
\item $\Gamma(L)$ is closed under the bracket $\llbracket\cdot,\cdot\rrbracket_\eta$.
\end{itemize}
In this case the restriction of $\llbracket\cdot,\cdot\rrbracket_\eta$ to $\Gamma(L)$ is a Lie bracket, and the anchor map
\begin{align*}
p_T:\quad L\quad&\longrightarrow TM\\
(v,\alpha)&\longmapsto v
\end{align*}
gives $L$ the structure of a Lie algebroid over $M$.

\begin{remark}
Note that the twist $\eta$ is only determined up to a closed form that vanishes on the distribution $p_T(L)$. In other words, suppose that $\eta,\eta'\in\Omega^3(M)$ are closed $2$-forms which agree on the distribution $p_T(L)$. Then $L$ is an $\eta$-twisted Dirac structure if and only if it is an $\eta'$-twisted Dirac structure.
\end{remark}

\begin{example}
Let $\omega\in\Omega^2(M)$ be a $2$-form on $M$ and let $\eta=-d\omega$. Then the graph of $\omega^\flat$ defines an $\eta$-twisted Dirac structure
\[L_\omega\coloneqq \{(v,\omega^\flat(v))\in \mathbb{T}M\mid v\in TM\}\]
on $M$. A Dirac structure $L$ on $M$ is induced by a $2$-form in this way if and only if it satisfies the transversality condition
\[L\cap T^*M=0.\]
This occurs if and only if the anchor map $p_T$ is surjective, and in this case the Dirac structure $L$ is called \emph{presymplectic} or \emph{nondegenerate}. In particular, there is a one-to-one correspondence between symplectic forms on $M$ and non-twisted Dirac structures which are transverse to both of the subbundles $TM$ and $T^*M$ of $\mathbb{T}M$.
\end{example}

More generally, if $L$ is an $\eta$-twisted Dirac structure on $M$ the image of the anchor map $p_T$ is an integrable generalized distribution in $TM$. It corresponds to a foliation of $M$ by nondegenerate leaves, and each nondegenerate leaf $\O$ carries a presymplectic form $\omega_{\mathcal{O}}$ such that
\[d\omega_{\mathcal{O}}=-\eta_{\vert\mathcal{O}}.\]

\begin{example}
The graph of any bivector field $\pi\in\mathcal{X}^2(M)$ defines a Lagrangian subbundle
\[L_\pi\coloneqq \{(\pi^\#(\alpha),\alpha)\in \mathbb{T}M\mid \alpha\in T^*M\}.\]
This is an $\eta$-twisted Dirac structure if and only if the Schouten--Nijenhuis bracket of $\pi$ with itself satisfies
\[[\pi,\pi]=\pi^\#(\eta),\]
In this case, $\pi$ is called an \emph{$\eta$-twisted Poisson structure}.

The \emph{kernel} of a Dirac structure $L_M$ is
\[\ker L_M\coloneqq L_M\cap TM.\]
In particular, there is a one-to-one correspondence between Poisson bivectors and non-twisted Dirac structures with trivial kernel. In this case, the induced nondegenerate foliation of $M$ is exactly the foliation by symplectic leaves.
\end{example}

When the Lie algebroid $L_M$ is integrable, it integrates to a \emph{twisted presymplectic groupoid} \cite{bur.cra.wei.zhu:04}. This is a Lie groupoid
\[(\mathsf{s},\mathsf{t}) : \mathcal{G} \rightrightarrows M\]
together with a multiplicative $2$-form $\Omega\in\Omega^2(\mathcal{G})$ and a closed $3$-form $\eta\in\Omega^3(M)$ such that
\begin{itemize}[topsep=1.5pt]
\item $d\Omega = \mathsf{s}^*\eta - \mathsf{t}^*\eta$,
\item $\dim \mathcal{G} = 2 \dim X$, and
\item $\ker(\Omega) \cap \ker(d\mathsf{s}) \cap \ker(d\mathsf{t}) = 0$.
\end{itemize}
In this case, letting
\[\rho : \Lie(\mathcal{G}) \longrightarrow TX\]
be the anchor map and defining
\begin{align*}
\label{yv16ri54}
\phi : \Lie(\mathcal{G}) &\longrightarrow T^*M\\
		a&\longmapsto \Omega(a, \cdot)_{\vert TM},
\end{align*}
the Dirac structure $L_M$ is the image of the map
\begin{equation}
\label{grpdmaps}
(\rho, \phi) : \Lie(\mathcal{G}) \longrightarrow TM \oplus T^*M,
\end{equation}
and the restriction $\Lie(\mathcal{G}) \longrightarrow L_M$ is an isomorphism of Lie algebroids.

Let $\mathfrak{g}$ be a (real or complex) Lie algebra equipped with an invariant, nondegenerate bilinear form. The \emph{Cartan $3$-form} of $\mathfrak{g}$ is the element $\eta\in\wedge^3\mathfrak{g}^*$ defined by
\begin{equation*}
\eta(x,y,z)=\frac{1}{2}(x,[y,z])\quad \quad\text{for all }x,y,z\in\mathfrak{g},
\end{equation*}
and we abuse notation by writing $\eta\in\Omega^3(G)$ for the corresponding bi-invariant $3$-form on the group $G$ which integrates $\mathfrak{g}$. Under the isomorphism
\begin{equation*}
\mathfrak{g}\cong\mathfrak{g}^*
\end{equation*}
induced by the inner product, the Cartan $3$-form corresponds to a $3$-tensor
\[\chi\in\wedge^3\mathfrak{g}.\]

Suppose that $\mathfrak{g}$ acts on a manifold $M$ through the action map
\begin{align*}
\rho_M:\mathfrak{g}&\longrightarrow TM \\
\xi&\longmapsto \xi_M.
\end{align*}
A \emph{quasi-Poisson} structure on $M$ is a $\mathfrak{g}$-invariant bivector $\pi\in\mathcal{X}^2(M)$ with the property that
\[[\pi,\pi]=\chi_M.\]
This bivector is \emph{Hamiltonian} if there exists a group-valued moment map
\[\Phi:M\longrightarrow G,\]
equivariant relative to the conjugation action of $G$ on itself, which satisfies a multiplicative analogue of the usual moment map condition \cite[Definition 2.2]{ale.kos.mei:02}.

By \cite[Theorem 3.16]{bur.cra:05}, such a manifold corresponds to a $-\Phi^*\eta$-twisted Dirac structure given by
\begin{equation}
\label{qpdirac}
L=\{(\pi^\#(\alpha)+\xi_M,C^*(\alpha)+\Phi^*\sigma(\xi))\mid\alpha\in T^*M, \xi\in\mathfrak{g}\}.
\end{equation}
Here
\begin{align*}
\sigma:\mathfrak{g}&\longrightarrow T^*G\\
		\xi&\longmapsto \tfrac{1}{2}(\xi^L+\xi^R)^\vee,
		\end{align*}
where $x^\vee$ is the dual of $x\in TG$ under the identification $TG\cong T^*G$ given by the nondegenerate bilinear form, and
\[C^*=\Id-\frac{1}{4}e_{iM}\otimes\Phi^*(\theta_i^L-\theta_i^R),\]
where $\{e_i\}$ is an orthonormal basis of $\mathfrak{g}$ and $\theta_i^L$ and $\theta_i^R$ are the corresponding left- and right-invariant $1$-forms on $G$. In particular, the projection of $L$ onto $TM$ is the image of the bundle map
\begin{align*}
T^*M\oplus\mathfrak{g}&\longrightarrow TM \\
		(\alpha,\xi)&\longmapsto\pi^\#(\alpha)+\xi_M.
		\end{align*}
The foliation integrating this distribution is the foliation of $M$ by \emph{quasi-Hamiltonian} leaves.

\begin{example}
\label{cd}
The group $G$ has a natural quasi-Poisson structure relative to the conjugation action \cite[Proposition 3.1]{ale.kos.mei:02}, called the \emph{Cartan--Dirac structure}. It can be viewed as a multiplicative analogue of the classical Kirillov--Kostant--Souriau Poisson structure on $\mathfrak{g}^*$. Its moment map is the identity, and the corresponding Dirac structure is
\begin{equation}
\label{cdexp}
L_G\coloneqq\left\{\left(\xi^L-\xi^R,\sigma(\xi)\right)\mid\xi\in\mathfrak{g}\right\},
\end{equation}
where $\xi^L$ and $\xi^R$ are the left- and right-invariant vector fields on $G$ induced by the Lie algebra element $\xi$ \cite[Example 3.4]{bur.cra:05}.

Note that there is a natural isomorphism
\[L_G\cong G\rtimes\mathfrak{g}\]
between the Cartan--Dirac structure and the action Lie algebroid on $G$ corresponding to the conjugation action. In particular, projecting onto the tangent bundle, the nondegenerate leaves of the Cartan--Dirac structure are the conjugacy classes of $G$.
\end{example}

\begin{example}
\label{fusdouble}
The quasi-Hamiltonian \emph{internal fusion double} $D(G)\coloneqq G\times G$ \cite[Example 5.3]{ale.kos.mei:02} is a nondegenerate quasi-Poisson $G\times G$-manifold with respect to the action
\[(g,h)\cdot (a,b)=(gah^{-1},hbh^{-1}).\]
The associated moment map is given by
\begin{align*}
\Phi:D(G)&\longrightarrow G\times G\\
	(a,b)&\longmapsto (aba^{-1},b^{-1}),
	\end{align*}
and so the quasi-Poisson structure on $D(G)$ is a multiplicative counterpart of the canonical symplectic structure on the cotangent bundle of $T^*G$. Moreover, viewed as a groupoid over $G$ with source and target maps given by the two components of $\Phi$, the double $D(G)$ is the presymplectic groupoid integrating the Cartan--Dirac structure $L_G$.
\end{example}

\subsection{Operations on Dirac structures}
Let $M$ and $N$ be two manifolds and let $f:M\longrightarrow N$ be a smooth map. If $L_N$ is an $\eta_N$-twisted Dirac structure on $N$, its \emph{pullback} along $f$ is the Lagrangian distribution
\[f^*L_N\coloneqq \{(v,f^*\alpha)\in\mathbb{T}M\mid (f_*v,\alpha)\in L_N\}.\]
If this distribution is a smooth subbundle, it is a $f^*\eta_N$-twisted Dirac structure on $M$. When $(M,L_M)$ and $(N,L_N)$ are Dirac manifolds, the map $f$ is \emph{backward-Dirac} or \emph{b-Dirac} if
\[L_{M}=f^*L_{N}.\]
Pullbacks of Dirac structures generalize the pullback of differential forms, and symplectomorphisms, for instance, are b-Dirac maps.

If $L_M$ is an $\eta_M$-twisted Dirac structure on $M$, its \emph{pushforward} along $f$ at the point $p\in M$ is the Lagrangian subspace
\[f_*L_{M,p}\coloneqq \{(f_*v,\beta)\in\mathbb{T}_{f(p)}N\mid (v,f^*\beta)\in L_{M,p}\}.\]
In order for this to define a distribution on $N$, the Dirac structure $L_M$ must be \emph{$f$-invariant}---that is, we must have
\[f_*L_{M,p}=f_*L_{M,q}\]
whenever $f(p)=f(q)$. In this case we denote the pushforward by $f_*L_M$. When $(M,L_M)$ and $(N,L_N)$ are Dirac manifolds, the map $f$ is \emph{forward-Dirac} or \emph{f-Dirac} if
\[L_{N}=f_*L_{M}.\]
This generalizes the pushforward of vector fields---in particular, all Poisson and quasi-Poisson maps are f-Dirac.

When $f$ is a diffeomorphism, it is b-Dirac if and only if it is f-Dirac, and in this case it is called a \emph{Dirac diffeomorphism}. In particular, in the case $(M,L_M)=(N,L_N)$ this occurs if and only if $f^*\eta_M=\eta_M$ and the automorphism
\[(f_*,f^{*-1}):\mathbb{T}M\longrightarrow\mathbb{T}M\]
preserves $L_M$.

Let $(M,L_M)$ and $(X,L_X)$ be two Dirac manifolds. An f-Dirac map $\Phi:(M,L_M)\longrightarrow (X,L_X)$ is called a \emph{Dirac realization} \cite{bur.cra:05} (or \emph{strong} f-Dirac map \cite{bur.cra:09}) if the Dirac structure on $M$ is transverse to the kernel of $\Phi_*$---that is, if
\begin{equation}
\label{strong}
\ker \Phi_*\cap L_M=0.
\end{equation}
Equivalently, $\Phi$ is a Dirac realization if and only if it induces an isomorphism between $\ker L_M$ and $\ker L_X$.

Condition \eqref{strong} implies that, for any $(v,\alpha)\in L_X$, there exists a unique $w\in TM$ such that
\[\Phi_*w=v\quad\text{and}\quad (w,\Phi^*\alpha)\in L_M.\]
This defines a Lie algebroid action \cite[Corollary 3.12]{bur.cra:05} of $L_X$ on $M$ via
\begin{align*}
\rho_M:L_X\times_XM&\longrightarrow TM\\
		(v,\alpha)&\longmapsto w.
\end{align*}

\begin{example}
\label{ex:poisson1}
\cite[Example 2.4]{bur.cra:09} Let $X$ be a Poisson manifold and let $\Phi:M\longrightarrow X$ be a Dirac realization. Because $\Phi$ induces an isomorphism on Dirac kernels, $M$ is also a Poisson manifold and therefore $\Phi$ is a Poisson map. It follows that Dirac realizations of $X$ are in one-to-one correspondence with Poisson maps to $X$.

In particular, let $X=\mathfrak{g}^*$ be the dual of a Lie algebra, equipped with the canonical Kirillov--Kostant--Souriau Poisson structure. Since a Poisson map $\Phi:(M,\pi_M)\longrightarrow (\mathfrak{g}^*,\pi_{\mathrm{KKS}})$ is a Dirac realization, it induces a $\mathfrak{g}$-action
\[\rho_M:\mathfrak{g}\longrightarrow TM.\]
defined by
\[\rho_M=\pi_M^\#\circ \Phi^*,\]
for which $\Phi$ is a moment map. In other words, Dirac realizations of $(\mathfrak{g}^*,\pi_{\mathrm{KKS}})$ are in one-to-one correspondence with Hamiltonian Poisson $\mathfrak{g}$-spaces.
\end{example}

\begin{example}
\label{ex:qpoisson1}
\cite[Theorem 3.16]{bur.cra:05}
If $G$ is a  Lie group whose Lie algebra $\mathfrak{g}$ carries an invariant, nondegenerate bilinear form, any quasi-Poisson moment map is a Dirac realization of $G$ relative to the Cartan--Dirac structure \cite[Proposition 3.19]{bur.cra:05}. Conversely, any Dirac realization
\[\Phi:(M,L_M)\longrightarrow (G, L_G)\]
equips $M$ with an action of $\mathfrak{g}$ and a quasi-Poisson bivector whose associated moment map is $\Phi$ \cite[Proposition 3.20]{bur.cra:05}. Therefore, Dirac realizations of $(G, L_G)$ are in one-to-one correspondence with Hamiltonian quasi-Poisson $\mathfrak{g}$-spaces.
\end{example}

If $L_1$ and $L_2$ are Dirac structures on $M$ twisted by $\eta_1$ and $\eta_2$ respectively, their \emph{sum} is defined to be
\[L_1+ L_2\coloneqq\{(x,\alpha+\beta)\mid (x,\alpha)\in L_1\text{ and }(x,\beta)\in L_2\}.\]
If it is a smooth bundle, then it is a $(\eta_1+\eta_2)$-twisted Dirac structure on $M$ \cite[Definition 2.16]{gua:11}. Similarly, we can define $L_1- L_2$ to be the sum of the Dirac structure $L_1$ and the \emph{opposite} Dirac structure
\begin{equation}
\label{opposite}
-L_2\coloneqq\{(x,-\beta)\mid (x,\beta)\in L_2\}.
\end{equation}
In particular, if $\gamma\in\Omega^2(M)$ is a $2$-form, the \emph{gauge transformation} of $L_1$ by $\gamma$ is
\begin{align*}
\tau_\gamma L_1&\coloneqq L_1+L_\gamma\\
				&=\{(v,\alpha+\gamma^\flat(v))\mid (v,\alpha)\in L\},
				\end{align*}
which is a $(\eta-d\gamma)$-twisted Dirac structure on $M$. In this case we say that the Dirac structures $L_1$ and $\tau_\gamma L_1$ are \emph{gauge-equivalent}. Gauge transformations commute with Dirac pullbacks and pushforwards in the sense that, for any smooth map $f:M\longrightarrow N$ between Dirac manifolds $(M,L_M)$ and $(N,L_N)$, and any $\gamma\in\Omega^2(N)$,
\begin{equation}
\label{gaugecom}
f^*\tau_\gamma L_N=\tau_{f^*\gamma}f^*L_N\quad\text{and}\quad\tau_\gamma f_*L_M=f_*\tau_{f^*\gamma}L_M
\end{equation}
whenever both sides are defined.

\subsection{Quotients of Dirac structures}
Let $(M, L_M)$ be an $\eta$-twisted Dirac manifold. A local vector field $Z$ on $M$ is a \emph{Dirac vector field} if $\L_Z\eta = 0$ and if the Lie derivative $\L_Z$ preserves the sheaf of sections of $L_M.$

\begin{proposition}\label{mplljqhp}
A local vector field $Z$ on $M$ is Dirac if and only if the flow of $Z$ consists of local Dirac automorphisms.
\end{proposition}
\begin{proof}
Let $Z$ be a local vector field, let $\varphi_t$ be its flow, and consider the local automorphism
\[\Phi_t \coloneqq (\varphi_{t*}, \varphi_{-t}^*):\mathbb{T}M\longrightarrow\mathbb{T}M.\]
Note that $\L_Z\eta = 0$ if and only if $\varphi_t^*\eta = \eta$ for all $t$, so it is sufficient to show that
\[\L_Z\Gamma(L_M) \subset \Gamma(L_M) \quad \iff\quad \Phi_t(L_M) = L_M\text{ for all $t$}.\]

Let $V$ be the vector field on $\mathbb{T}M$ whose flow is $\Phi_t$. Since $\Phi_t$ consists of local bundle automorphisms, $V$ corresponds \cite[Propositon 2.2]{mac.xu:98} to the first-order linear differential operator
\begin{align*}
D_V : \Gamma(\mathbb{T}M) &\longrightarrow \Gamma(\mathbb{T}M)\\
				(v,\alpha)&\longmapsto (\L_Zv,\L_Z\alpha).
				\end{align*}
Then $Z$ is a Dirac vector field if and only if $D_V$ preserves the sheaf of sections of $L_M$, which occurs if and only if $V$ is tangent to $L_M$ \cite[Propositon 2.5]{mac.xu:98}, which is the case if and only if $\Phi_t(L_M)$ preserves $L_M$ for all $t$.
\end{proof}

A distribution $D \subseteq TM$ is called a \emph{Dirac distribution} if every vector in $D$ extends to a Dirac local section of $D$. Such a distribution is \emph{regular} if $D \cap L_M$ has constant rank.

\begin{remark}
\label{mzq8z1k7}
Suppose that $D \subset TM$ is a distribution contained in $L_M \cap \ker \eta$. Then for every local section $Z$ of $D$ we have
\[(\L_Zv, \L_Z\alpha) = \llbracket (Z, 0), (v, \alpha) \rrbracket_\eta,\]
so $Z$ is a Dirac vector field by integrability of $L_M$. Therefore $D$ is a regular Dirac distribution.

