2.1 Fundamental conventions

In this manuscript, we work exclusively over $\mathbb {C}$ ; it is the base field underlying all objects for which a base field is presupposed (e.g., vector spaces, manifolds, and algebraic varieties). The reader should interpret all differential-geometric notions as being in the holomorphic category.

2.2 Conventions for Lie groupoids

Let $\mathcal G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; X$ be a groupoid object in the category of complex manifolds, with source, target, and identity bisection maps denoted $\textsf {s}:\mathcal G\longrightarrow X$ , $\textsf {t}:\mathcal G\longrightarrow X$ , and $\textsf {1}:X\longrightarrow \mathcal G$ , respectively. Given a submanifold $S\subseteq X$ , we write $\mathcal G\big \vert _S$ for the restriction $\textsf {s}^{-1}(S)\cap \textsf {t}^{-1}(S)\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; S$ . Let us adopt the simplified notation $\mathcal G_x:= \mathcal G\vert _{\{x\}}$ for the isotropy group of $x\in X$ . We also observe the following convention for groupoid multiplication: given $g,h\in \mathcal G$ , the product $gh\in \mathcal G$ is defined if and only if $\textsf {s}(g)=\textsf {t}(h)$ . Multiplication thereby takes the form of a map $\textsf {m}:\mathcal G{\, }_{\textsf {s}}{\!\!}\times _{\textsf {t}}\,\mathcal G\longrightarrow \mathcal G$ .

One calls $\mathcal G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; X$ a Lie groupoid if $\textsf {s}$ and $\textsf {t}$ are submersions. Assume this to be the case, and let $M$ be a manifold. An action of $\mathcal G$ on $M$ consists of holomorphic maps $\mu :M\longrightarrow X$ and

$$ \begin{align*}\mathcal{A}:\mathcal G{}_{\textsf{s}}\times_{\mu}\,M\longrightarrow M,\quad (g,p)\mapsto g\cdot p\end{align*} $$

that satisfy the natural generalizations of the left group action axioms. The associated action groupoid is $\mathcal G{\, }_{\textsf {s}}{\!\!}\times _{\mu }\,M\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; M$ , with source $(g,p)\mapsto p$ , target $(g,p)\mapsto g\cdot p$ , identity bisection $p\mapsto (\textsf {1}(p),p)$ , and multiplication $(g,p)(h,q)=(gh,q)$ . We denote the action groupoid by $\mathcal G\ltimes M$ .

2.3 Quasi-symplectic groupoids

A quasi-symplectic groupoid is a triple $(\mathcal G,\omega ,\phi )$ of a Lie groupoid $\mathcal G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; X$ , multiplicative $2$ -form $\omega \in \Omega ^2(\mathcal G)$ , and $3$ -form $\phi \in \Omega ^3(X)$ satisfying $\mathrm {d}\omega =\textsf {s}^*\phi -\textsf {t}^*\phi $ , $\dim \mathcal G=2\dim X$ , and $\ker \omega _x\cap \ker \mathrm {d}\textsf {s}_x\cap \ker \mathrm {d}\textsf {t}_x=\{0\}$ for all $x\in X$ , where one uses the identity bisection to regard $x$ as belonging to $\mathcal G$ . We let $\overline {\mathcal G}$ denote the “opposite” $(\mathcal G,-\omega ,-\phi )$ of a quasi-symplectic groupoid $(\mathcal G,\omega ,\phi )$ .

Consider a quasi-symplectic groupoid $(\mathcal G \;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; X, \omega , \phi )$ and manifold $M$ equipped with a 2-form $\gamma \in \Omega ^2(M)$ . Suppose that $\mathcal G$ acts on $M$ along a holomorphic map $\mu : M \longrightarrow X$ . This action is called Hamiltonian [Reference Xu49] if it satisfies the following properties:

• • $\mu ^*\phi = -\mathrm {d}\gamma $ ;

• • the graph of the action, that is, $\{(g, m, g \cdot m) : (g, m) \in \mathcal G{\, }_{\textsf {s}}{\!\!}\times _{\mu }\,M\}$ , is isotropic in $\mathcal G \times M \times M$ with respect to $(\omega , \gamma , -\gamma )$ ;

• • $\ker \mathrm {d}\mu \cap \ker \gamma = 0$ .