In general, not every local section of a Dirac distribution is Dirac. For example, if $(M, \omega)$ is symplectic, then a local vector field $Z$ is Dirac if and only if $\L_Z\omega = 0$.
In particular, every Hamiltonian vector field is Dirac, so $D = TM$ is a regular Dirac distribution. However, there are many local sections $Z$ of $TM$ which do not satisfy $\L_Z\omega = 0$.
\end{remark}

When $D$ is an involutive distribution on $M$, we will say that the leaf space $M/D$ \emph{is a manifold} if it has the structure of a smooth manifold such that the quotient map $\pi:M\longrightarrow M/D$ is a smooth submersion. The next proposition shows that, when $D$ is regular and Dirac, under suitable assumptions the Dirac structure on $M$ descends to a Dirac structure on the corresponding quotient.

\begin{proposition}
\label{bgfkcgj1}
Let $D \subseteq TM$ be an involutive regular Dirac distribution contained in $\ker \eta$, and suppose that the leaf space $Q \coloneqq M/D$ is a manifold with quotient map
\[\pi : M \longrightarrow Q.\]
\begin{enumerate}[label=\textup{(\arabic*)}]
\item\label{tpcp6ia5}
The pushforward
\[L_Q \coloneqq \pi_*L_M\]
is a $\theta$-twisted Dirac structure on $Q$, where $\theta$ is uniquely characterized by the property that $\pi^*\theta = \eta$.
\item\label{15i0o7u8}
If $(X,L_X)$ is another Dirac manifold and $f : (M,L_M) \longrightarrow (X, L_X)$ is an f-Dirac map such that $D$ is contained in the kernel of $f_*$, then $f$ descends to an f-Dirac map
\[\bar{f} : (Q,L_Q) \longrightarrow (X,L_X).\]
If $D$ contains $\ker f_* \cap L_M$, then $\bar{f}$ is a Dirac realization.
\item\label{gpxy5jhi}
If $D$ is contained in the kernel of $L_M$, then the f-Dirac map
\[\pi : (M,L_M) \longrightarrow (Q, L_Q)\]
is also b-Dirac.
\item\label{r0dvmlu4}
If $D$ is contained in the kernel of $L_M$, the nondegenerate leaves of $Q$ are the images of nondegenerate leaves of $M$. In particular, if $M$ is nondegenerate and $L_M = L_\omega$, then $L_Q$ is also nondegenerate, and the corresponding 2-form $\bar{\omega}$ is uniquely characterized by the property that $\pi^*\bar{\omega} = \omega$.
\end{enumerate}
\end{proposition}

\begin{proof}
\ref{tpcp6ia5}
Since $D$ is contained in $\ker \eta$ and $d\eta = 0$, there is a unique closed 3-form $\zeta$ on $Q$ such that $\pi^*\zeta = \eta$. By \cite[Proposition 1.13]{bur:13}, it then suffices to show that $L_M$ is $\pi$-invariant. Let $p$ and $q$ be points of $M$ such that $\pi(p)=\pi(q)$, and suppose that $Z$ is a Dirac local section of $D$ whose flow $\varphi$ at some time satisfies
\[\varphi(p)=q.\]
Let $(\pi_*v,\alpha)$ be an element of $\pi_*L_{M,p}$, so that $(v, \pi^*\alpha) \in L_{M, p}$. Then, since the local automorphism $(\varphi_*, \varphi^{-1*})$ of $\mathbb{T}M$ preserves $L_M$ by Proposition \ref{mplljqhp}, we have
\[(\varphi_*v, \varphi^{-1*}\pi^*\alpha)=(\varphi_*v,\pi^*\alpha) \in L_{M, q}.\]
It follows that
\[(\pi_*v, \alpha)=(\pi_*\varphi_*v,\alpha)\in \pi_*L_{M,q},\]
so $\pi_*L_{M,p}$ is contained in $\pi_*L_{M,q}$. Since these are vector spaces of the same dimension, it follows that they are equal and therefore $L_M$ is $\pi$-invariant.

\ref{15i0o7u8}
Suppose that $D$ is contained in $\ker f_*$ and let $\bar{f} : Q \longrightarrow X$ be the induced map. Then
\[\bar{f}_*L_Q = \bar{f}_* \pi_* L_M = f_* L_M = L_X,\]
so $\bar{f}$ is f-Dirac. Suppose moreover that $D$ contains $\ker f_* \cap L_M$. Then
\[\ker\bar{f}_*\cap L_Q=\ker\bar{f}_*\cap\pi_* L_M=\pi_*(\ker f\cap L_M)=0,\]
so $\bar{f}$ is a Dirac realization.

\ref{gpxy5jhi}
Suppose that $D$ is contained in the kernel of $L_M$ and let
\[(u,\pi^*\alpha)\in\pi^*\pi_* L_M,\]
so that $(\pi_*u, \alpha)\in \pi_*L_M$. Then there is an element $v$ in $TM$ such that $\pi_*v=\pi_*u$ and $(v,\pi^*\alpha)\in L_M$, so $u - v$ is contained in $D$ and therefore also in the kernel of $L_M$. It follows that
\[(u, \pi^*\alpha) = (v, \pi^*\alpha) + (u - v, 0) \in L_M,\]
so $\pi^*\pi_*L_M$ is contained in $L_M.$ Since both sides are distributions on $M$ of the same rank, this implies that they are equal. Therefore $\pi$ is b-Dirac.

\ref{r0dvmlu4}
Suppose once again that $D$ is contained in the kernel of $L_M$. Then
\[p_T(\pi_*L_M)=\pi_*p_T(L_M\cap (TM\oplus\im\pi^*))=\pi_*p_T(L_M),\]
where the second equality follows from the fact that $\pi$ is b-Dirac by part \ref{gpxy5jhi}. In particular, if $L_M$ is nondegenerate and $L_M=L_\omega$, this implies that $L_Q$ is also nondegenerate. In this case, whenever $Z$ is a Dirac vector field we have
\[\L_Z\omega = 0\quad\text{and}\quad \imath_Z\omega = 0,\]
where the first equality follows by definition and the second follows from the fact that $D$ is in the kernel of $L_M$. Therefore there is a unique 2-form $\bar{\omega}$ on $Q$ such that $\pi^*\bar{\omega} = \omega$, and so $L_Q=L_{\bar{\omega}}$.
\end{proof}

\begin{remark}
\label{strata1}
More generally, $M=\sqcup M^i$ may be a stratified space in which each stratum $M^i$ is a Dirac manifold. Then we can apply Theorem \ref{bgfkcgj1} to any collection $D=\{D^i\subset TM^i\}$ of involutive regular Dirac distributions for which the leaf spaces $M^i/D^i$ are smooth manifolds. We then obtain a stratified quotient space
\[M/D\coloneqq\bigsqcup M^i/D^i\]
whose strata are also Dirac manifolds.
\end{remark}

\vspace{.1in}

\section{Dirac reduction}
\label{second}
\subsection{Reduction levels of Dirac manifolds}
Let $(X,L_X)$ be an $\eta$-twisted Dirac manifold.

\begin{definition}
\label{levels}
A \emph{reduction level} of $X$ is a pair $(S,\gamma)$ consisting of a submanifold $\imath:S\lhook\joinrel\longrightarrow X$ and a 2-form $\gamma\in\Omega^2(S)$ such that
\begin{itemize}[topsep=1.5pt]
\item the distribution
\begin{align*}
A_{S,\gamma} &\coloneqq L_X \cap \imath_*L_{\gamma}\\
		&\,\,= \{(v, \alpha) \in L_X \mid v \in TS \text{ and } \imath^*\alpha=\gamma^\flat(v)\}
		\end{align*}
has constant rank, and
\item $p_T(A_{S, \gamma}) \subset \ker(\imath^*\eta + d\gamma)$.
\end{itemize}
Proposition \ref{subalg} shows that in this case $A_{S,\gamma}$ is a Lie algebroid, called the \emph{stabilizer subalgebroid} of $(S,\gamma)$.
\end{definition}

\begin{remark}
\label{uvxtvk5e}
If $\gamma,\gamma'\in\Omega^2(S)$ are such that the difference $\gamma-\gamma'$ vanishes along the distribution $TS\cap p_T(L_X)$, then
\[A_{S,\gamma}=A_{S,\gamma'}.\]
Therefore, in the special case when $\eta_{\vert TS\cap T\O} = 0$ for all nondegenerate leaves $\O$ we simply write $S$ for the reduction level $(S,0)$ and $A_{S}$ for the corresponding stabilizer subalgebroid.
\end{remark}

Definition \ref{levels} has the following generalization, which we will use throughout what follows.

\begin{definition}
\label{genlevels}
A \emph{generalized reduction level} of $X$ is a triple $(S,\gamma,\varphi)$ consisting of a submanifold $\imath:S\lhook\joinrel\longrightarrow X$, a $2$-form $\gamma\in\Omega^2(S)$, and a smooth map $\varphi:S\longrightarrow Z$ of constant rank to a Dirac manifold $(Z,L_Z)$ such that
\begin{itemize}[topsep=1.5pt]
\item $\varphi_*(\tau_{-\gamma}\imath^*L_X) = L_Z$,
\item
the distribution
\begin{align*}
A_{S,\gamma,\varphi} &\coloneqq L_X \cap \imath_* L_\gamma\cap (\ker \varphi_* \oplus T^*X)\\
					&= \{(v, \alpha) \in L_X \mid v \in \ker \varphi_* \text{ and } \imath^*\alpha=\gamma^\flat(v)\}
					\end{align*}
has constant rank, and
\item $p_T(A_{S,\gamma,\varphi}) \subset \ker(\imath^*\eta + d\gamma)$.
\end{itemize}
Proposition \ref{subalg} shows that in this case $A_{S,\gamma,\varphi}$ is a Lie algebroid, and we call it the \emph{stabilizer subalgebroid of} $(S, \gamma, \varphi)$.
\end{definition}

Note that, while the Lagrangian distribution $\imath^*L_X$ is not necessarily a smooth bundle and therefore may fail to be a Dirac structure on $S$, in the first condition of Definition \ref{genlevels} we can nevertheless define its gauge transform by $-\gamma$ and its pushforward along $\varphi$ pointwise.

\begin{remark}
When $Z$ is a single point, Definition \ref{genlevels} specializes to the ordinary reduction levels of Definition \ref{levels} and we use the same notation. Moreover, as in Remark \ref{uvxtvk5e}, in the case where $\eta_{\vert TS\cap T\O} = 0$ for all nondegenerate leaves $\O$ we suppress $\gamma$ to write $(S,\varphi)$ for the generalized reduction level $(S,\gamma,\varphi)$ and $A_{S,\varphi}$ for the stabilizer subalgebroid $A_{S,\gamma,\varphi}$.
\end{remark}

\begin{proposition}
\label{subalg}
The distribution $A_{S,\gamma,\varphi}$ is a Lie subalgebroid of $\imath^*L_X$.
\end{proposition}

\begin{proof}
Since $A_{S,\gamma,\varphi}$ is an intersection of vector bundles and has constant rank, it is a smooth subbundle of $\imath^*L_X$. It remains to show that for any two sections $(v, \alpha)$ and $(w, \beta)$ of $L_X$ which restrict to sections of $A_{S,\gamma,\varphi}$, the bracket
\[\llbracket(v, \alpha), (w, \beta) \rrbracket_\eta\]
also restricts to a section of $A_{S,\gamma,\varphi}$. Since $v$ and $w$ are contained in $\ker \varphi_*$, so is $[v, w]$. Moreover,
\begin{align*}
\imath^*(\imath_{v\wedge w}\eta) = -\imath_{v\wedge w} d\gamma
				&= \imath_w d(\imath_v \gamma)-\L_v (\imath_w \gamma) + \imath_{[u, v]}\gamma \\
				&=  \imath_wd(\imath^*\alpha)-\L_v(\imath^*\beta)+  \imath_{[u, v]}\gamma,
\end{align*}
so
\[\gamma^\flat([u, v])= \L_v(\imath^*\beta)-\imath_wd(\imath^*\alpha)+\imath_{v\wedge w}(\imath^*\eta)\]
and therefore $\llbracket(u, \alpha), (v, \beta) \rrbracket_\eta$ restricts to a section of $A_{S,\gamma,\varphi}$.
\end{proof}

\subsection{Examples of reduction levels}
Many reduction procedures in Poisson, quasi-Poisson, and more general Dirac geometry occur at ordinary reduction levels. Before we prove the general reduction Theorem \ref{main}, we give a series of familiar examples.

\begin{example}
\label{ex:poisson2}
If $Z$ is a point and $\eta_{\vert TS\cap T\O} = 0$, the corresponding distribution is
\[A_S=L_X\cap (TS\oplus TS^\circ).\]
In particular, when $X$ is a Poisson manifold, all coisotropic submanifolds and, more generally, all pre-Poisson submanifolds \cite{cat.zam:09} are reduction levels of $X$.
\end{example}

\begin{example}
\label{point}
Any point $x\in X$ is a reduction level, and its stabilizer subalgebroid
\[A_x=L_{X,x}\]
is the isotropy Lie algebra of $L_X$ at $x$.
\end{example}

\begin{example}
Suppose that the Dirac manifold $(X,L_X)$ is untwisted, and that $\O\subset X$ is a nondegenerate leaf. Then
\begin{align*}
A_\O&=L_X\cap (T\O\oplus T\O^\circ)\\
					&=L_X\cap (0\oplus T\O^\circ) \\
					&=p_T(L_X)^\circ\cap T\O^\circ =0\oplus T\O^\circ,
					\end{align*}
where the first equality follows from Example \ref{ex:poisson2} and the second follows since $L_X$ is Lagrangian. Therefore $A_\O$ has constant rank and $\O$ is a reduction level of $X$. In particular, in the case when $X=\mathfrak{g}^*$ is the dual of a Lie algebra and $\O\subset\mathfrak{g}^*$ is a coadjoint orbit we obtain
\[A_{\O}=\{(\xi,x)\in\mathcal{O}\times\mathfrak{g}\mid \ad_x^*(\xi)=0\}.\]
\end{example}

\begin{example}
\label{orbit}
Let $\O$ be a nondegenerate leaf of $X$ with presymplectic form $\omega$. Then
\[d\omega=-\imath^*\eta\]
and
\[A_{\O,\omega}=L_X\cap\imath_*L_{\omega}=L_X\cap\imath_*\imath^*L_X.\]
Since the inclusion map of a presymplectic leaf is both b-Dirac and f-Dirac \cite[Example 3.5]{bur.cra:05},
\[A_{\O,\omega}=L_{X\vert\O}\]
is simply the restriction of the bundle $L_X$ to $\O$. In particular, it has constant rank, and therefore $(\O,\omega)$ is a reduction level of $X$. In the special case when $X=\mathfrak{g}^*$ is the dual of a Lie algebra and $\O\subset\mathfrak{g}^*$ is a coadjoint orbit equipped with the Kirillov--Kostant--Souriau symplectic form $\omega$, the stabilizer subalgebroid is the action Lie algebroid
\[A_{\O,\omega}=\mathcal{O}\rtimes\mathfrak{g}.\]
\end{example}

\begin{lemma}
\label{ptrans}
Suppose that $\imath:S\lhook\joinrel\longrightarrow X$ is a submanifold such that
\begin{itemize}
\item $S$ intersects every nondegenerate leaf of $X$ transversally,
\item $\eta_{\vert S\cap\O}=0$ for every nondegenerate leaf $\O$ of $X$, and
\item the kernel of $\imath^*L_X$ has constant rank.
\end{itemize}
Then $S$ is a reduction level of $X$ with stabilizer subalgebroid $A_S=\ker\imath^*L_X$.
\end{lemma}
\begin{proof}
By \cite[Lemma 1.18]{bal:21}, transversality implies that
\[L_X\cap TS^\circ=0,\]
and in particular that $\imath^*L_X$ is a non-twisted Dirac structure on $S$. Moreover, we have
\[\ker\imath^* L_X\cong\frac{L_X\cap(TS\oplus TS^\circ)}{L_X\cap TS^\circ}\cong L_X\cap(TS\oplus TS^\circ).\]
It follows that $A_S=\ker\imath^*L_X$ has constant rank, and therefore that $S$ is a reduction level.
\end{proof}

In particular, a submanifold $\imath:S\lhook\joinrel\longrightarrow X$ is called a \emph{Poisson transversal} if $S$ intersects every nondegenerate leaf $\O$ of $X$ transversally, and the restriction of the presymplectic form $\omega_{\O}$ to the intersection $S\cap\O$ is a symplectic form. In this case, the pullback $\imath^*L_X$ is a Poisson structure, so its kernel is trivial. In view of Lemma \ref{ptrans}, this means that $S$ is a reduction level of $X$ with trivial stabilizer subalgebroid.

Generalized reduction levels appear in various examples from geometric representation theory, as well as in symplectic \cite{gui.jef.sja:02} and quasi-Hamiltonian implosion \cite{hur.jef.sja:06}, where reduction takes place along the faces of the dominant Weyl chamber.

\begin{example}
\label{unips}
Let $\mathfrak{g}$ be a reductive complex Lie algebra, and equip it with a nondenerate invariant symmetric bilinear form. The resulting isomorphism $\mathfrak{g}\cong\mathfrak{g}^*$ gives $\mathfrak{g}$ a Kirillov--Kostant--Souriau Poisson structure. Let $\mathfrak{p}$ be a parabolic subalgebra of $\mathfrak{g}$ let $\mathfrak{u}$ be its nilradical. Then
\begin{align*}
A_{\mathfrak{u}}&= L_{\mathfrak{g}}\cap (T\mathfrak{u}\oplus T\mathfrak{u}^\circ)\\
			&=\{(\xi,x)\in\mathfrak{p}\times\mathfrak{g}\mid x\in\mathfrak{p}\}\\
			&=\mathfrak{u}\rtimes\mathfrak{p}
			\end{align*}
has constant rank, so $\mathfrak{u}$ is a reduction level of $\mathfrak{g}$ with stabilizer subalgebroid $\mathfrak{u}\rtimes\mathfrak{p}$.
\end{example}

\begin{example}
\label{parabs}
In the same setting as Example \ref{unips}, let $\imath:\mathfrak{p}\lhook\joinrel\longrightarrow\mathfrak{g}$ be the inclusion of a parabolic subalgebra with Levi decomposition
\[\mathfrak{p}=\l +\mathfrak{u},\]
so that the nondegenerate form on $\mathfrak{g}$ restricts to a nondegenerate invariant symmetric bilinear form on $\l$. Consider the quotient map
\[\varphi:\mathfrak{p}\longrightarrow\l,\]
which fits into the commutative diagram
\begin{equation*}
\begin{tikzcd}[column sep=large, row sep=large]
\mathfrak{p}			\arrow[r, "\varphi"] \arrow[d, hook, swap, "\imath"]			&\l	\\
\mathfrak{g},		\arrow[ru, swap, "f"]							&
\end{tikzcd}
\end{equation*}
where the diagonal map is induced by the restriction $\mathfrak{g}^*\longrightarrow\l^*$ and is therefore a Poisson map.

Identifying $L_{\mathfrak{g}}$ with the action Lie algebroid $\mathfrak{g}\rtimes\mathfrak{g}$, the kernel of the distribution $\imath^*L_{\mathfrak{g}}$ is precisely
\[\mathfrak{p}\rtimes\mathfrak{u},\]
and therefore $\varphi_*\imath^*L_{\mathfrak{g}}=f_*L_{\mathfrak{g}}=L_{\l}$. Moreover, the stabilizer subalgebroid is
\begin{align*}
A_{\mathfrak{p},\varphi}&= L_{\mathfrak{g}}\cap (T\mathfrak{p}\oplus T\mathfrak{p}^\circ)\cap (\ker\varphi_*\oplus T^*\mathfrak{g})\\
			&=\{(\xi,x)\in\mathfrak{p}\times\mathfrak{g}\mid x\in\mathfrak{p}\cap\mathfrak{u}\}\\
			&=\mathfrak{p}\rtimes\mathfrak{u}.
			\end{align*}
If follows that $(\mathfrak{p},\varphi)$ is a generalized reduction level of $\mathfrak{g}$.
\end{example}

\begin{example}
In a multiplicative parallel to Example \ref{parabs}, let $G$ be a complex reductive group and equip its Lie algebra $\mathfrak{g}$ with an invariant nondegenerate bilinear form, so that $G$ has a Cartan--Dirac structure. Let $\imath:P\lhook\joinrel\longrightarrow G$ be the inclusion of a parabolic subgroup with unipotent radical $U$, and write
\[\varphi:P\longrightarrow L\]
for the projection of $P$ onto its Levi $L\coloneqq P/U$. We write $\mathfrak{p}$, $\l$, and $\mathfrak{u}$ for the Lie algebras of $P$, $L$, and $U$ respectively.

The nondegenerate form on $\mathfrak{g}$ restricts to a nondegenerate form on the Lie algebra $\l=\text{Lie }L$, and we compute
\begin{align*}
\varphi_*\imath^*L_G&=\varphi_*\{(x^R-x^L, x^{R\vee}+x^{L\vee})\mid x\in\mathfrak{p}\}\\
				&=\{(x^R-x^L, x^{R\vee}+x^{L\vee})\mid x\in\l\}\\
				&=L_L.
				\end{align*}
Moreover, using the identification \eqref{cdexp}
\begin{align*}
A_{(P,\varphi)}&=L_G\cap (\ker\varphi_*\oplus TP^\circ)\\
			&=\{(\xi^R-\xi^L,\sigma(\xi))\mid \xi\in\mathfrak{u}\\
			&=P\rtimes\mathfrak{u}
			\end{align*}
is a distribution of constant rank, and the Cartan $3$-form vanishes along
\[p_T(A_{P,\varphi})=\{x^R-x^L\mid x\in\mathfrak{u}\}\]
since the nilradical $\mathfrak{u}$ satisfies $(\mathfrak{u},[\mathfrak{u},\mathfrak{u}])=0$. Therefore $(P,\varphi)$ is a generalized reduction level of $G$ with stabilizer subalgebroid $P\rtimes\mathfrak{u}$. This is a special case of a more general class of reduction levels of Cartan--Dirac structures discussed in Section \ref{qpsec}.
\end{example}

\begin{example}
\label{faces}
Let $\mathfrak{g}$ be the Lie algebra of a compact Lie group, choose a maximal Cartan subalgebra $\t$ and a set of simple roots $\Delta$, and denote by $\t^*_+\subset\t^*$ the corresponding closed dominant Weyl chamber. This chamber is a disjoint union
\[\t^*_+=\bigcup_{J\subset\Delta}\sigma_J\]
of faces defined by
\[\sigma_J\coloneqq\{\xi\in\t^*_+\mid \xi(\alpha^\vee)=0\text{ if and only if }\alpha\in J\},\]
where we write $\alpha^\vee$ for the coroot corresponding to the simple root $\alpha$. Note that any two elements of $\sigma_J$ have the same centralizer under the coadjoint action, which we denote by $\mathfrak{g}_J$.

Writing $\imath:\sigma_J\lhook\joinrel\longrightarrow\mathfrak{g}^*$ and $\varphi:\sigma_J\lhook\joinrel\longrightarrow\t^*$ for the inclusion maps, we see that
\[\varphi_*\imath^*L_{\mathfrak{g}^*}=L_{\t^*}\]
are both the zero Poisson structure on $\t^*$, since $\sigma_J$ is a cross-section for the coadjoint orbits that it meets. Moreover,
\begin{align*}
A_{\sigma_J,\varphi}&= L_{\mathfrak{g}^*}\cap (T\sigma_J\oplus T\sigma_J^\circ)\cap (\ker\varphi_*\oplus T^*\mathfrak{g}^*) \\
			&=\{(\xi,x)\in\sigma_J\times\mathfrak{g}\mid \ad_x^*\xi\in (T_\xi\sigma_J\cap 0), x\in T_\xi\sigma_J^\circ\}\\
			&=\{(\xi,x)\in\sigma_J\times\mathfrak{g}\mid x\in[\mathfrak{g}_\xi,\mathfrak{g}_\xi]\}\\
			&=\sigma_J\rtimes[\mathfrak{g}_J,\mathfrak{g}_J],
			\end{align*}
where the third equality follows, for instance, from the proof of \cite[Lemma 4.27]{cro.may:21}. Therefore $(\sigma_J,\varphi)$ is a generalized reduction level of $\mathfrak{g}^*$ with stabilizer subalgebroid $\sigma_J\rtimes [\mathfrak{g}_J,\mathfrak{g}_J]$, and we can view the closed dominant Weyl chamber $(\t^*_+,\varphi)$ as a stratified reduction level of $\mathfrak{g}^*$.
\end{example}

\begin{example}
\label{alcove}
Let $G$ be a simply-connected compact Lie group whose Lie algebra carries an invariant nondegenerate symmetric bilinear form, so that $G$ has a Cartan--Dirac structure. As in Example \ref{faces}, write $\mathfrak{g}$ for its Lie algebra, $T$ for a fixed choice of maximal torus with Lie algebra $\t$, and $\Delta$ for a fixed set of simple roots. For each subset $J\subset\Delta$ define
\begin{equation}
\label{expstrata}
S_J\coloneqq\{t\in T\mid \alpha(t)=1\text{ if and only if }\alpha\in J\}
\end{equation}
to be the image of the corresponding face of the dominant alcove $\overline{\mathcal{A}}$ under the exponential map. Note that any two elements in the submanifold $S_J$ of $G$ have the same centralizer, whose Lie algebra is precisely the subalgebra $\mathfrak{g}_J$ of Example \ref{faces}.

Writing once again $\imath:S_J\lhook\joinrel\longrightarrow G$ and $\varphi:S_J\lhook\joinrel\longrightarrow T$ for the inclusion maps, we see that $\varphi_*\imath^*L_{G}$ is the zero Poisson structure on $T$ because $S_J$ intersects each conjugacy class in a unique point. Moreover,
\begin{align*}
A_{S_J,\varphi}&= L_G\cap (TS_J\oplus TS_J^\circ)\cap (\ker\varphi_*\oplus T^*G) \\
			&=\{(g,x)\in S_J\times\mathfrak{g}\mid x^R-x^L\in (TS_J\cap 0), x^{R\vee}+x^{L\vee}\in TS_J^\circ\}\\
			&=\{(g,x)\in S_J\times\mathfrak{g}\mid x\in[\mathfrak{g}_g,\mathfrak{g}_g]\}\\
			&=S_J\rtimes[\mathfrak{g}_J,\mathfrak{g}_J],
			\end{align*}
where once again the third equality follows from the proof of \cite[Lemma 4.27]{cro.may:21}. Since the Cartan $3$-form $\eta$ vanishes on the projection
\[p_T(A_{S_J,\varphi})=0,\]
we conclude that $(S_J,\varphi)$ is a generalized reduction level of $G$ with stabilizer subalgebroid $S_J\rtimes [\mathfrak{g}_J,\mathfrak{g}_J]$. We can therefore view image of the closed dominant alcove $(\overline{\mathcal{A}},\varphi)$ as a stratified reduction level of the compact Lie group $G$.
\end{example}

\subsection{Reducing Dirac realizations}
Let $(M, L_M)$ be a Dirac manifold together with a Dirac realization
\[\Phi:M\longrightarrow X.\]
Via \eqref{strong}, there is a Lie algebroid action
\[\rho_M:L_{X}\times_{X}M\longrightarrow TM.\]
Let $(S,\gamma,\varphi)$ be a generalized reduction level of $X$ and suppose that $S$ has clean intersection with $\Phi$, so that the preimage
\[C \coloneqq \Phi^{-1}(S)\]
is a smooth submanifold of $M$. We then have a Cartesian diagram
\begin{equation}
\label{diagram}
\begin{tikzcd}[row sep=large]
C			\arrow[r, hook, "\jmath"] \arrow[d, swap, "\Phi_C"]			&M	\arrow[d, "\Phi"]\\
S		\arrow[r, hook, "\imath"]							&X.
\end{tikzcd}
\end{equation}
The action of $L_{X}$ on $M$ restricts to an action of the stabilizer subalgebroid $A_{S,\gamma,\varphi}$ on $C$, with action map
\[\rho_C : A_{S,\gamma,\varphi} \times_S C \longrightarrow TC.\]

\begin{proposition}
\label{restriction}
Suppose that the isotropy algebras of $\rho_C$ have constant dimension along $C$. Then
\begin{equation}
\label{twistpull}
L_C \coloneqq \tau_{-\Phi_C^*\gamma}\jmath^*L_M
\end{equation}
is a $\Phi_C^*(\imath^*\eta + d\gamma)$-twisted Dirac structure on $C$ which satisfies
\[\Phi_{C*}L_C=\tau_{-\gamma}\imath^*L_X.\]
\end{proposition}

\begin{proof}
Since $\rho_M(v,\alpha)=0$ implies that $v=0$, the isotropy subalgebras of $\rho_C$ can be written
\begin{equation}
\label{eq:dim}
\ker\rho_C=\ker\rho_M\cap L_X\cap \imath_*L_\gamma\cap \ker \varphi_*=\ker\rho_M\cap L_X\cap TS^\circ.
\end{equation}

To show that $L_C$ is a Dirac structure, we must show that $\jmath^*L_M$ is a smooth bundle. There is a natural isomorphism
\[\jmath^*L_M\cong\frac{L_M\cap(TC\oplus T^*M)}{L_M\cap TC^\circ},\]
and by \cite[Section 3.1]{cou:90} it suffices to show that $L_M\cap TC^\circ$ has constant rank. We have
\begin{align*}
L_M\cap TC^\circ&=\{(0,\beta)\in L_M\mid \beta\in TC^\circ\}\\
					&=\{(0,\Phi^*\alpha)\in L_M\mid \alpha\in TS^\circ\}.
					\end{align*}
					%
Because $\Phi$ is f-Dirac, $(0,\Phi^*\alpha)\in L_M$ implies that $(0,\alpha)\in L_{X}$. At each point of $C$ we get a natural isomorphism
\[L_M\cap TC^\circ\cong \frac{\ker\rho_M\cap L_X\cap TS^\circ}{\ker\Phi^*}.\]
Since $C$ has clean intersection with $\Phi$, the bottom of this quotient has constant dimension along $S$. Moreover, in view of \eqref{eq:dim} and the assumption of the proposition, the top also has constant dimension along $S$. Therefore the dimension of the left-hand side is constant along $C$, and it follows that $L_C$ is a Dirac structure.

To compute the twist we chase \eqref{diagram}. Since $\Phi$ is a Dirac realization and $\tau_{\Phi_C^*\gamma}$ is a gauge transform, the Dirac structure $L_C$ is twisted by
\[\jmath^*\Phi^*\eta + d\Phi_C^*\gamma = \Phi_C^*(\imath^*\eta + d\gamma).\]
Moreover,
\begin{align*}
\Phi_{C*}L_C&=\Phi_{C*}\tau_{-\Phi_C^*\gamma}\jmath^*L_M \\
			&=\tau_{-\gamma}\Phi_{C*}\jmath^*L_M \\
			&=\tau_{-\gamma}\imath^*L_X,
			\end{align*}
where the second equality follows from \eqref{gaugecom} and the third from \cite[Lemma 1.16]{bal:21}
\end{proof}

\begin{proposition}
\label{distribution}
The orbit distribution of the action map $\rho_C$ satisfies
\[\im \rho_C = \ker L_C \cap \ker(\varphi_*\circ\Phi_{C*}).\]
\end{proposition}

\begin{proof}
We begin by showing the forward inclusion. Let $(v, \alpha) \in A_{S,\gamma,\varphi}$ and let $u= \rho_C(v, \alpha)$, so that $u$ is the unique element of $TM$ satisfying
\[\Phi_*u=v \quad \text{and} \quad (u, \Phi^*\alpha) \in L_M.\]
Then $u \in \ker (\varphi_*\circ\Phi_{C*})$. Moreover,
\[\jmath^*\Phi^*\alpha = \Phi_C^*\imath^*\alpha = \Phi_C^*\gamma^\flat(v)=(\Phi_C^*\gamma)^\flat(u),\]
so
\[(u, 0) = (u, \jmath^*\Phi^*\alpha - (\Phi_C^*\gamma)^\flat(u)) \in L_C,\]
and therefore $u\in\ker L_C$.

Conversely, let $u \in \ker L_C \cap \ker(\varphi_*\circ\Phi_{C*})$ and write $v=\Phi_*u$. By Proposition \ref{restriction}, $\Phi_{C*}L_C = \tau_{-\gamma}\imath^*L_X$ and therefore
\[(v,0)\in\tau_{-\gamma}\imath^*L_X.\]
This means that there exists a $1$-form $\alpha\in T^*X$ such that
\[(v,\alpha)\in L_{X}\quad\text{and}\quad\imath^*\alpha=\gamma^\flat(v),\]
so in particular $(v,\alpha)\in A_{S,\gamma,\varphi}$ and $(u,\Phi^*\alpha)\in L_M$. Therefore $u\in TM$ satisfies
\[\Phi_*u=v\quad\text{and}\quad (u,\Phi^*\alpha)\in L_M,\]
implying that $u\in\im\rho_C$.
\end{proof}

\begin{theorem}
\label{main}
Suppose that the orbit space $Q\coloneqq C/A_{S,\gamma,\varphi}$ of the action $\rho_C$ is a manifold with quotient map $\pi:C\longrightarrow Q.$
\begin{enumerate}[label=\textup{(\arabic*)}]
\item\label{8b2exmfb}
The pushforward
\[L_Q\coloneqq \pi_*L_C\]
is a $\theta$-twisted Dirac structure on $Q$, where $\theta$ is uniquely characterized by the condition
\[\pi^*\theta=\Phi_C^*(\imath^*\eta + d\gamma).\]
\item\label{bq5k3wpc}
The composition $\varphi \circ \Phi_C : C \longrightarrow Z$ descends to a Dirac realization
\[\overline{\varphi}:Q\longrightarrow Z.\]
\item\label{8b2exmfa}
The $f$-Dirac map $\pi : (C, L_C) \longrightarrow (Q, L_Q)$ is also b-Dirac.
\item\label{4s6edzna}
The nondegenerate leaves of $L_Q$ are the images under $\pi$ of the nondegenerate leaves of $L_C$. In particular, if $M$ is nondegenerate and $L_M=L_\omega$, then $L_Q$ is also nondegenerate, and the presymplectic form $\bar{\omega}$ determining it is uniquely characterized by the equation
\[\pi^*\bar{\omega} = \jmath^*\omega - \Phi_C^*\gamma.\]
\end{enumerate}
\end{theorem}

\begin{proof}
\ref{8b2exmfb}
By Theorem \ref{bgfkcgj1}\ref{tpcp6ia5}, to check that $L_Q$ is a $\zeta$-twisted Dirac structure on $Q$ it suffices to check that the orbit distribution $\im \rho_C$ on $C$ is a regular Dirac distribution. By Remark \ref{mzq8z1k7}, this follows from Proposition \ref{distribution}.

\ref{bq5k3wpc} Note that the composition $\varphi\circ\Phi_C$ is f-Dirac by Proposition \ref{restriction}. Moreover, by Proposition \ref{distribution} we have
\[\im \rho_C \subset \ker (\varphi\circ\Phi_C)_*\quad\text{ and }\quad\ker (\varphi\circ\Phi_C)_* \cap L_C \subset \im \rho_C,\]
so Theorem \ref{bgfkcgj1}\ref{15i0o7u8} implies that $f$ descends to a Dirac realization $Q \longrightarrow Z$.