A quasi-symplectic groupoid $(\mathcal G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; X,\omega ,\phi )$ is called symplectic if $\phi =0$ and $\omega $ is non-degenerate. It turns out that a Lie groupoid $\mathcal G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; X$ and $2$ -form $\omega \in \Omega ^2(\mathcal G)$ constitute a symplectic groupoid if and only if the graph of groupoid multiplication is isotropic in $\mathcal G\times \mathcal G\times \overline {\mathcal G}$ . The Lie algebroid of $\mathcal G$ is then $T^*X\longrightarrow X$ , along with an anchor map that defines a Poisson structure on $X$ . An integration of a Poisson manifold $X$ is a symplectic groupoid $\mathcal G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; X$ that induces the given Poisson structure on $X$ .

We now discuss a paradigm example of a symplectic groupoid. To this end, let $G$ be a Lie group with Lie algebra $\mathfrak {g}$ . Use the left trivialization to identify $T^*G$ with $G\times \mathfrak {g}^*$ . While the former is symplectic, the latter may be viewed as the action groupoid of the coadjoint action. These structures are compatible in the sense that $T^*G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\;\mathfrak {g}^*$ is a symplectic groupoid integrating the canonical Poisson structure on $\mathfrak {g}^*$ . The source $\textsf {s}:T^*G\longrightarrow \mathfrak {g}^*$ and target $\textsf {t}:T^*G\longrightarrow \mathfrak {g}^*$ are given by $\textsf {s}(g,\xi )=\mathrm {Ad}_g^*(\xi )$ and $\textsf {t}(g,\xi )=\xi $ , respectively, where $\mathrm {Ad}^*:G\longrightarrow \mathrm {GL}(\mathfrak {g}^*)$ is the coadjoint representation of $G$ . One calls $T^*G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\;\mathfrak {g}^*$ the cotangent groupoid of $G$ . Symplectic Hamiltonian $G$ -spaces are equivalent to symplectic manifolds equipped with Hamiltonian actions of $T^*G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\;\mathfrak {g}^*$ [Reference Mikami and Weinstein39, Example 3.9].

2.4 Integrating quotients of symplectic manifolds

Let $(M, \omega )$ be a symplectic manifold. Suppose that a Lie group $G$ acts on $M$ in a Hamiltonian fashion with moment map $\mu : M \longrightarrow \mathfrak {g}^*$ . Equip the total space of the pair groupoid $\mathrm {Pair}(M)=M \times M\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; M$ with the symplectic structure on $M\times \overline {M}$ , where $\overline {M}$ results from negating the symplectic form on $M$ ; this yields a symplectic groupoid integrating $M$ . The diagonal action of $G$ on $M \times M$ is then Hamiltonian with moment map

$$\begin{align*}M \times M \longrightarrow \mathfrak{g}^*, \quad (p, q) \mapsto \mu(p) - \mu(q). \end{align*}$$

If the $G$ -action on $M$ is free and proper, then

$$ \begin{align*}\mathrm{Pair}(M)\mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{0}}G=(M \times_{\mathfrak{g}^*} M) / G\end{align*} $$

is a Lie groupoid over $M/G$ . We also know $\mathrm {Pair}(M)\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{0}}G$ and $M/G$ to be symplectic and Poisson, respectively. These considerations give context for the following result.

2.5 Symplectic reduction along a submanifold

Let $\mathcal G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; X$ be a symplectic groupoid. Write $\sigma \in H^0(X,\wedge ^2(TX))$ for the Poisson bivector field on $X$ , and $\sigma ^{\vee }:T^*X\longrightarrow TX$ for its contraction with cotangent vectors. One calls a submanifold $S\subseteq X$ pre-Poisson if

$$ \begin{align*}L_S:=(\sigma^{\vee})^{-1}(TS)\cap\mathrm{ann}_{T^*X}(TS)\longrightarrow S\end{align*} $$

has constant fiber dimension, in which case $L_S$ is Lie subalgebroid of $(T^*X,\sigma ^{\vee })$ [Reference Cattaneo, Zambon, Dito, Lu, Maeda and Weinstein18, Reference Cattaneo and Zambon19]. Suppose that $\mathcal {H}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; S$ is a Lie subgroupoid of $\mathcal G$ over a pre-Poisson submanifold $S\subseteq X$ . We call $\mathcal {H}$ a stabilizer subgroupoid if $\mathcal {H}$ is isotropic in $\mathcal G$ and has $L_S$ as its Lie algebroid [Reference Crooks and Mayrand21]. Noteworthy special cases of these definitions arise for $\mathcal G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; X$ the (left-trivialized) cotangent groupoid $T^*G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\;\mathfrak {g}^*$ of a Lie group $G$ . The submanifold $\{\xi \}\subseteq \mathfrak {g}^*$ is pre-Poisson for all $\xi \in \mathfrak {g}^*$ and admits $G_{\xi }\longrightarrow \{\xi \}$ as a stabilizer subgroupoid. This observation foreshadows a generalization of Marsden–Weinstein reduction [Reference Marsden and Weinstein37]; the following are some details.