Parts \ref{8b2exmfa} and \ref{4s6edzna} are a direct consequence of parts \ref{gpxy5jhi} and \ref{r0dvmlu4} of Proposition \ref{bgfkcgj1}, since Proposition \ref{distribution} implies that $\im \rho_C$ is contained in the kernel of $L_C$.
\end{proof}

\begin{remark}
In particular, in view of condition \eqref{strong}, if $Z$ is a single point and $(S,\gamma)$ is an ordinary reduction level, Theorem \ref{main} gives a Poisson structure on the reduced space $C/A_{S,\gamma}$.
\end{remark}

\begin{remark}
Suppose that the Dirac manifold $(X,L_X)$ is integrable with presymplectic groupoid $\mathcal{G} \rightrightarrows X$. If the Lie subalgebroid $A_{S,\gamma,\varphi}$ integrates to a source-connected groupoid
\[\mathcal{G}_{S,\gamma,\varphi}\rightrightarrows S\]
which acts on $C$, the orbits of $\mathcal{G}_{S,\gamma,\varphi}$ coincide with those of $A_{S,\gamma,\varphi}$ and Theorem \ref{main} gives a Dirac structure on the quotient $C/\mathcal{G}_{S,\gamma,\varphi}$. In the nondegenerate case, an extension of this result to certain actions of groupoids which are not source-connected is proved in Section \ref{fifth}.
\end{remark}

\begin{remark}
\label{strata2}
More generally, we can apply Theorem \ref{main} to the case when $S=\sqcup S_i$ is a stratified subpace of $X$ whose strata are compatible generalized reduction levels, as in Examples \ref{faces} and \ref{alcove}. That is, suppose that there exist $2$-forms $\gamma_i\in\Omega^2(S_i)$ and a map $\varphi:S\longrightarrow Z$ to a Dirac manifold $(Z,L_Z)$ such that for each $i$ the restriction
\[\varphi_i:S_i\longrightarrow Z\]
is a smooth map of constant rank and the triple $(S_i,\gamma_i,\varphi_i)$ is a generalized reduction level. If each of these reduction levels satisfies the hypotheses of Theorem \ref{main}, we obtain a stratified reduced space
\[\bigsqcup \Phi^{-1}(S_i)/A_{S_i,\gamma_i,\varphi_i}\]
which is a topological quotient of $\Phi^{-1}(S)$ and which is equipped with a continuous map to $Z$ that restricts to a Dirac realization along every stratum.
\end{remark}

Suppose that $(Y,L_Y)$ is a $\zeta$-twisted Dirac structure and consider the product Dirac manifold $(X\times Y, L_X\times L_Y)$. If $S$ is a generalized reduction level of $X$ relative to a Dirac manifold $(Z,L_Z)$, then $S\times Y$ is a generalized reduction level of $X\times Y$ relative to $(Z\times Y, L_Z\times L_Y.)$. Theorem \ref{main} then has the following immediate corollary, which we will use repeatedly to study quasi-Poisson reduction in Section \ref{qpsec}.

\begin{corollary}
\label{cormain}
Let
\[(\mu,\nu):M\longrightarrow X\times Y\]
be a Dirac realization such that the generalized reduction level $S$ of $X$ intersects $\mu$ cleanly, and let $C= \mu^{-1}(S)$. Suppose that the orbit space $Q\coloneqq C/A_{S,\gamma,\varphi}$ is a manifold with quotient map $\pi:C\longrightarrow Q.$ Then the pushforward
\[L_Q\coloneqq \pi_*L_C\]
is a $\theta$-twisted Dirac structure on $Q$, where $\theta$ is uniquely characterized by the condition
\[\pi^*\theta=\mu_C^*(\imath^*\eta + d\gamma)+\jmath^*\nu^*\zeta,\]
and the map $(\mu,\nu)$ descends to a Dirac realization
\[(\overline{\varphi},\overline{\nu}):Q\longrightarrow Z\times Y.\]
\end{corollary}

For nondegenerate Dirac realizations, we give a simple freeness criterion that determines when the reduction level $S$ is transverse to the Dirac realization map. It is a generalization of the classical fact that, when $M$ is a Hamiltonian $G$-space, the point $0\in\mathfrak{g}^*$ is a regular value of the moment map if and only if the action of $G$ on the $0$-fiber is locally free. To state it, first note that the condition $\rho_C(v, \alpha)=0$ implies that $v = 0$, so we may view $\ker \rho_C$ as a subset of $T^*X$.

\begin{proposition}
\label{freeness}
Suppose that $M$ is a nondegenerate Dirac manifold, or more generally that $L_M\cap TC^\circ=0$. Then
\[TS + \im \mu_* = (\ker \rho_C)^\circ.\]
In particular, $A_{S,\gamma,\varphi}$ acts freely on $C$ if and only if $S$ intersects $\mu$ transversally.
\end{proposition}
\begin{proof}
We have
\begin{align*}
u \in (\ker \rho_C)^\circ &\quad\Leftrightarrow\quad \alpha(u) = 0\text{ for all }\alpha \in T^*X\text{ such that }\imath^*\alpha = 0\text{ and }(0, \mu^*\alpha) \in L_M\\
					&\quad\Leftrightarrow\quad \alpha(u) = 0\text{ for all }\alpha \in T^*X\text{ such that }\imath^*\alpha = 0\text{ and }\mu^*\alpha=0\\
					&\quad\Leftrightarrow\quad u\in (TS^\circ\cap\ker\mu^*)^\circ\\
					&\quad\Leftrightarrow\quad u\in TS +\im\mu_*,
\end{align*}
where the second equivalence follows from the assumption $L_M \cap TC^\circ = 0$.
\end{proof}

Since the pullback of a Dirac realization to a nondegenerate leaf is still a Dirac realization, by restricting Proposition \ref{freeness} to nondegenerate leaves we obtain the following immediate corollary.

\begin{corollary}
If $S$ intersects the restriction of $\mu$ to each nondegenerate leaf transversally, then the algebroid $A_{S,\gamma,\varphi}$ acts freely on $C$.
\end{corollary}

\subsection{Familiar examples of reduction}
The reduction procedure of Theorem \ref{main} recovers a number of familiar constructions in symplectic and Poisson geometry.

\begin{example}(Symplectic reduction along a submanifold.)
Let $X$ be a Poisson manifold. A Dirac realization of $X$ is simply a Poisson map $\Phi:M\longrightarrow X$ from a Poisson manifold $M$. By Example \ref{ex:poisson2}, whenever $S\subset X$ is a pre-Poisson submanifold which intersects $\Phi$ cleanly and such that the quotient
\[Q=\Phi^{-1}(S)/A_S\]
exists, Theorem \ref{main} equips $Q$ with a reduced Poisson structure. In particular, if $M$ is a symplectic manifold, the reduced structure on $Q$ is also symplectic and was studied in \cite{cro.may:21}.
\end{example}

\begin{example}
\label{pointred}
(Dirac reduction at a point.)
Let $X$ be a Dirac manifold and let $\Phi:M\longrightarrow X$ be a Dirac realization. In view of Example \ref{point}, whenever $x\in X$ is a regular value of $\Phi$ and the quotient
\[Q_x\coloneqq \Phi^{-1}(x)/A_x\]
is a manifold, we recover the usual Poisson structure obtained by reducing a Dirac realization at a point \cite[Theorem 4.11]{bur.cra:05}.
\end{example}

\begin{example}
\label{orbitred}
(Dirac reduction along a leaf.)
Let $X$ be an $\eta$-twisted Dirac manifold and let $\O$ be a nondegenerate leaf of $X$ with presymplectic form $\omega$. By Example \ref{orbit}, $(\O,\omega)$ is a reduction level of $X$ with stabilizer subalgebroid
\[A_{\O,\omega}=L_{X\vert\O}.\]
Given a Dirac realization $\Phi:M\longrightarrow X$, the reduced space
\[Q_\O\coloneqq \Phi^{-1}(\O)/L_{X\vert\O}\]
is Poisson.

Moreover, fixing a point $x\in\O$ gives a local diffeomorphism
\[\psi:Q_x\longrightarrow Q_\O\]
between the Dirac reduction of $M$ at $x\in X$ and the space $Q_\O$, which fits into the diagram
\begin{equation*}
\begin{tikzcd}[row sep=large, column sep=large]
\Phi^{-1}(x)\arrow[r, hook, "\imath"]\arrow[d,"q"]	&\Phi^{-1}(\O)\arrow[r, hook, "\jmath"]\arrow[d, "q"]	&M\\
Q_x\arrow[r, "\psi"]	&\,Q_\O.	&
\end{tikzcd}
\end{equation*}
Chasing this diagram, we get
\begin{align*}
\psi^*q_{*}\tau_{\Phi^*\omega}\jmath^*L_M&=q_{*}\imath^*\tau_{\Phi^*\omega}\jmath^*L_M\\
		&=q_{*}\tau_{\Phi^*\omega}\imath^*\jmath^*L_M\\
		&=q_{*}\imath^*\jmath^*L_M\\
\end{align*}
where the first equality follows from \cite[Lemma 1.16]{bal:21}, the second from \eqref{gaugecom}, and the third from the fact that the restriction of $\Phi^*\omega$ to $\Phi^{-1}(x)$ vanishes. In other words, $\varphi$ is a local Dirac diffeomorphism between the reduced structure
\[q_{*}\tau_{\Phi^*\omega}\jmath^*L_M\]
on $Q_\O$ and the structure
\[q_{*}\imath^*\jmath^*L_M\]
on the reduction $Q_x$ of Example \ref{pointred}.
\end{example}

In the special case of Hamiltonian Poisson and quasi-Poisson manifolds, the reduction of Example \ref{orbitred} specializes to reduction along group orbits.

\begin{example}
(Marsden--Weinstein reduction along a coadjoint orbit.)
Let $\mathfrak{g}$ be a Lie algebra and let $\mathcal{O}$ be a coadjoint orbit in $\mathfrak{g}^*$ with Kirillov--Kostant--Souriau symplectic form $\omega$. By Example \ref{orbit}, $(\O,\omega)$ is a reduction level of $\mathfrak{g}^*$ with stabilizer subalgebroid
\[A_{\O,\omega}=\O\rtimes\mathfrak{g}.\]
A Dirac realization of $\mathfrak{g}^*$ is a Hamiltonian Poisson $\mathfrak{g}$-manifold $M$ with moment map $\Phi:M\longrightarrow\mathfrak{g}^*$, and the image of the action of $A_{\O,\omega}$ on $\Phi^{-1}(\O)$ is a regular Dirac distribution. If the space
\[\Phi^{-1}(\mathcal{O})/A_{\O,\omega}=\Phi^{-1}(\mathcal{O})/\mathfrak{g}\]
is a manifold then, by Example \ref{orbitred}, its Poisson structure coincides with the usual Marsden--Weinstein reduction \cite{mar.wei:74} along a coadjoint orbit.
\end{example}

\begin{example}
(Quasi-Poisson reduction along a conjugacy class.)
Let $G$ be a Lie group whose Lie algebra carries a nondegenerate inner product. Then $G$ has a Cartan--Dirac structure, and we let $\mathcal{O}\subset G$ be a conjugacy class with quasi-Hamiltonian $2$-form $\omega$. By Example \ref{orbit}, $(\mathcal{O},\omega)$ is a reduction level of $G$ with stabilizer subalgebroid
\[A_{\mathcal{O},\omega}=\mathcal{O}\rtimes\mathfrak{g}.\]
A Dirac realization of $G$ is a Hamiltonian quasi-Poisson $G$-manifold $M$ with moment map $\Phi:M\longrightarrow G$, and the image of the action of $\O\rtimes\mathfrak{g}$ on $\Phi^{-1}(\O)$ is a regular Dirac distribution. If the reduced space
\[\Phi^{-1}(\mathcal{O})/A_{\mathcal{O},\omega}=\Phi^{-1}(\mathcal{O})/\mathfrak{g}\]
is a manifold, Theorem \ref{main} equips it with a Poisson structure. By Example \ref{orbitred}, this Poisson structure coincides with the quasi-Poisson reduction of \cite[Theorem 5.1]{ale.mal.mei:98} and \cite[Theorem 6.1]{ale.kos.mei:02}.
\end{example}

\begin{example}
\label{steinslice}
(Steinberg slices in quasi-Poisson manifolds.)
Let $G$ be a simply-connected, semisimple complex Lie group equipped with the Cartan-Dirac structure. The \emph{Steinberg cross-section} of $G$ is an affine space $\Sigma\subset G$ which consists entirely of regular elements and intersects every regular conjugacy class of $G$ exactly once and transversally \cite[Theorem 1.4]{ste:65}. By Lemma \ref{ptrans}, $\Sigma$ is a reduction level of $G$ with trivial stabilizer subalgebroid. If $\Phi:M\longrightarrow G$ is a quasi-Poisson moment map, we have
\[T\Sigma+\im\Phi_*\supset T\Sigma+p_T(L_G)=TG\]
and therefore $\Sigma$ is transverse to $\Phi$. It then follows from Theorem \ref{main} that the pullback of the Dirac structure on $X$ to the preimage
\[\Phi^{-1}(\Sigma)\]
is a Poisson structure, recovering \cite[Theorem 2.3]{bal:21}.
\end{example}

\begin{example}
\label{unired}
Let $G$ be a reductive complex Lie group with Lie algebra $\mathfrak{g}$, let $\mathfrak{p}$ be a parabolic subalgebra of $\mathfrak{g}$ with nilradical $\mathfrak{u}$, and write $P$ for the corresponding parabolic subgroup of $G$. Example \ref{unips} shows that $\mathfrak{u}$ is a reduction level of $\mathfrak{g}$. If $\Phi:M\longrightarrow\mathfrak{g}$ is a Hamiltonian $G$-space, Theorem \ref{main} then gives a Poisson structure on the quotient
\[\Phi^{-1}(\mathfrak{u})/P.\]
This structure coincides with the following familiar Hamiltonian reduction. Since $M$ is also Hamiltonian for the action of $P$, there is a commutative diagram of moment maps
\begin{equation*}
\begin{tikzcd}[column sep=tiny]
M\arrow[rrd, "\Phi"]\arrow[dd,swap,"\Phi_P"]	&&\\
					&&\mathfrak{g}\arrow[ld]\\
\,\,\quad\mathfrak{p}^*\cong&\hspace{-.15in}\mathfrak{g}/\mathfrak{u}.	&
\end{tikzcd}
\end{equation*}
Taking Hamiltonian reduction relative to $\Phi_P$, we obtain a Poisson structure on the quotient
\[\Phi_P^{-1}(0)/P=\Phi^{-1}(\mathfrak{u})/P,\]
which agrees with the Poisson structure given by Theorem \ref{main}. In particular, when $M=T^*G$ is the cotangent bundle of $G$, we obtain the usual symplectic structure on the cotangent bundle
\[T^*(G/P)\cong G\times_P\mathfrak{u}\]
of the partial flag variety $G/P$.
\end{example}

\begin{example}
\label{parared}
Keeping the notation of Example \ref{unired}, let $\mathfrak{p}=\l+\mathfrak{u}$ be a Levi decomposition and
\[\varphi:\mathfrak{p}\longrightarrow\l\]
be the quotient map. By Example \ref{parabs}, $(\mathfrak{p},\varphi)$ is a generalized reduction level of $\mathfrak{g}$. If $\Phi:M\longrightarrow\mathfrak{g}$ is a Hamiltonian $G$-space, Theorem \ref{main} shows that the quotient
\[\overline{\varphi}:\Phi^{-1}(\mathfrak{p})/U\longrightarrow \l\]
is a Hamiltonian Poisson $L$-space. As before, we may restrict to the action of $U$ to obtain the commutative diagram of moment maps
\begin{equation*}
\begin{tikzcd}[column sep=tiny]
M\arrow[rrd, "\Phi"]\arrow[dd,swap,"\Phi_U"]	&&\\
					&&\mathfrak{g}\arrow[ld]\\
\,\,\quad\mathfrak{u}^*\cong&\hspace{-.15in}\mathfrak{g}/\mathfrak{p}	&
\end{tikzcd}
\end{equation*}
and reduce along $\Phi_U$. This gives a Poisson structure on the quotient
\[\Phi^{-1}_U(0)/P=\Phi^{-1}(\mathfrak{p})/U\]
which agrees with the structure obtained from Theorem \ref{main}. When $M=T^*G$, we obtain in this way the usual symplectic structure on the cotangent bundle
\[T^*(G/U)\cong G\times_U\mathfrak{p}\]
of the base affine space $G/U$.
\end{example}

\begin{example}
\label{sympimp}
(Symplectic implosion)
Let $\mathfrak{g}$ be the Lie algebra of a compact Lie group as in Example \ref{faces}, and fix a choice of closed dominant Weyl chamber
\[\t^*_+=\bigcup_{J\subset\Delta}\sigma_J\subset\t^*.\]
The inclusion map $\imath:\t^*_+\lhook\joinrel\longrightarrow\t^*$ makes $\t^*_+$ a stratified generalized reduction level in the sense of Remark \ref{strata2}. If $\Phi:M\longrightarrow \mathfrak{g}^*$ is a symplectic Hamiltonian $G$-space, the reduced space
\[\bigsqcup_J\Phi^{-1}(\sigma_J)/[K_J,K_J]\]
is a union of submanifolds of \cite[Theorem 5.10]{gui.jef.sja:02}. In view of Remark \ref{strata1}, this gives it the structure of a stratified Hamiltonian $T$-space, with moment map given by the composition $\imath\circ\Phi$, recovering the structure of the \emph{imploded cross-section} of $M$ developed in \cite[Theorem 2.10]{gui.jef.sja:02}.
\end{example}

\begin{example}
\label{qhamimp}
(Quasi-Hamiltonian implosion.)
Let $G$ be a simply-connected compact Lie group equipped with a Cartan-Dirac quasi-Poisson structure, as in Example \ref{alcove}. Let $\mathcal{C}\subset\t^*$ be an open Weyl chamber, let $\mathcal{C}^\vee$ be the chamber in $\t$ dual to $\mathcal{C}$, and let $\mathcal{A}\subset\mathcal{C}^\vee$ the unique open alcove such that $0\in\bar{\mathcal{A}}$. The exponential image $\exp(\bar{\mathcal{A}})\subset T$ has the property that it intersects each conjugacy class in $K$ in a single point, and it decomposes as a disjoint union
\[\exp(\bar{\mathcal{A}})=\bigsqcup_\sigma S_J\]
of the strata defined in \eqref{expstrata}. The inclusion map $\imath:\exp(\bar{\mathcal{A}})\lhook\joinrel\longrightarrow T$ gives $\exp(\bar{\mathcal{A}})$ the structure of a stratified generalized reduction level as in Remark \ref{strata2}. If $\Phi:M\longrightarrow G$ is a quasi-Hamiltonian $G$-space, the reduced space
\[\bigsqcup_J\Phi^{-1}(S_J)/[K_J,K_J]\]
is once again a union of smooth manifolds \cite[Corollary 3.12]{hur.jef.sja:06}, so it has the structure of a stratified quasi-Hamiltonian $T$-space with moment map $\imath\circ\Phi$. This recovers the quasi-Hamiltonian imploded cross-section of \cite[Theorem 3.17]{hur.jef.sja:06}.
\end{example}

\section{Fiber products and universality}
\label{third}
\subsection{Fiber products of Dirac realizations}
If $(X,L_X)$ is an $\eta$-twisted Dirac manifold, we define $\overline{X}$ to be the manifold $X$ equipped with the opposite Dirac structure
\[L_{\overline{X}}\coloneqq -L_X\]
defined in \eqref{opposite}. Then the product $X\times\overline{X}$ carries the $(\eta,-\eta)$-twisted Dirac structure $L_{X\times\overline{X}}= L_X\times L_{\overline{X}}$, and we write $X_\Delta\subset X\times\overline{X}$ for the diagonal submanifold of this space.

\begin{lemma}
The subspace $X_\Delta$ is a reduction level of $X\times\overline{X}$.
\end{lemma}
\begin{proof}
The pullback of $(\eta,-\eta)$ to $X_\Delta$ vanishes since the tangent bundle $TX_\Delta\subset TX\times T\overline{X}$ consists of diagonal pairs of tangent vectors. Moreover
\[L_{X\times\overline{X}}=\{((u,\alpha),(v,\beta))\mid (u,\alpha),(v,-\beta)\in L_X\},\]
so it follows that
\[A_\Delta\coloneqq L_{X\times\overline{X}}\cap (TX_\Delta\oplus TX_\Delta^\circ)=\{((u,\alpha),(u,-\alpha))\in L_{X\times\overline{X}}\}\]
has constant rank.
\end{proof}

If $\Phi:M\longrightarrow X$ and $\Psi:N\longrightarrow \overline{X}$ are two Dirac realizations with the property that the diagonal $X_\Delta$ has clean intersection with $(\Phi,\Psi)$, then the fiber product $M\times_XN$ is a smooth manifold and we obtain the following special case of Theorem \ref{main}.

\begin{proposition}
\label{fiberprod}
Suppose that the quotient $Q\coloneqq (M\times_XN)/A_\Delta$ is a manifold. Then the Dirac structure on $M\times N$ descends to a Poisson structure on $Q$.
\end{proposition}

Letting $\Psi$ be the identity map then induces a Poisson structure on the quotient $M/L_X$.

\begin{proposition}
\label{quotients}
Let $\Phi:M\longrightarrow X$ be a Dirac realization, and suppose that the quotient $M/L_X$ is a manifold with quotient map $\pi:M\longrightarrow M/L_X.$ Then the Dirac structure $\pi_*(L_M - \Phi^*L_X)$ on $M/L_X$ is Poisson, and its symplectic leaves are the reductions
\[\Phi^{-1}(\O)/L_X\]
of $L_M$ along the reduction levels $(\O,\omega)$ given by the nondegenerate leaves of $X$.
\end{proposition}
\begin{proof}
Consider the Dirac realization $(\Phi,\Id):M\times\overline{X}\longrightarrow X\times\overline{X}$. The reduction level $X_\Delta\subset X\times\overline{X}$ is transverse to $(\Phi,\Id)$, and fits into the diagram
\begin{equation*}
\begin{tikzcd}[row sep=large, column sep=large]
M\arrow{r}{(\Id,\Phi)}[swap]{\sim}\arrow[d,swap,"\Phi"]	&M\times_X\overline{X}\arrow[r, hook, "\jmath"]\arrow[d, "\Phi_\Delta"]	&M\times\overline{X}\arrow[d, "\Phi\times\Id"]\\
X\arrow{r}[swap]{\sim}						&X_\Delta		\arrow[r, hook]							&X\times\overline{X}.
\end{tikzcd}
\end{equation*}
Here the right-hand square is Cartesian, and the horizontal arrows in the left-hand square intertwine the actions of $L_X$ and $A_\Delta$, so that we obtain a second commutative diagram
\begin{equation*}
\begin{tikzcd}[row sep=large, column sep=large]
M\arrow{r}{(\Id,\Phi)}[swap]{\sim}\arrow[d,swap,"\pi"]	&M\times_X\overline{X}\arrow[d, "\pi_\Delta"]\\
M/L_X\arrow{r}{\varphi}[swap]{\sim}						&(M\times_X\overline{X})/A_\Delta.
\end{tikzcd}
\end{equation*}
By Theorem \ref{main}, we obtain a Poisson structure on $M/L_X$ given by
\[\varphi^*\pi_*\jmath^*(L_M\times L_{\overline{X}})=\pi_*(\Id,\Phi)^*\jmath^*(L_M\times L_{\overline{X}}) =\pi_*(L_M-\Phi^*L_X),\]
where the second equality follows from \cite[Lemma 1.16]{bal:21}.

Suppose now that $M$ is nondegenerate with presymplectic form $\Omega$, and that
\[X=\sqcup(\O,\omega)\]
is the partition of $X$ into nondegenerate leaves. Then presymplectic foliation of $M\times\overline{X}$ is given by
\[M\times\overline{X}=\sqcup(M\times\O,\Omega+\omega),\]
and by Theorem \ref{main}\ref{4s6edzna} the nondegenerate leaves of $M/L_X$ are all of the form
\[\varphi^{-1}(\pi_\Delta(M\times_\O\overline{\O}))=\Phi^{-1}(\O)/L_X.\]
Considering the diagram
\begin{equation*}
\begin{tikzcd}[row sep=large, column sep=large]
\Phi^{-1}(\O)\arrow[r, hook, "{\jmath_\O}"]\arrow[d, swap, "\pi"]	&M\arrow[d, "\pi"]\\
\Phi^{-1}(\O)/L_X\arrow[r, hook, "{\imath_\O}"]			&M/L_X,
\end{tikzcd}
\end{equation*}
the nondegenerate structure on each leaf is
\[\imath_\O^*\pi_*(L_M-\Phi^*L_X)=\pi_*\jmath_\O^*(L_M-\Phi^*L_X)=\pi_*\tau_{-\Phi^*\omega}\jmath^*L_M=\pi_*L_{\Phi^{-1}(\O)},\]
where $L_{\Phi^{-1}(\O)}$ is the Dirac structure defined on $\Phi^{-1}(\O)$ by \eqref{twistpull} when it is viewed as the pullback of the reduction level $(\O,\omega)$ of $X$. Therefore the symplectic structure on $\Phi^{-1}(\O)/L_X$ is precisely the one obtained by reduction at $(\O,\omega)$ in Example \ref{orbitred}.
\end{proof}

\begin{example}
\label{parags}
Let $G$ be a complex reductive Lie group with Lie algebra $\mathfrak{g}$, and let $\mathfrak{p}$ be a parabolic subalgebra of $\mathfrak{g}$ with Levi decomposition $\mathfrak{p}=\l+\mathfrak{u}$. By Example \ref{parared}, for any Hamiltonian $G$-manifold $\Phi:M\longrightarrow\mathfrak{g}$, the reduction
\[\Phi^{-1}(\mathfrak{p})/U\]
has the structure of a Hamiltonian Poisson $L$-space. By Proposition \ref{quotients}, we may quotient further by the action of $L$ to obtain a Poisson manifold whose partitition into symplectic leaves is given by
\[\Phi^{-1}(\mathfrak{p})/P=\bigsqcup\Phi^{-1}(\O\mathfrak{u})/P,\]
where each $\O$ is an adjoint orbit in $\l$ and $\O\mathfrak{u}$ denotes its preimage under $\varphi$. In particular, when $M=T^*G$, we obtain the usual partition into symplectic leaves
\[G\times_P\mathfrak{p}=\bigsqcup G\times_P\O\mathfrak{u}\]
of the parabolic Grothendieck--Springer resolution.
\end{example}

\subsection{Hamiltonian and quasi-Hamiltonian reduction}
Suppose that $M$ and $N$ are Hamiltonian Poisson $\mathfrak{g}$-manifolds with moment maps $\mu:M\longrightarrow\mathfrak{g}^*$ and $\nu:N\longrightarrow\mathfrak{g}^*.$ Then the Poisson structure on the product space $M\times N$ is Hamiltonian for the diagonal action of $\mathfrak{g}$, and the corresponding moment map is
\begin{align*}
\Phi:M\times N&\longrightarrow \quad\mathfrak{g}^*\\
	(m,n)&\longmapsto\mu(m)+\nu(n).
	\end{align*}
In this case the Dirac reduction of the Poisson map
\[(\mu,-\nu):M\times N\longrightarrow\mathfrak{g}^*\times\overline{\mathfrak{g}^*}\]
along $\mathfrak{g}^*_\Delta$ described in Proposition \ref{fiberprod} is precisely the Hamiltonian reduction
\[\Phi^{-1}(0)/\mathfrak{g}\]
of $M\times N$ at the point $0$ in $\mathfrak{g}^*$.

This observation has the following quasi-Poisson counterpart. Let $G$ be a Lie group whose Lie algebra $\mathfrak{g}$ carries an invariant nondegenerate symmetric bilinear form, and suppose that $(M,\pi)$ is a Hamiltonian quasi-Poisson $G\times G$-manifold with moment map
\[(\Phi_1,\Phi_2):M\longrightarrow G\times G.\]
Once again letting $\{e_i\}\subset\mathfrak{g}$ be an orthonormal basis, we define the $2$-tensor
\[\psi\coloneqq \frac{1}{2}\sum(e_i,0)\wedge(0,e_i)\in\wedge^2(\mathfrak{g}\oplus\mathfrak{g}).\]
The bivector
\begin{equation}
\label{fusion}
\pi_{\text{fus}}\coloneqq \pi+\psi_M.
\end{equation}
defines a Hamiltonian quasi-Poisson structure on $M$ relative to the diagonal action of $G$, and the corresponding moment map is the product $\Phi_1\Phi_2$ of the original moment map components \cite[Section 5]{ale.kos.mei:02}. The triple
\[(M,\pi_{\text{fus}},\Phi_1\Phi_2)\]
is called the \emph{internal fusion} of the quasi-Poisson $G\times G$-manifold $M$.

If $(M,\pi_M,\Phi_M)$ and $(N,\pi_N,\Phi_N)$ are Hamiltonian quasi-Poisson $G$-manifolds, the product $M\times N$ is naturally a quasi-Poisson $G\times G$-manifold. The \emph{fusion product} of $M$ and $N$, denoted $M\circledast N$, is the Hamiltonian quasi-Poisson $G$-manifold
\[(M\circledast N, (\pi_M+\pi_N)_{\text{fus}}, \Phi_M\Phi_N).\]
By \cite[Theorem 6.1]{ale.kos.mei:02}, if $1$ is a regular value of the moment map $\Phi_M\Phi_N$ and the reduction
\[(\Phi_M\Phi_N)^{-1}(1)/G,\]
is a manifold, then it inherits a Poisson structure.

On the other hand, let $\overline{G}$ be the group $G$ equipped with the opposite Cartan--Dirac structure
\[L_{\overline{G}}=\left\{\left((\xi^L-\xi^R),-\sigma(\xi)\right)\mid\xi\in\mathfrak{g}\right\}.\]
Then $\Phi_N^{-1}:N\longrightarrow\overline{G}$ is a Dirac realization, and we may consider the fiber product
\[M\times_GN=\{(m,n)\in M\times N\mid \Phi_M(m)=\Phi^{-1}_N(n)\}=(\Phi_M\Phi_N)^{-1}(1).\]
Its quotient by the action of $G$, if it exists, has a Poisson structure obtained via Proposition \ref{fiberprod}. These two constructions are related as follows.

\begin{proposition}\label{ooysgjbr}
The Dirac structure on the reduction of the Dirac realization $M\times N\longrightarrow G\times\overline{G}$ along the diagonal reduction level $G_\Delta$ agrees with the quasi-Poisson reduction of $M\circledast N$ at the identity.
\end{proposition}
\begin{proof}
Consider the inclusion $\jmath:M\times_GN\lhook\joinrel\longrightarrow M\times N$ and let $\pi:M\times_GN\longrightarrow (M\times_GN)/G$ be the quotient map. We must show the equality of Dirac structures
\[\pi_*\jmath^*L_{M\circledast N}=\pi_*\jmath^*(L_M\times L_N),\]
and for this it suffices to prove that
\[\jmath^*L_{M\circledast N}=\jmath^*(L_M\times L_N).\]
Since these are generalized distributions of the same rank, it is enough to show that the left-hand side is contained in the right-hand side.

Suppose that $(v,\jmath^*\gamma)\in \jmath^*L_{M\circledast N}.$ According to \eqref{qpdirac} and \eqref{fusion}, this implies that
\[v=\pi_M^\#(\alpha)+\pi_N^\#(\beta)+\rho_M(\xi-\xi_\beta)+\rho_N(\xi+\xi_\alpha)\]
and
\[\gamma=C^*_{M\circledast N}(\alpha+\beta)+(\Phi_M\Phi_N)^*\sigma(\xi).\]
Here $\alpha\in T^*M$, $\beta\in T^*N$, $\xi\in \mathfrak{g}$, the maps $C^*_{M\circledast N}$ and $\sigma$ are as defined in \eqref{qpdirac}, and we write
\[\xi_\alpha\coloneqq \sum\alpha(e_{iM})e_i\quad\text{and}\quad\xi_\beta\coloneqq \sum\beta(e_{iN})e_{i}.\]
It is sufficient to show that the pullback $\jmath^*\gamma$ agrees with
\[\jmath^*(C^*_M(\alpha)+C^*_N(\beta)+\Phi_M^*\sigma(\xi-\xi_\beta)+\Phi_N\sigma(\xi+\xi_\alpha)).\]

The fact that
\begin{equation}
\label{xyz}
\jmath^*\circ(\Phi_M\Phi_N)^*=0
\end{equation}
immediately gives
\[\jmath^*\gamma=\jmath^*C^*_{M\circledast N}(\alpha+\beta)+\jmath^*(\Phi_M\Phi_N)^*\sigma(\xi)=\jmath^*\alpha+\jmath^*\beta.\]
Moreover, as in the proof of \cite[Lemma 2.11]{bal:21}, composition \eqref{xyz} implies that $\xi_\alpha=-\xi_\beta$ and that
\[\jmath^*\Phi_N^*(\theta_i^L)=-\jmath^*\Phi_M(\theta^R)\quad\text{and}\quad\jmath^*\Phi_N^*(\theta_i^R)=-\jmath^*\Phi_M(\theta^L).\]
We therefore obtain
\[\jmath^*(C^*_M(\alpha)+C^*_N(\beta))+\jmath^*(\Phi_M^*\sigma(\xi-\xi_\beta)+\Phi_N\sigma(\xi+\xi_\alpha))=\jmath^*\alpha+\jmath^*\beta,\]
proving the desired equality. It follows that $(v,\jmath^*\gamma)\in \jmath^*(L_M\times L_N)$, and therefore that $\jmath^*L_{M\circledast N}=\jmath^*(L_M\times L_N)$.
\end{proof}

\subsection{Reduction of a presymplectic groupoid}
\label{7vb0zcqj}
Now suppose that $(X,L_X)$ is integrable, and let
\[(\mathsf{s},\mathsf{t}):\mathcal{G}\rightrightarrows X\]
be the source-simply-connected presymplectic groupoid which integrates it. By \cite[Example 2.5]{bur.cra:09}, the map
\[(\mathsf{s},\mathsf{t}):\mathcal{G}\longrightarrow \overline{X}\times X\]
is a Dirac realization. If $(S,\gamma)$ is a reduction level of $X$, from Theorem \ref{main} we obtain a Dirac structure on the reduced space
\[\mathcal{G}_S\coloneqq t^{-1}(S)/A_{S,\gamma}\]
such that the source map descends to a Dirac realization
\[\bar{\mathsf{s}}:\mathcal{G}_S\longrightarrow \overline{X}.\]

Let $(\mu,\nu):M\longrightarrow X\times Y$ be a Dirac realization. Under the clean intersection hypothesis of Proposition \ref{fiberprod}, the quotient $(M\times_X\mathcal{G}_S)/A_\Delta$ is a Dirac realization of $Y$. For every $x\in X$, let $e_x\in\mathcal{G}$ be the identity arrow at $x$. Then the natural b-Dirac map
\begin{align*}
M&\longrightarrow M\times_X\mathcal{G} \\
m&\longmapsto (m, e_{\mu(m)})
\end{align*}
restricts to a b-Dirac map
\begin{equation}
\label{restriction-2}
\mu^{-1}(S)\longrightarrow M\times_X\mathsf{t}^{-1}(S).
\end{equation}

\begin{proposition}
\label{univred}
The map \eqref{restriction-2} descends to a Dirac diffeomorphism
\[\overline{F}:\mu^{-1}(S)/A_{S,\gamma}\longrightarrow (M\times_X\mathcal{G}_S)/A_\Delta.\]
which commutes with the Dirac realization maps to $Y$.
\end{proposition}
\begin{proof}
Let $\mathcal{H}$ be a source-simply-connected groupoid integrating $A_{S,\gamma}$. Suppose that $(m,e_{\mu(m)})$ and $(m',e_{\mu(m')})$ are conjugate under the action of $\mathcal{G}_\Delta\times \mathcal{H}$. Then there exist $g\in \mathcal{G}$ and $h\in \mathcal{H}$ such that
\[\mathsf{t}(h)=\mathsf{s}(g)=\mu(m),\quad g\cdot m=m',\qquad\text{and}\qquad g\cdot e_{\mu(m)}\cdot h= e_{\mu(m')},\]
which implies that $g=h^{-1}$ and therefore that $g\in\mathcal{H}$. It follows that $m$ and $m'$ are in the same $\mathcal{H}$-orbit on $\mu^{-1}(S)$, and therefore $\overline{F}$ is injective.

Since $\mathcal{G}$ acts transitively on the fibers of the restriction
\[\mathsf{s}:\mathsf{t}^{-1}(S)\longrightarrow X,\]
every element of $M\times_X \mathsf{t}^{-1}(S)$ is $\mathcal{G}_\Delta$-conjugate to an element of the form $(m,e_x)$. Moreover, if $(m,e_x)\in M\times_X \mathsf{t}^{-1}(S)$, this implies that $\mu(m)=x\in S$ and therefore $(m, e_x)$ is in the image of \eqref{restriction-2}. It follows that $\overline{F}$ is surjective.

It remains to show that the diffeomorphism F is a Dirac diffeomorphism. Consider the commutative diagram
\begin{equation*}
\begin{tikzcd}[column sep=large, row sep=large]
M_S		\arrow[r, "F"]\arrow[rd, swap, "\overline{F}"]			&M\times_X\mathcal{G}_S \arrow[d]\\
						&\quad(M\times_X\mathcal{G}_S)/{A_\Delta}.
\end{tikzcd}
\end{equation*}
The vertical quotient map is b-Dirac by Theorem \ref{main}\ref{8b2exmfa}. The map $F$ descends from \eqref{restriction-2} via the diagram
\begin{equation*}
\begin{tikzcd}[column sep=large, row sep=large]
\Phi^{-1}(S)		\arrow[r]\arrow[d]		& M\times_X\mathsf{t}^{-1}(S)\arrow[d]\\
M_S			\arrow[r, "F"]				&M\times_X\mathcal{G}_S,
\end{tikzcd}
\end{equation*}
where the vertical arrows are quotients by the action of $\mathcal{H}$. Since the vertical maps are f-Dirac and \eqref{restriction-2} is b-Dirac, it follows from \cite[Lemma 1.16]{bal:21} that $F$ is also b-Dirac. Then, since $\overline{F}$ is the composition of two b-Dirac maps, it is also b-Dirac and therefore a Dirac diffeomorphism.
\end{proof}

Because of Proposition \ref{univred}, we call $\mathcal{G}_S$ the \emph{universal reduced space} associated to the reduction level $(S,\gamma)$. It generalizes the symplectic universal reduced spaces of \cite[Section 4.3]{cro.may:21}, and specializes to a number of examples which already appear in the literature.

\begin{example}
\begin{enumerate}[leftmargin=20pt]
\item When $X=\k^*$ is the dual of a compact Lie algebra and $S=\t^*_+$ is the closure of the dominant Weyl chamber as in Example \ref{sympimp}, the universal reduced space is the universal imploded cross-section of \cite[Theorem 4.9]{gui.jef.sja:02}.

\item When $X=K$ is a compact Lie group and $S$ is the closed alcove $S=\overline{\mathcal{A}}$ of Example \ref{qhamimp}, Proposition \ref{ooysgjbr} implies that the universal reduced space is the quasi-Hamiltonian universal imploded cross-section of \cite[Theorem 4.6]{hur.jef.sja:06}.