Suppose that a symplectic groupoid $\mathcal G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; X$ acts on a symplectic manifold $(M,\omega )$ in a Hamiltonian fashion with moment map $\mu :M\longrightarrow X$ . Let $\mathcal {H}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; S$ be a stabilizer subgroupoid of $\mathcal G$ over a pre-Poisson submanifold $S\subseteq X$ . Note that $\mathcal {H}$ acts on $\mu ^{-1}(S)$ . Consider the inclusion $j:\mu ^{-1}(S)\longrightarrow M$ and quotient $\pi :\mu ^{-1}(S)\longrightarrow \mu ^{-1}(S)/\mathcal {H}$ . If $\mathcal {H}$ acts freely and properly on $\mu ^{-1}(S)$ , then $\mu $ is transverse to $S$ , and the manifold $\mu ^{-1}(S)/\mathcal {H}$ carries a unique symplectic form $\overline {\omega }$ satisfying $\pi ^*\overline {\omega }=j^*\omega $ . The symplectic manifold

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

is called the symplectic reduction of M by $\mathcal G$ along S with respect to $\mathcal {H}$ [Reference Crooks and Mayrand21]. We use the simpler notation $M \mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S}} \mathcal G$ if $\mathcal {H}$ is source-connected; all source-connected integrations of $L_S$ yield the same symplectic manifold.

The preceding construction is more general than described above; it also holds for smooth manifolds, complex analytic spaces, complex algebraic varieties, and affine schemes. Another feature of this construction is that it generalizes many of the approaches to symplectic reduction that have developed over the last 50 years. We refer the interested reader to [Reference Crooks and Mayrand21, Reference Crooks and Mayrand23] for precise details. A more leisurely introduction can be found in [Reference Crooks, Gao, Pound and Thompson20].

2.6 Some Lagrangian relations

Let $X$ be a Poisson manifold. Recall that a submanifold $S\subseteq X$ is called a Poisson transversal if it intersects each symplectic leaf of $X$ transversely and in a symplectic submanifold of that leaf. The Poisson structure on $X$ then induces a Poisson structure on $S$ [Reference Frejlich and Mărcuţ27, Lemma 3]. One also knows that the inverse image of a Poisson transversal under a Poisson map is a Poisson transversal [Reference Frejlich and Mărcuţ27, Lemma 7]. It is also clear that the Poisson transversals in a symplectic manifold are the symplectic submanifolds of that manifold. These considerations feature in the following result.

Proof Write $Y$ for the submanifold in (i) and $\pi :\mu ^{-1}(S)\longrightarrow M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S,\mathcal {H}}}\mathcal G$ for the quotient map. Given $(m,[m])\in Y$ , one finds that

$$ \begin{align*}T_{(m,[m])}Y=\{(v,\mathrm{d}\pi_m(v)):v\in(\mathrm{d}\mu_m)^{-1}(T_{\mu(m)}S)\}\subseteq T_m M\oplus T_{[m]}(M\mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S,\mathcal{H}}}\mathcal G).\end{align*} $$

Let us therefore fix two vectors $(v_1,\mathrm {d}\pi _m(v_1))$ and $(v_2,\mathrm {d}\pi _m(v_2))$ in $T_{(m,[m])}Y$ . Write $\overline {\omega }$ for the symplectic forms on $M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S,\mathcal {H}}}\mathcal G$ and $j:\mu ^{-1}(S)\longrightarrow M$ for the inclusion. Since $\pi ^*\overline {\omega }=j^*\omega $ , we have

$$ \begin{align*}\overline{\omega}_{[m]}(\mathrm{d}\pi_m(v_1),\mathrm{d}\pi_m(v_2))=\omega_m(v_1,v_2).\end{align*} $$

The calculation

$$ \begin{align*}\omega_m(v_1,v_2)-\overline{\omega}_{[m]}(\mathrm{d}\pi_m(v_1),\mathrm{d}\pi_m(v_2))=0\end{align*} $$

then shows $Y$ to be isotropic in $M\times \overline {M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S,\mathcal {H}}}\mathcal G}$ , verifying (i).