\item When $X=G$ is a simply-connected semisimple complex Lie group and $S=\Sigma$ is the Steinberg cross-section from Example \ref{steinslice}, Proposition \ref{ooysgjbr} implies that the universal reduced space is the one-sided Steinberg slice of \cite[Proposition 2.17]{bal:21}.\\
\end{enumerate}
\end{example}

\section{Reduction of quasi-Poisson manifolds}
\label{qpsec}
In this section we give some applications of Theorem \ref{main} to the setting of quasi-Poisson manifolds. We develop a procedure for quasi-Poisson reduction relative to the action of a certain class of subgroup, as well as a quasi-Poisson analogue to Whittaker reduction based on transversal spaces introduced by Steinberg.

\subsection{Quasi-Poisson reduction by a subgroup}
Let $G$ be a Lie group and suppose that its Lie algebra carries an invariant, nondegenerate symmetric bilinear form, so that $G$ has a Cartan--Dirac structure.

\begin{definition}
\label{reducible}
A Lie subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ is \emph{reducible} if its annihilator $\mathfrak{h}^\perp$ is also a Lie subalgebra. We denote by $H$ and $H^\perp$ the connected subgroups of $G$ which integrate $\mathfrak{h}$ and $\mathfrak{h}^\perp$, and we call such subgroups \emph{reducible} as well.
\end{definition}

\begin{remark}
The invariance of the bilinear form implies that $\mathfrak{h}$ is reducible if and only if it satisfies the condition
\[[\mathfrak{h},\mathfrak{h}^\perp]\subset\mathfrak{h}\cap\mathfrak{h}^\perp.\]
\end{remark}

Note that $H$ is a reducible subgroup whenever $\mathfrak{h}$ is coisotropic, as well as whenever $\mathfrak{h}$ is isotropic and contains $[\mathfrak{h}^\perp,\mathfrak{h}^\perp]$. In particular, any subgroup whose Lie algebra is Lagrangian is reducible. When $G$ is complex and reductive, the following subgroups are examples of reducible subgroups:
\begin{itemize}[topsep=1.5pt]
\item parabolic subgroups
\item the unipotent radical of any parabolic subgroup
\item the derived subgroup of any parabolic subgroup
\end{itemize}

When $\mathfrak{h}$ is reducible, the intersection $\mathfrak{h}\cap\mathfrak{h}^\perp$ is a normal Lie subalgebra of $\mathfrak{h}$, and the quotient
\[\overline{\h}\coloneqq \mathfrak{h}/(\mathfrak{h}\cap\mathfrak{h}^\perp)\]
inherits an invariant nondegenerate symmetric bilinear form. Therefore the Lie group
\[\overline{H}\coloneqq H/(H\cap H^\perp)\]
can also be equipped with a Cartan--Dirac structure. We write $\varphi:H\longrightarrow\overline{H}$ for the quotient map.

\begin{proposition}
\label{plevels}
The pair $(H,\varphi)$ is a generalized reduction level of $G$ whose stabilizer subalgebroid is the action Lie algebroid $H\rtimes\mathfrak{h}^\perp$.
\end{proposition}
\begin{proof}
Consider the diagram
\begin{equation*}
\begin{tikzcd}[column sep=large, row sep=large]
H \arrow[hook]{r}{\imath} \arrow{d}{\varphi} & G \\
\bar{H}.
\end{tikzcd}
\end{equation*}
We begin by showing that $\varphi_* \imath^*L_G = L_{\bar{H}}$. An element of $\varphi_*\imath^*L_G$ at a point $\varphi(h)$ is of the form $(\varphi_*v, \alpha)$, where $v = \xi^L-\xi^R$ and $\varphi^*\alpha=\sigma(\xi)$ for some $\xi \in \mathfrak{g}$ satisfying
\[(\Ad_{h}\xi - \xi) \in \mathfrak{h} \quad \text{ and } \quad (\Ad_{h}\xi + \xi) \in (\mathfrak{h} \cap \mathfrak{h}^\perp)^\perp = \mathfrak{h} + \mathfrak{h}^\perp.\]
In particular, this implies that $\xi \in \mathfrak{h} + \mathfrak{h}^\perp$. Write
\[\xi = \xi_{\mathfrak{h}} + \xi_{\mathfrak{h}^\perp}\]
for $\xi_\mathfrak{h} \in \mathfrak{h}$ and $\xi_{\mathfrak{h}^\perp} \in \mathfrak{h}^\perp$, and let $\overline{\xi}_\mathfrak{h}$ be the image of $\xi_\mathfrak{h}$ under the quotient map to $\overline{\h}.$ Reducibility implies that $\Ad_{h} \xi_{\mathfrak{h}^\perp} - \xi_{\mathfrak{h}^\perp} \in \mathfrak{h} \cap \mathfrak{h}^\perp$, so
\[\varphi_*(\xi^L-\xi^R)=\overline{\xi}_\mathfrak{h}^L-\overline{\xi}_\mathfrak{h}^R.\]
Similarly, $\Ad_{h}\xi_{\mathfrak{h}^\perp} + \xi_{\mathfrak{h}^\perp} \in \mathfrak{h}^\perp$ and therefore
\[\alpha = \frac{1}{2}\left(\overline{\xi}_\mathfrak{h}^L+\overline{\xi}_\mathfrak{h}^R\right)=\sigma(\overline{\xi}_\mathfrak{h}).\]
It follows that $(\varphi_*v, \alpha) \in L_{\bar{H}}$, and this implies the desired equality of Dirac structures.

To compute the Lie algebroid, we see that at any point $h\in H$ it is given by
\begin{align*}
A_{H,\varphi}&=L_G\cap (T(H\cap H^\perp)\oplus TH^\circ)\\
		&=\left\{(\xi^L-\xi^R,\sigma(\xi))\mid \Ad_h\xi-\xi\in\mathfrak{h}\cap\mathfrak{h}^\perp, \Ad_h\xi+\xi\in\mathfrak{h}^\perp\right\}\\
		&=\left\{(\xi^L-\xi^R,\sigma(\xi))\mid \xi\in\mathfrak{h}^\perp\right\},
		\end{align*}
where the last equality follows from the reducibility condition. Therefore the rank of this distribution is constant, and under the identification of Example \ref{cd} it coincides with the action Lie algebroid
\[A_{H,\varphi}\cong H\rtimes\mathfrak{h}^\perp.\]
In particular, if $\eta\in\Omega^3(G)$ is the Cartan $3$-form we have
\[\eta_{\vert p_T(A_{H,\varphi})}=\eta_{\vert\mathfrak{h}\cap\mathfrak{h}^\perp}=0.\]
Therefore $(H,\varphi)$ is a generalized reduction level of $G$.
\end{proof}

Applying Corollary \ref{cormain} in this setting, we obtain the following theorem.

\begin{theorem}
\label{qpred}
Let $(\mu,\nu):M\longrightarrow K\times G$ be a quasi-Poisson $K\times G$-manifold, and suppose that $\mathfrak{h}$ is a reducible Lie subalgebra of $\mathfrak{g}$ whose integrating subgroup $H\subset G$ intersects $\nu$ cleanly. If the quotient
\[Q=\nu^{-1}(H)/H^\perp\]
is a manifold, the maps $\varphi$ and $\mu$ descend to give a quasi-Poisson $K\times\overline{H}$-structure
\[(\overline{\mu},\overline{\varphi}):Q\longrightarrow K\times\overline{H}.\]
\end{theorem}

Suppose now that $G$ is a complex reductive group, let $P\subset G$ be a parabolic subgroup, and let $U\subset P$ be its unipotent radical. Then $P^\perp=U$, so $P$ and $U$ are both reducible, and we may apply Theorem \ref{qpred}.

\begin{example}
The subspace $U$ of $G$ is an ordinary reduction level. If $(\mu,\nu):M\longrightarrow K\times G$ is a quasi-Poisson $K\times G$-manifold, the reduction
\[\nu^{-1}(U)/P\longrightarrow K\]
is a quasi-Poisson $K$-manifold. In particular, reducing the internal fusion double gives a quasi-Hamiltonian $G$-structure on $G\times_UP$. The corresponding moment map is
\begin{align*}
G\times_PU &\longrightarrow G\\
[g:u]&\longmapsto gug^{-1},
\end{align*}
and this space is a multiplicative counterpart of the cotangent bundle of the partial flag variety $G/P$ described in Example \ref{unired}.
\end{example}

\begin{example}
Letting $L\coloneqq P/U$ we have a quotient map
\[\varphi:P\longrightarrow L.\]
In view of Proposition \ref{plevels}, for any quasi-Poisson $K\times G$-manifold $(\mu,\nu):M\longrightarrow K\times G$, the reduction
\[\nu^{-1}(P)/U\longrightarrow K\times L\]
is a quasi-Poisson $K\times L$-manifold. In particular, applying this to the internal fusion double $D(G)$ of Example \ref{fusdouble} we obtain space $G\times_UP$, which has a quasi-Hamiltonian $G\times L$-structure with moment map
\begin{align*}
G\times_UP&\longrightarrow G\times L\\
	[g:p]&\longmapsto (pU, gpg^{-1}).
	\end{align*}
This quasi-Hamiltonian manifold, which was first introduced in \cite[Theorem 9]{boa:11}, is a multiplicative analogue of the cotangent bundle of the quasiaffine variety $G/U$ described in Example \ref{parared}.
\end{example}

In view of Corollary \ref{qpred}, reduction at the level $(H,\varphi)$ produces a quasi-Poisson $K\times\overline{H}$-manifold. Taking a quotient by the action of $\overline{H}$ and applying Proposition \ref{quotients} gives a further corollary.

\begin{theorem}
\label{qpred2}
Let $(\mu,\nu):M\longrightarrow K\times G$ be a quasi-Poisson $K\times G$-manifold, and let $\mathfrak{h}$ be a reducible Lie subalgebra of $\mathfrak{g}$ whose integrating subgroup $H\subset G$ intersects $\mu$ cleanly. Suppose that the quotient $\nu^{-1}(H)/HH^\perp$ is a manifold.
\begin{enumerate}
\item The map $\mu$ descends to a quasi-Poisson moment map
\[\overline{\mu}:\nu^{-1}(H)/HH^\perp\longrightarrow K.\]
\item For every conjugacy class $\O\subset\overline{H}$, let $\O_{H\cap H^\perp}$ denote its preimage in $H$. Then
\[\nu^{-1}(\O_{H\cap H^\perp})/HH^\perp\]
is a quasi-Poisson submanifold of $\nu^{-1}(H)/HH^\perp$.
\item If $M$ is quasi-Hamiltonian, then
\[\nu^{-1}(H)/HH^\perp=\bigsqcup\nu^{-1}(\O_{H\cap H^\perp})/HH^\perp\]
is the foliation of this quotient into quasi-Hamiltonian leaves.
\end{enumerate}
\end{theorem}

\begin{example}
Letting $G$ be a complex reductive group and $H=P$ be a parabolic subgroup with unipotent radical $U$, Corollary \ref{qpred2} gives the bundle $G\times_PP$ the structure of a Hamiltonian quasi-Poisson $G$-space with moment map
\begin{align*}
\Phi:G\times_PP&\longrightarrow G\\
	[g:p]&\longmapsto gpg^{-1}.
	\end{align*}
The partition of this space into quasi-Hamiltonian leaves is given by
\[G\times_PP=\bigsqcup G\times_P\O U\]
and is indexed by conjugacy classes $\O\subset L$. This recovers the quasi-Hamiltonian $G$-structures on $G\times_P\O U$ constructed in \cite[Section 4]{boa:11}, and it is a multiplicative analogue to Example \ref{parags}.

In particular, when $P=B$ is a Borel subgroup, $U$ is the maximal unipotent subgroup of $G$ and $L=T$ is a torus. We obtain a quasi-Poisson structure on the multiplicative Grothendieck--Springer resolution
\[G\times_BB=\bigsqcup G\times_BtU\]
with quasi-Hamiltonian leaves given by twisted unipotent bundles indexed by elements $t\in T$.
\end{example}

If $\mathfrak{h}$ is reducible, then the sum $\mathfrak{h}+\mathfrak{h}^\perp$ is also a reducible subalgebra of $\mathfrak{g}$. If we make the additional assumption that $[\mathfrak{h}^\perp,\mathfrak{h}^\perp]\subset\mathfrak{h}$, then the subgroup $HH^\perp$ together with the map
\[\psi:HH^\perp\longrightarrow HH^\perp/H^\perp\cong \overline{H}\]
gives another type of reduction level of $G$, which coincides with that of Proposition \ref{plevels} when $\mathfrak{h}^\perp\subset \mathfrak{h}$. Parabolic subalgebras of a complex reductive Lie algebra, their commutator subalgebras, and maximal nilpotent subalgebras are all examples of reducible subalgebras satisfying this additional condition.

\begin{proposition}
Suppose that $\mathfrak{h}$ is reducible and that
\begin{equation}
\label{perps}
[\mathfrak{h}^\perp,\mathfrak{h}^\perp]\subset\mathfrak{h}.
\end{equation}
Then the pair $(HH^\perp,\psi)$ is a generalized reduction level of $G$ whose stabilizer subalgebroid is the action Lie algebroid $HH^\perp\rtimes(\mathfrak{h}\cap\mathfrak{h}^\perp)$.
\end{proposition}
\begin{proof}
Factoring the map $\psi$ gives the diagram
\begin{equation*}
\begin{tikzcd}[]
&HH^\perp \arrow[ld,swap, "\varphi"]\arrow[hook]{rr}{\imath} \arrow{dd}{\psi} && G \\
HH^\perp/(H\cap H^\perp)\arrow[rd, swap, "q"]&&&\\
&\bar{H},&&
\end{tikzcd}
\end{equation*}
where $\varphi$ is the map obtained from viewing $HH^\perp$ as a reducible subgroup as in Proposition \ref{plevels}. Both quotients of $HH^\perp$ inherit invariant nondegenerate symmetric bilinear forms from $G$, so that we can equip them with Cartan--Dirac structures for which the map $q$ is f-Dirac. We therefore have
\[\psi_*\imath^* L_G=q_*\varphi_*\imath^* L_G=L_{\overline{H}},\]
where the second equality follows from Proposition \ref{plevels}.

Moreover, at any point $g\in HH^\perp$,
\begin{align*}
A_{HH^\perp,\psi}&=L_G\cap (TH^\perp\oplus T(HH^\perp)^\circ)\\
		&=\left\{(\xi^R-\xi^L,\sigma(\xi))\mid \xi-\Ad_g\xi\in\mathfrak{h}^\perp, \xi+\Ad_g\xi\in\mathfrak{h}\cap\mathfrak{h}^\perp\right\}\\
		&=\left\{(\xi^R-\xi^L,\sigma(\xi))\mid \xi\in\mathfrak{h}\cap\mathfrak{h}^\perp\right\},
		\end{align*}
where the last equation follows from the assumption \eqref{perps}. Therefore $A_{HH^\perp,\psi}\cong HH^\perp\rtimes (\mathfrak{h}\cap\mathfrak{h}^\perp)$ has constant rank. Once again
\[\eta_{p_T(A_{HH^\perp,\psi})}=\eta_{\mathfrak{h}\cap\mathfrak{h}^\perp}=0,\]
and so we conclude that $(HH^\perp,\psi)$ is a generalized reduction level of $G$.
\end{proof}

Applying Corollary \ref{cormain}, we obtain the following reduction.

\begin{proposition}
\label{qpred3}
Let $(\mu,\nu):M\longrightarrow K\times G$ be a quasi-Poisson $K\times G$-manifold, and suppose that $\mathfrak{h}$ is a reducible Lie subalgebra of $\mathfrak{g}$ which satisfies \eqref{perps} and such that $HH^\perp\subset G$ intersects $\nu$ cleanly. If the quotient
\[Q=\nu^{-1}(HH^\perp)/H\cap H^\perp\]
is a manifold, the maps $\varphi$ and $\mu$ descend to give a quasi-Poisson $K\times\overline{H}$-structure
\[(\overline{\mu},\overline{\varphi}):Q\longrightarrow K\times\overline{H}.\]
\end{proposition}

We may once again quotient further by the action of $\overline{H}$.

\begin{proposition}
\label{qpred4}
Let $(\mu,\nu):M\longrightarrow K\times G$ be a quasi-Poisson $K\times G$-manifold, and let $\mathfrak{h}$ be a reducible Lie subalgebra of $\mathfrak{g}$ which satisfies \eqref{perps} and such that $HH^\perp\subset G$ intersects $\nu$ cleanly. Suppose that the quotient $\mu^{-1}(HH^\perp)/H$ is a manifold.
\begin{enumerate}
\item The map $\mu$ descends to a quasi-Poisson moment map
\[\overline{\mu}:\nu^{-1}(HH^\perp)/H\longrightarrow K.\]
\item For every conjugacy class $\O\subset\overline{H}$, let $\O H^\perp$ denote its preimage in $HH^\perp$. Then
\[M \sll{\O} H \coloneqq \nu^{-1}(\O H^\perp)/H\]
is a quasi-Poisson submanifold of $\nu^{-1}(HH^\perp)/H$.
\item If $M$ is quasi-Hamiltonian, then
\[\nu^{-1}(HH^\perp)/H=\bigsqcup\nu^{-1}(\O H^\perp)/H\]
is the foliation of this quotient into quasi-Hamiltonian leaves.
\end{enumerate}
\end{proposition}

\begin{example}
Let $P\subset G$ be a parabolic subgroup, let $U$ be its unipotent radical, and consider the reducible subgroup $H=[P,P]$. In this case we have
\[H^\perp=Z_LU, \quad HH^\perp=P, \quad H\cap H^\perp=U,\quad\text{and}\quad \overline{H}=[L,L],\]
where $Z_L$ is the center of the Levi component of $P$. We may then reduce the fusion product $D(G)\longrightarrow G\times G$ as in Proposition \ref{qpred4}. This gives a quasi-Poisson structure along the moment map
\begin{align*}
G\times_{[P,P]}P&\longrightarrow G\\
		[g:p]&\longmapsto gpg^{-1}
		\end{align*}
whose quasi-Hamiltonian foliation is indexed by the conjugacy classes of $[L,L]$ and is given by
\[G\times_{[P,P]}P=\bigsqcup G\times_{[P,P]}\O L_ZU.\]
\end{example}

\subsection{Multiplicative Whittaker reduction}
Let $G$ be a simply-connected, semisimple complex group equipped with the Cartan--Dirac structure. Fix a maximal torus $T$ and a Borel subgroup $B$ containing it. We write $U$ for the unipotent radical of $B$, $U^-$ for the unipotent radical of the opposite Borel $B^-$, and $\mathfrak{g}$, $\t$, $\b^\pm$, and $\mathfrak{u}^\pm$ for the corresponding Lie algebras.

Let $c \in N_G(T)$ be a representative of a Coxeter element of minimal length in the Weyl group of $G$. Then the Steinberg cross-section of Example \ref{steinslice} is given by
\[\Sigma=Uc\cap cU^-.\]
This cross-section is a slice \cite[Section 8.9]{ste:65} for the action of $U$ on the affine space
\[\Theta \coloneqq UcU,\]
in the sense that the conjugation action induces an isomorphism
\begin{equation}
\label{cross}
U\times\Sigma\xlongrightarrow{\sim} \Theta.
\end{equation}

In this section, we will show that $\Theta$ is a reduction level of $G$, and that the resulting reduced spaces coincide with the Steinberg slices of Example \ref{steinslice}. This is a multiplicative parallel to Whittaker reduction, which can be obtained both by taking the pullback of the Kostant slice under a moment map as well as by Hamiltonian reduction with respect to the action of the maximal unipotent subgroup. We will need the following simple lemma.

\begin{lemma}
\label{vsf93e3e}
For all $g \in \Theta$, we have
\[(\Ad_{g^{-1}} - 1)\t \cap (\Ad_{g^{-1}} \mathfrak{u} + \mathfrak{u}) = 0.\]
\end{lemma}

\begin{proof}
Let $g = ucv$ for $u,v \in U$ and let $\xi \in \t$ be such that $\Ad_{g^{-1}}\xi - \xi \in \Ad_{g^{-1}}\mathfrak{u} + \mathfrak{u}$. Then
\[\Ad_{c^{-1}u^{-1}}\xi - \Ad_v \xi \in \Ad_{c^{-1}} \mathfrak{u} + \mathfrak{u} \subset \mathfrak{u}^- + \mathfrak{u}.\]
We can also write
\[\Ad_{c^{-1}u^{-1}} \xi - \Ad_v \xi =(\Ad_{c^{-1}}\xi - \xi) + (\Ad_{c^{-1}}(\Ad_{u^{-1}} \xi - \xi) - (\Ad_v \xi  - \xi)),\]
where the first parenthesis is contained in $\t$ and the second is contained in $\mathfrak{u}^-+\mathfrak{u}$. Therefore $\Ad_{c^{-1}}\xi - \xi = 0$, and since Coxeter elements have no non-zero fixed points \cite[Section 3.16]{hum:90} it follows that $\xi = 0$.
\end{proof}

At every point $g\in\Theta$
\[T_g\Theta = \{x^R + y^L \mid x, y \in \mathfrak{u}\}.\]
We can therefore define a $2$-form $\gamma\in\Omega^2(\Theta)$ by
\[\gamma_g(x_1^R + y_1^L, x_2^R + y_2^L) = \tfrac{1}{2}(\Ad_g x_2, y_1)-\tfrac{1}{2}(\Ad_g x_1, y_2).\]
Since the Killing form is $G$-invariant and since $\mathfrak{u}$ is isotropic, it follows that $\gamma$ is well-defined.

\begin{proposition}
The pair $(\Theta,\gamma)$ is a reduction level of $G$ with stabilizer subalgebroid $\Theta\rtimes\mathfrak{u}$.
\end{proposition}
\begin{proof}
First we show that the differential of $-\gamma$ agrees with the restriction of the Cartan $3$-form $\eta$ to $\Theta$. Fix a point $g\in\Theta$, let $V_1,V_2,V_3\in T_g\Theta$, and write $V_i = x_i^R + y_i^L$ for $x_i, y_i \in \mathfrak{u}$. Then
\begin{align*}
2d\gamma(V_1, V_2, V_3) &= 2V_1 \gamma(V_2, V_3) + 2\gamma(V_1, [V_2, V_3]) + c.p. \\
					&= (\Ad_g[x_1, x_2], y_3) + ([y_1, y_2], \Ad_gx_3) + c.p.,
					\end{align*}
and
\[2\eta(V_1, V_2, V_3) = (x_1 + \Ad_{g^{-1}}y_1, [x_2 + \Ad_{g^{-1}}y_2, x_3 + \Ad_{g^{-1}} y_3])
= -2d\gamma(V_1, V_2, V_3).\]

Now we compute the distribution
\[A_{\Theta,\gamma}=L_G\cap\imath_*L_{\gamma}.\]
In view of Example \ref{cd}, a pair $(g,\xi) \in \Theta\times \mathfrak{g}$ lies in $A_{\Theta, \gamma}$ if and only if
\[\xi - \Ad_{g}\xi \in \Ad_{g} \mathfrak{u} + \mathfrak{u}\]
and
\[\gamma_g(\xi^R - \xi^L, x^R + y^L)= \frac{1}{2}(\Ad_{g} \xi + \xi, \Ad_{g}x + y)\quad\text{ for all $x,y\in\mathfrak{u}$}.\]
The second condition simplifies to
\[(\xi, x+y)=0\quad\text{ for all $x,y\in\mathfrak{u}$},\]
which occurs if and only if $\xi\in\mathfrak{u}^\perp=\b$, and by Lemma \ref{vsf93e3e} this is equivalent to $\xi\in \mathfrak{u}$. Therefore
\[A_{\Theta,\gamma}=\Theta\rtimes\mathfrak{u}.\qedhere\]
\end{proof}

Now let $K$ be a complex Lie group whose Lie algebra has an invariant inner product, and let $M$ be a quasi-Poisson $G\times K$-manifold with moment map
\[(\mu,\nu):M\longrightarrow G\times K.\]
It was shown in \cite[Theorem 2.3]{bal:21}, and also follows from Theorem \ref{main}, that the quasi-Poisson structure on $M$ pulls back to a Dirac structure on $\mu^{-1}(\Sigma)$ which is quasi-Poisson for the action of $K$.

Note that the affine space $\Theta$ is transverse to the moment map $\mu$, so the preimage $\mu^{-1}(\Theta)$ is a smooth manifold. Moreover, in view of \eqref{cross}, there is an isomorphism
\begin{equation}
\label{steiso}
\chi:\mu^{-1}(\Sigma)\xlongrightarrow{\sim}\mu^{-1}(\Theta)/U.
\end{equation}
In particular, the quotient $\mu^{-1}(\Theta)/U$ is alway a manifold. Since $(\Theta,\gamma)$ is a reduction level of $G$ and since the action of $\Theta\rtimes\mathfrak{u}$ on $\mu^{-1}(\Theta)$ integrates to an action of the unipotent subgroup $U$, Theorem \ref{main} gives the quotient $\mu^{-1}(\Theta)/U$ the structure of a quasi-Poisson $K$-manifold. We now show that $\chi$ is a quasi-Poisson isomorphism.

\begin{proposition}
The map \eqref{steiso} is an isomorphism of Hamiltonian quasi-Poisson manifolds.
\end{proposition}
\begin{proof}
Since it is clear that $\chi$ intertwines the quasi-Poisson moment maps of the two spaces, by \cite[Proposition 3.24]{bur.cra:05} it is sufficient to show that $\chi$ is f-Dirac. Consider the commutative diagram
\begin{equation*}
\begin{tikzcd}[row sep=large]
\mu^{-1}(\Sigma)\arrow[r, hook, "\imath"]\arrow[rd, swap, "\chi"]	&\mu^{-1}(\Theta)\arrow[r, hook, "\jmath"]\arrow[d, "q"]	&M\\
&\mu^{-1}(\Theta)/U.	&
\end{tikzcd}
\end{equation*}
The quasi-Poisson structure on $\mu^{-1}(\Sigma)$ is given by the Dirac pullback
\[L_\Sigma\coloneqq\imath^*\jmath^*L_M,\]
and the quasi-Poisson structure on the quotient $\mu^{-1}(\Theta)/U$ is the Dirac bundle
\[L_\Theta\coloneqq q_*\tau_{\mu^*\eta}\jmath^*L_M.\]

Chasing the diagram, we see that
\begin{align*}
\chi^*L_\Theta&=\imath^*q^*q_*\tau_{\mu^*\eta}\jmath^*L_M \\
			&=\imath^*\tau_{\mu^*\eta}\jmath^*L_M\\
			&=\tau_{\mu^*\eta}\imath^*\jmath^*L_M\\
			&=L_\Sigma.
			\end{align*}
Here the second equality follows from the fact that the quotient map $q$ is both b-Dirac and f-Dirac by Theorem \ref{main}\ref{8b2exmfa}, the third equality follows from \eqref{gaugecom}, and the fourth follows from the fact that the restriction of $\eta$ to the slice $\Sigma$ vanishes \cite[Lemma 2.2]{bal:21}. This implies that the isomorphism $\chi$ is b-Dirac and therefore also f-Dirac.
\end{proof}

\subsection{The multiplicative Moore--Tachikawa TQFT}
For each complex reductive group $G$, Moore and Tachikawa \cite{moo.tac:12} conjectured the existence of a certain 2-dimensional topological quantum field theory (TQFT) valued in a category $\mathrm{HS}$ of holomorphic symplectic varieties---that is, a symmetric monoidal functor
\[
\eta_G : \mathrm{Bod}_2 \longrightarrow \mathrm{HS}.
\]
Here $\mathrm{Bod}_2$ is the category whose objects are 1-dimensional compact manifolds and whose morphisms are cobordisms between them.
The objects of $\mathrm{HS}$ are complex reductive groups and a morphism from $G$ to $H$ is an affine symplectic variety with a Hamiltonian action of $G \times H$.
Two morphisms $M : G \longrightarrow H$ and $N : H \longrightarrow K$ can be composed by taking the symplectic reduction of $M \times N$ by the diagonal $G$-action.

The Moore--Tachikawa conjecture states that, for each complex reductive group $G$, there is such a functor $\eta_G$ which sends the cap to the symplectic manifold $G \times \mathcal{S}$, where $\mathcal{S}$ is the Kostant slice. The non-triviality of this conjecture is exemplified by the case $G = \mathrm{SL}(3, \mathbb{C})$, where $\eta_G$ maps the pair of pants to the minimal nilpotent orbit of the exceptional Lie algebra $E_6$. Such a functor is completely determined by the symplectic varieties to which it maps $n$-punctured spheres. These symplectic varieties are known as the \emph{Moore--Tachikawa varieties}, and in type $A$ they have been explicitly constructed \cite{bra.fin.nak:19, gin.kaz:21}. In general Lie type, smooth open dense subvarieties of these spaces can be realized as Hamiltonian reductions relative to the action of a group scheme known as the universal centralizer \cite{gin.kaz:21, bie:21}. In \cite{cro.may:21}, it was shown that this is a special case of a more general procedure of symplectic reduction along a submanifold.

One can formulate a multiplicative analogue of this conjecture, where the morphisms in the target category are quasi-Hamiltonian $G \times H$-varieties. The conjecture then requires a functor which sends the cap to the quasi-Hamiltonian $G$-space $G \times \Sigma$ \cite[Example 2.8]{bal:21}, where $\Sigma$ is the Steinberg cross-section discussed in Example \ref{steinslice}. The quasi-Hamiltonian varieties corresponding to the $n$-punctured spheres under this conjectural functor are multiplicative analogues of the Moore--Tachikawa varieties. In this section, we construct smooth open dense subvarieties of these spaces and we show that they satisfy the corresponding gluing laws.

Let $G$ be a simply-connected complex semisimple group and let $G_{\ad}$ be its adjoint form. Consider the double
\[\mathbf{D} \coloneqq G_{\ad} \times G,\]
which is the quotient of the internal fusion double $D(G)$ of Example \ref{fusdouble} by the left action of the center of $G$. The quasi-Hamiltonian structure on $D(G)$ induces a quasi-Hamiltonian structure on $\mathbf{D}$. Moreover, since the conjugation action of $G$ on itself descends to an action of $G_{\ad}$, the double $\mathbf{D}$ is in fact a twisted presymplectic groupoid integrating the Cartan--Dirac structure on $G$. For any element $a \in G$, we write
\[Z_{\ad}(a) \coloneqq \{g \in G_{\ad} \mid gag^{-1} = a\}\]
for its isotropy group with respect to this action, and $\mathfrak{g}_a$ for the corresponding Lie subalgebra of $\mathfrak{g}$.

Let $m$ and $n$ be non-negative integers, let $\Sigma \subset G$ be the Steinberg cross-section, and embed it into $G^m\times G^n$ as the submanifold
\[\Sigma_{m, n} \coloneqq \{(a, b) \in G^m \times G^n \mid a_1 = \cdots = a_m = b_1^{-1} = \cdots = b_n^{-1} \in \Sigma\}.\]

\begin{proposition}
The submanifold $\Sigma_{m, n}$ is a reduction level of $G^{m + n}$ with stabilizer subalgebroid
\[A_{m,n}\coloneqq \{(s, x, y) \in \Sigma_{m,n}\times \mathfrak{g}^m \times \mathfrak{g}^n \mid x_1 + \cdots + x_m = y_1 + \cdots + y_n\text{ and }x_i, y_i\in\mathfrak{g}_s\}.\]
\end{proposition}

\begin{proof}
Note that the restriction of $\eta$ to the Steinberg cross-section vanishes. Therefore it is sufficient to compute the distribution
\[A_{m,n}=L_{G^{m+n}}\cap (T\Sigma_{m,n}\oplus T\Sigma_{m,n}^\circ).\]
Let $\iota:G\longrightarrow G$ be the inversion map. At every point $s\in\Sigma_{m,n}$, the fiber of $A_{m,n}$ consists of the points $(x,y)\in\mathfrak{g}^m\times\mathfrak{g}^n$ such that
\[(x_1^R - x_1^L)_s = \cdots = (x_m^R - x_m^L)_s = \iota_*(y_1^R - y_1^L)_{s^{-1}} = \cdots = \iota_*(y_n^R - y_n^L)_{s^{-1}} \in T\Sigma,\]
and
\[(x^R + x^L, y^R + y^L) \in T\Sigma_{m, n}^\circ.\]

Because $\Sigma$ intersects each conjugacy class transversally, all the terms in the first condition must vanish and so we obtain
\[\Ad_s (x_i) = x_i\quad \text{ and} \quad\Ad_s (y_i) = (y_i) \quad\text{ for all } i.\]
In particular, $x_i,y_i\in\mathfrak{g}_s$. The second condition then implies that $(x^L,y^L) \in T\Sigma_{m, n}^\circ$, and this occurs if and only if
\[x_1+\ldots+x_m=y_1+\ldots+y_m.\]
It follows that $A_{m,n}$ has constant rank, and therefore $\Sigma_{m,n}$ is a reduction level.
\end{proof}

Consider the source-connected subgroupoid of $\mathbf{D}^{m+n}=\mathbf{D}^m\times \mathbf{D}^n$ given by
\[Z_{m, n} \coloneqq \{(s, g, h) \in \Sigma_{m,n}\timesG_{\ad}^m \times G_{\ad}^n \mid g_1 \cdots g_m = h_1 \cdots h_n\text{ and }g_i, h_i \in Z_{\ad}(s)\}\]
which integrates $A_{m,n}$. We may view $\mathbf{D}^{m+n}$ as a quasi-Hamiltonian $G^{m+n} \times G^{m+n}$-space with moment map
\begin{align*}
(\mu,\nu):\mathbf{D}^{m+n}&\longrightarrow G^{m+n}\times G^{m+n}\\
		(g,h)&\longmapsto (ghg^{-1},h^{-1}).
		\end{align*}
Applying Corollary \ref{cormain} and reducing at the reduction level $\Sigma_{m,n}$ of the second factor of $G^{m+n}$, we obtain the quasi-Hamiltonian $G^{m + n}$-space
\[\mathfrak{Z}_{m, n} \coloneqq \nu^{-1}(\Sigma_{m,n})/Z_{m, n}\cong (G^{m+n}\times\Sigma)/Z_{m,n},\]
with moment map
\begin{align*}
\mathfrak{Z}_{m,n}&\longrightarrow \qquad G^{m+n} \\
(g,h,s)&\longmapsto (g_isg_i^{-1},h_is^{-1}h_i^{-1}).
\end{align*}
Note that this quotient is always a manifold, since $Z_{m,n}$ is a closed subgroupoid of $G^{m+n}\times\Sigma$ and therefore acts properly. We call it an \emph{open multiplicative Moore--Tachikawa variety}.

\begin{example}
Taking $m=1$ and $n=0$ gives the Steinberg slice
\[\mathfrak{Z}_{0, 1} = G_{\ad} \times \iota(\Sigma),\]
which is a quasi-Hamiltonian $G$-space studied in \cite[Section 2.2]{bal:21}. Taking $m=n=1$ gives
\[\mathfrak{Z}_{1, 1}\cong G_{\ad} \times G^{\mathrm{r}},\]
which is the restriction of the groupoid $\mathbf{D}$ to the regular locus.
\end{example}

Under the multiplicative Moore--Tachikawa correspondence, the space $\mathfrak{Z}_{m,n}$ should be thought of as an open dense subset of the variety associated with the genus-0 cobordism from $m$ circles to $n$ circles. Accordingly, the composition of $\mathfrak{Z}_{m, n}$ and $\mathfrak{Z}_{n, k}$, which is given by quasi-Hamiltonian reduction of the fusion product relative to the diagonal $G_{\ad}^n$-action, should be isomorphic to $\mathfrak{Z}_{m, k}$. The rest of this section is devoted to the proof of this fact.

Let $m, n, \widetilde{m}, \widetilde{n}$ be positive integers. Consider the quasi-Hamiltonian moment maps
\[(\mu,\mu_0):\mathfrak{Z}_{m,n}\longrightarrow G^{m+n-1}\times G\quad\text{and}\quad(\widetilde{\mu},\widetilde{\mu}_0):\mathfrak{Z}_{\widetilde{m},\widetilde{n}}\longrightarrow G^{\widetilde{m}+\widetilde{n}-1}\times G,\]
where $\mu_0$ and $\widetilde{\mu}_0$ are defined by
\[\mu_0(g,h,s)=g_1sg_1^{-1}\quad\text{and}\quad\widetilde{\mu}_0(g,h,s)=h_1s^{-1}h_1^{-1}.\]
Using these two maps to reduce along the diagonal as in Proposition \ref{fiberprod}, we see that the open Moore--Tachikawa spaces satisfy the following recursion.

\begin{proposition}
There is a canonical isomorphism of quasi-Hamiltonian $G_{\ad}^{m + \widetilde{m} - 1} \times G_{\ad}^{n + \widetilde{n} - 1}$-spaces
\[\mathfrak{Z}_{m+\widetilde{m}-1,n+\widetilde{n}-1}\cong(\mathfrak{Z}_{m,n}\times_G\mathfrak{Z}_{\widetilde{m},\widetilde{n}})/G,\]
where the fiber product on the right is taken along the maps $\mu_0$ and $\widetilde{\mu}_0^{-1}$.
\end{proposition}

\begin{proof}
Since $\Sigma$ is a slice for the conjugation action of $G$ on $G^{\mathrm{r}}$,
\[\mathfrak{Z}_{m,n}\times_G\mathfrak{Z}_{\widetilde{m},\widetilde{n}}= \{(g, h, s, \widetilde{g}, \th, \widetilde{s})\mid s=\widetilde{s}\text{ and } \tilde{g}_1^{-1} h_1 \in G^s\}.\]
We can therefore define the $G^{m + \widetilde{m} - 1} \times G^{n + \widetilde{n} - 1}$-equivariant map
\begin{align*}
F: \mathfrak{Z}_{m + \widetilde{m} - 1, n + \widetilde{n} - 1} &\longrightarrow \mathfrak{Z}_{m,n}\times_G\mathfrak{Z}_{\widetilde{m},\widetilde{n}}\\
		(g,h,s)&\longrightarrow(g,\bar{h},s,\bar{g},h,s),
		\end{align*}
where for any tuple $a=(a_1,\ldots,a_k)$ of elements of $G$ we define $\bar{a}\coloneqq (1,a_1,\ldots, a_k).$ Then $F$ descends to an isomorphism
\[\overline{F}:\mathfrak{Z}_{m + \widetilde{m} - 1, n + \widetilde{n} - 1} \longrightarrow (\mathfrak{Z}_{m,n}\times_G\mathfrak{Z}_{\widetilde{m},\widetilde{n}})/G.\]

To show that $\overline{F}$ is a Dirac diffeomorphism, consider the commutative diagram
\begin{equation*}
\begin{tikzcd}[column sep=large, row sep=large]
\mathfrak{Z}_{m + \widetilde{m} - 1, n + \widetilde{n} - 1}		\arrow[r, "F"]\arrow[rd, swap, "\overline{F}"]			&\mathfrak{Z}_{m,n}\times_G\mathfrak{Z}_{\widetilde{m},\widetilde{n}} \arrow[d]\\
						&(\mathfrak{Z}_{m,n}\times_G\mathfrak{Z}_{\widetilde{m},\widetilde{n}})/G.
\end{tikzcd}
\end{equation*}
The vertical map is b-Dirac by Theorem \ref{main}\ref{8b2exmfa}, and the horizontal map $F$ descends from a diagram
\begin{equation*}
\begin{tikzcd}[column sep=large, row sep=large]
G^{m+m\widetilde{m}-1}\times G^{n+\widetilde{n}-1}\times\Sigma		\arrow[r]\arrow[d]		& G^{m+n}\times_G G^{\widetilde{m}+\widetilde{n}}\times \Sigma\arrow[d]\\
\mathfrak{Z}_{m + \widetilde{m} - 1, n + \widetilde{n} - 1}			\arrow[r, "F"]				&\mathfrak{Z}_{m,n}\times_G\mathfrak{Z}_{\widetilde{m},\widetilde{n}}.
\end{tikzcd}
\end{equation*}
Here the vertical arrows are quotients by the actions of $Z_{m + \widetilde{m} - 1, n + \widetilde{n} - 1}$ and $Z_{m,n}\times Z_{\widetilde{m},\widetilde{n}}$, and are therefore f-Dirac. Since the top horizontal map is b-Dirac, it follows from \cite[Lemma 1.16]{bal:21} that $F$ is also b-Dirac. Then, since $\overline{F}$ is the composition of two b-Dirac maps, it is also b-Dirac and therefore a Dirac diffeomorphism.
\end{proof}

In view of Proposition \ref{ooysgjbr}, we can give an interpretation of this recursive ``gluing'' condition in terms of fusion products.

\begin{corollary}
Then there is a canonical isomorphism of quasi-Hamiltonian $G_{\ad}^{m + \widetilde{m} - 1} \times G_{\ad}^{n + \widetilde{n} - 1}$-spaces
\[\left(\mathfrak{Z}_{m,n}\circledast\mathfrak{Z}_{\widetilde{m},\widetilde{n}}\right)\sll{1}G\longrightarrow \mathfrak{Z}_{m+\widetilde{m}-1,n+\widetilde{n}-1},\]
where the fusion product on the left is taken along the maps $\mu_0$ and $\widetilde{\mu}_0$.
\end{corollary}

\noindent This corollary corresponds to the fact that the by taking the genus-0 cobordism from $m$ circles to $n$ circles, the genus-0 cobordism from $\widetilde{m}$ circles to $\widetilde{n}$-circles, and gluing an outgoing circle of the first with an incoming circle of the second, we obtain the genus-0 cobordism from $m + \widetilde{m} - 1$ circles to $n + \widetilde{n} - 1$ circles.

\section{Interpretation in shifted symplectic geometry}
\label{fifth}
In the case when $(S,\gamma)$ is an ordinary reduction level of $X$ and $\Phi:M\longrightarrow X$ is a Dirac realization from a nondegenerate Dirac manifold $M$, the reduction result of Theorem \ref{main} has an interpretation in shifted symplectic geometry. We explain it below, and as a corollary we obtain a generalization of Theorem \ref{main} in this special case which allows us to reduce by the action of stabilizer subgroupoids that are not necessarily source-connected.

Suppose that the Dirac manifold $X$ is integrated by the presymplectic groupoid $\mathcal{G}\rightrightarrows X$, and write $\Omega$ for the corresponding presymplectic form on $\mathcal{G}$. A \emph{Hamiltonian $\mathcal{G}$-space} \cite{xu:04} is a nondegenerate Dirac manifold $M$ with presymplectic form $\omega\in\Omega^2(M)$ and an action of $\mathcal{G}$ along a map $\Phi:M\longrightarrow X$ such that
\begin{itemize}[topsep=1.5pt]
\item $\Phi$ is a Dirac realization with respect to the $2$-form $-\omega\in\Omega^2(M)$, and
\item the graph of this action, which is given by
\[\Gamma = \{(g, x, g \cdot x) \in \mathcal{G} \times M \times M \mid \mathsf{t}(g) = \mu(x)\},\]
is isotropic with respect to $(\Omega, \omega, -\omega)$.
\end{itemize}

This notion generalizes both Hamiltonian and quasi-Hamiltonian manifolds relative to the action of a group $G$. When
\[\mathcal{G}=T^*G\rightrightarrows\mathfrak{g}^*\]
is the symplectic groupoid of the Kirillov--Kostant--Souriau Poisson structure on $\mathfrak{g}^*$, $M$ is a Hamiltonian $\mathcal{G}$-space if and only if it is Hamiltonian for the induced action of the group $G$. When
\[\mathcal{G}=D(G)\rightrightarrows G\]
is the quasi-Hamiltonian internal fusion double which integrates the Cartan--Dirac structure on $G$, a Hamiltonian $\mathcal{G}$-space is the same thing as a quasi-Hamiltonian $G$-manifold.

Symplectic and quasi-Hamiltonian reduction generalize to the case of Hamiltonian $\mathcal{G}$-spaces---at any point $x$ of $X$ which is a regular value of $\mu$, the quotient
\[M \sll{x} \mathcal{G} \coloneqq \mu^{-1}(x) / \mathcal{G}_x\]
has a natural reduced symplectic structure \cite[Theorem 3.18]{xu:04}, as long as it is a manifold. If the isotropy group $\mathcal{G}_x$ is connected, this reduction is also an example of Theorem \ref{main}. It is therefore natural to try to generalize Theorem \ref{main} by taking quotients with respect to groupoids that integrate the stabilizer subalgebroid $A_{S, \gamma}$ but that are not source-connected. To do this, we will need the following definition.

\begin{definition}
\label{gpoids}
Let $\mathcal{G} \rightrightarrows X$ be an $\eta$-twisted presymplectic groupoid with $2$-form $\Omega$, and let $(S,\gamma)$ be a reduction level. A \emph{stablizer subgroupoid} of $S$ in $\mathcal{G}$ is a Lie groupoid $\mathcal{H} \rightrightarrows S$ together with a Lie groupoid morphism $f : \mathcal{H} \longrightarrow \mathcal{G}$ such that
\begin{itemize}[topsep=1.5pt]
\item the map $f_* : \Lie(\mathcal{H}) \to \Lie(\mathcal{G})$ restricts to a Lie algebroid isomorphism $\Lie(\mathcal{H}) \cong A_{S, \gamma}$, and
\item $f^*\Omega = \mathsf{s}^*\gamma - \mathsf{t}^*\gamma$.
\end{itemize}
\end{definition}

\begin{remark}
If $X$ is a Poisson manifold and $\mathcal{H}$ is source-connected, the second condition of Definition \ref{gpoids} is automatically satisfied \cite[Proposition 7.2]{cat.zam:09}. It is also easy to see that, in general, this condition holds along the identity section of any groupoid.
\end{remark}

The usual reduction of Hamiltonian $\mathcal{G}$-spaces then has the following generalization.

\begin{theorem}\label{0rue6qjh}
Let $\mathcal{G} \rightrightarrows X$ be a twisted presymplectic groupoid and let $(S, \gamma)$ be a reduction level of $X$. Suppose that $\Phi : M \to X$ is a Hamiltonian $\mathcal{G}$-space such that $S$ has clean intersection with the moment map $\Phi$, and that $\mathcal{H} \rightrightarrows S$ is a stablizer subgroupoid of $S$ such that the quotient
\[M \sll{S} \mathcal{G} \coloneqq \Phi^{-1}(S) / \mathcal{H}\]
is a manifold. Then reduction equips $M \sll{S} \mathcal{G}$ with a natural symplectic structure.
\end{theorem}

This theorem is a simple consequence of interpreting our construction in the framework of shifted symplectic geometry \cite{pan.toe.vaq.vez:13}, as an intersection of two Lagrangians in a 1-shifted symplectic stack. Recall that there is an equivalence between 1-shifted symplectic 1-stacks and twisted presymplectic groupoids up to Morita equivalence \cite[Section 1.2.3]{cal:20}. Namely, given a Lie groupoid
\[\mathcal{G} \rightrightarrows X,\]
a 1-shifted symplectic structure on the quotient stack $[X/\mathcal{G}]$ consists precisely of a 2-form $\Omega$ on $\mathcal{G}$ and a 3-form $\eta$ on $X$ satisfying the definition of a presymplectic groupoid. Two twisted presymplectic groupoids correspond to the same 1-shifted symplectic stack if and only if they are Morita equivalent in the sense of \cite{xu:04}.

One of the main existence results for shifted symplectic structures is through the notion of Lagrangian morphisms. Given an $n$-shifted symplectic derived stack $Z$ and two Lagrangians $L_1 \to Z$ and $L_2 \to Z$, the fibre product $L_1 \times_Z L_2$ (in the sense of derived algebraic geometry) is $(n-1)$-shifted symplectic \cite[Theorem 2.9]{pan.toe.vaq.vez:13}. In the case where $Z = [X/\mathcal{G}]$ is a 1-shifted symplectic 1-stack, the notion of Lagrangians has the following simple description. Let $\mathcal{H} \rightrightarrows S$ be a Lie groupoid together with a Lie groupoid morphism $f : \mathcal{H} \to \mathcal{G}$, which gives a morphism of quotient stacks
\[[f] : [S/\mathcal{H}] \longrightarrow [X/\mathcal{G}].\]
An \emph{isotropic structure} on $[f]$ is a $2$-form $\gamma\in\Omega^2(S)$ such that $f^*\Omega = \mathsf{s}^*\gamma - \mathsf{t}^*\gamma\quad\text{and}\quad d\gamma = f^*\eta.$ This structure is \emph{Lagrangian} if the induced morphism
\begin{equation}
\label{ldm50cv0}
\mathbb{T}_f  \longrightarrow \mathbb{L}_{[S/\mathcal{H}]}
\end{equation}
is a quasi-isomorphism, where
\[\mathbb{T}_f \coloneqq f^*\mathbb{T}_{[X/\mathcal{G}]}[-1] \oplus \mathbb{T}_{[S/\mathcal{H}]}\]
is the relative tangent complex endowed with the differential obtained from the map $\mathbb{T}_{[S/\mathcal{H}]} \to f^*\mathbb{T}_{[X/\mathcal{G}]},$ and $\mathbb{L}_{[S/\mathcal{H}]}$ is the cotangent complex.

The notion of Hamiltonian $\mathcal{G}$-space has a simple interpretation \cite[Example 1.31]{cal:20} in this setting: given a groupoid action of $\mathcal{G} \rightrightarrows X$ along a smooth map $\Phi : M \longrightarrow X,$ a 2-form $\omega$ on $M$ gives it the structure of Hamiltonian $\mathcal{G}$-space if and only if $\omega$ is a Lagrangian structure on the morphism of quotient stacks
\[[\Phi] : [M/\mathcal{G}] \longrightarrow [X/\mathcal{G}].\]
To obtain 0-shifted symplectic structures---that is, ordinary symplectic structures---it then suffices to find another Lagrangian morphism $L \to [X/\mathcal{G}]$ and take the fibre product
\[L \times_{[X/\mathcal{G}]} [M/\mathcal{G}].\]
The main result of this section is to show that Lagrangian \emph{substacks} of $[X/\mathcal{G}]$ are precisely stablizer subgroupoids of twisted reduction levels.

\begin{theorem}\label{ex4wuat6}
Let $[X/\mathcal{G}]$ be a 1-shifted symplectic 1-stack and $\mathcal{H} \rightrightarrows S$ be a Lie subgroupoid of $\mathcal{G} \rightrightarrows X$. A 2-form $-\gamma\in\Omega^2(S)$ is a Lagrangian structure on
\[[S/\mathcal{H}] \longrightarrow [X/\mathcal{G}]\]
if and only if $(S, \gamma)$ is a reduction level and $\mathcal{H} \rightrightarrows S$ is a stabilizer subgroupoid of $(S, \gamma)$.
\end{theorem}

\begin{remark}
\label{abzwgl6w}
If $(S, \gamma)$ is a reduction level of $X$ and $\gamma$ extends to a $2$-form $\tilde{\gamma}\in\Omega^2(X)$, the stabilizer subalgebroid $A_{S,\gamma}$ is equal to the stabilizer subalgebroid of the reduction level $(S, 0)$ relative to the gauge-transformed Dirac structure $\tau_{\tilde{\gamma}}L_X.$ In the holomorphic or algebraic setting such a global extension may not exist, but this still allows us to reduce local results to the case where $\gamma = 0$, as we will do here.
\end{remark}

\begin{lemma}\label{19xp3wff}
Let $(S, \gamma)$ be a reduction level of $X$ and let $p_{T,A} : A_{S, \gamma} \longrightarrow TS$ be the anchor map. Then the sequences
\[\begin{tikzcd}[row sep = -1mm]
0 \arrow{r} & A_{S, \gamma} \arrow{r} & L_X \cap (TS \oplus T^*X) \arrow{r} & \ker(p_{T,A}^*) \arrow{r} & 0,\\
& & \hspace{2cm}(v, \alpha) \arrow[mapsto]{r} & \imath^*\alpha - i_v \gamma
\end{tikzcd}\]
and
\[\begin{tikzcd}
0 \arrow{r} & A_{S,\gamma} \cap T^*X \arrow{r} & T^*X \arrow{r} & (\im p_T + TS)^* \arrow{r} & 0
\end{tikzcd}\]
are exact.
\end{lemma}

\begin{proof}
Since this is a pointwise result, by Remark \ref{abzwgl6w} we may assume that $\gamma = 0$. For the first sequence, we need to show that the map $\varphi : (v, \alpha) \longmapsto \imath^*\alpha$ is surjective. By identifying
\[\ker(p_{T,A}) = p_T(A_S)^\circ / TS^\circ\]
and writing $p_{T^*}:L_X\longrightarrow T^*X$ for the second projection, the image of $\varphi$ is
\[(p_{T^*}(L_X \cap (TS \oplus T^*X)) + TS^\circ) / TS^\circ.\]
Since
\[p_T(A_S)^\circ = TS^\circ + p_T(L_X \cap (TX \oplus TS^\circ))^\circ,\]
the statement now follows from the fact that
\[p_{T^*}(L_X \cap (TS \oplus T^*X)) = p_T(L_X \cap (TX \oplus TS^\circ))^\circ.\]

For the second sequence, we need to show that
\[A_{S, \gamma} \cap T^*X = (\im p_T + TS)^\circ.\]
Note that $\alpha \in A_{S, \gamma} \cap T^*X$ if and only if $(0, \alpha) \in L_X$ and $\alpha \in TS^\circ$. By the isotropy of $L_X$, this is equivalent to $\alpha \in (\im p_T)^\circ$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{ex4wuat6}]
Suppose that $-\gamma$ is a Lagrangian structure and let $L \coloneqq \Lie(\mathcal{H})$. Since $\imath^*\eta = -d\gamma$, it suffices to show that $L = A_{S, \gamma}$. The map of complexes \eqref{ldm50cv0} is then given by
\begin{equation}\label{64x0xucj}
\begin{tikzcd}[row sep=large]
L \arrow{r}{\alpha} \arrow{d} & L_{X\vert S} \oplus TS \arrow{r}{\beta} \arrow{d}{\gamma} & TX_{\vert S} \arrow{d}{\delta} \\
0 \arrow{r} & T^*S \arrow{r}{\varepsilon} & L^*,
\end{tikzcd}
\end{equation}
where the maps are defined by
\[\alpha(l) = (l, \rho_L(l)),\quad \beta(a, v) = v - \rho(a),\quad\text{and}\quad \gamma(a, v) = \imath^*\sigma(a) - i_v\gamma,\]
$\delta$ is the dual of
\[L \longrightarrow L_{X\vert S} \xlongrightarrow{p_{T^*}} T^*X_{\vert S},\]
and $\varepsilon$ is the dual of the anchor map $p_{T,L}:L\longrightarrow TS$.

The middle map on cohomologies $\ker \beta / \im \alpha \to \ker \varepsilon$ then reduces to
\begin{align}\label{ekskha1y}
\varphi : p_T^{-1}(TS) / L &\longrightarrow \ker(p_{T,L}^*)\\
a&\longmapsto \imath^*p_{T^*}(a) - \gamma^\flat(p_T(a)).\nonumber
\end{align}
The assumption $\ker \varphi = 0$ implies that $L = A_{S}$. Therefore $S$ is a reduction level and $\mathcal{H}$ is a stabilizer subgroupoid of $S$.

Conversely, suppose that $(S, \gamma)$ is a reduction level and $\mathcal{H} \rightrightarrows S$ is a stabilizer subgroupoid of $S$. We need to show that \eqref{64x0xucj} is a quasi-isomorphism. Since $L = A_{S, \gamma}$, Lemma \ref{19xp3wff} shows that \eqref{ekskha1y} is an isomorphism. Showing that the third map on cohomologies is an isomorphism amounts to the exactness of the sequence
\[\begin{tikzcd}
0 \arrow{r} & \im \rho + TS \arrow{r} & TX|_S \arrow{r} \arrow{r} & (\ker \rho_L)^* \arrow{r} & 0,
\end{tikzcd}\]
which is the dual of the second sequence in Lemma \ref{19xp3wff}.
\end{proof}

Theorem \ref{0rue6qjh} then follows from the observation that
\[M \sll{S} \mathcal{H} =  [S/\mathcal{H}] \times_{[X / \mathcal{G}]} [M/\mathcal{G}].\]
Moreover, note that even in the absence of smoothness conditions we can still interpret $M \sll{S} \mathcal{H}$ as a 0-shifted symplectic derived stack.\\

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