In light of (i), $Y$ is Lagrangian in $M\times \overline {M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S,\mathcal {H}}}\mathcal G}$ if and only if $\dim Y=\frac {1}{2}(\dim (M\times \overline {M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S,\mathcal {H}}}\mathcal G})$ . We also have the identities

$$ \begin{align*} \dim Y = &\dim\mu^{-1}(S)=\dim M-\dim X+\dim S\quad\text{and}\quad \\ &\dim(M\mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{S,\mathcal{H}}}\mathcal G)=\dim\mu^{-1}(S)-\mathrm{rank}\hspace{2pt}L_S.\end{align*} $$

It is now straightforward to verify that $\dim Y=\frac {1}{2}(\dim (M\times \overline {M\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{S,\mathcal {H}}}\mathcal G})$ if and only if $\mathrm {rank}\hspace {2pt}L_S=\dim X-\dim S$ . The latter equation holds if and only if $L_S=\mathrm {ann}_{T^*X}(TS)$ , as $\mathrm {rank}\hspace {2pt}(\mathrm {ann}_{T^*X}(TS))=\dim X-\dim S$ . Letting $\sigma ^{\vee }:T^*X\longrightarrow TX$ denote the contraction of the Poisson structure $\sigma $ on $X$ , the condition $L_S=\mathrm {ann}_{T^*X}(TS)$ becomes $\mathrm {ann}_{T^*X}(TS)\subseteq (\sigma ^{\vee })^{-1}(TS)$ . This new condition is precisely the definition of $S$ being coisotropic in $X$ , completing the proof of (ii).

Part (iii) follows immediately from (ii).

There is a another class of Lagrangian submanifolds arising in the context of symplectic reduction along a submanifold. To this end, let a symplectic groupoid $\mathcal G \;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; X$ act in a Hamiltonian fashion on a symplectic manifold $M$ with moment map $\mu : M \longrightarrow X$ . The definition of a Hamiltonian action implies that

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

is a Lagrangian subgroupoid of $\mathrm {Pair}(M) \times \overline {\mathcal G}$ . More generally, consider a symplectic groupoid $\mathcal G \;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; X$ , a Lie group $G$ , and a Hamiltonian $\mathcal G\times T^*G$ -space $M$ with moment map $(\mu ,\nu ):M\longrightarrow X\times \mathfrak {g}^*$ . The equivalence between Hamiltonian $G$ -spaces and Hamiltonian $T^*G$ -spaces renders $M$ a Hamiltonian $G$ -space. Suppose that $G$ acts freely and properly on $M$ . Proposition 2.1 yields a symplectic groupoid

$$ \begin{align*}(\tilde{\textsf{s}}, \tilde{\textsf{t}}) :\mathrm{Pair}(M)\mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{0}}G=(M\times_{\nu}\overline{M})\;\substack{\longrightarrow\\[-9pt] \longrightarrow}\; M/G.\end{align*} $$

It is also clear that $\mu $ descends to a Poisson map $\overline {\mu }:M/G\longrightarrow X$ . We conclude that

$$\begin{align*}\mathcal{L}_{\overline{\mu}} := \{([g \cdot p, p], g) \in (M \times_\nu \overline{M})/G \times \overline{\mathcal G} : (g, p) \in \mathcal G \ltimes M\} \end{align*}$$

is a Lagrangian relation from $\mathrm {Pair}(M)\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{0}}G$ to $\mathcal G$ . The following result will be useful in Section 7.

Proof As $\overline {\mu }$ is a Poisson map, $\tilde {S}$ is a Poisson transversal in $M/G$ . This implies the first claim. For the second claim, it remains to check that $\mathcal {L}_{\overline {\mu }} \cap (\tilde {\textsf {t}}^{-1}(\tilde {S}) \times \textsf {t}^{-1}(S))$ is smooth and has the correct dimension.

Let us set

$$\begin{align*}\mathcal G \ltimes_S M/G := \{(g, [p]) \in \mathcal G \times M/G : \textsf{t}(g) = \mu(p) \in S\} = \textsf{t}^{-1}(S) \times_S \overline{\mu}^{-1}(S). \end{align*}$$

Note that we have an embedding

$$\begin{align*}\mathcal G \ltimes_S M/G \longrightarrow (M \times_\nu \overline{M})/G \times \overline{\mathcal G}, \quad (g, [p]) \longmapsto ([g \cdot p, p], g), \end{align*}$$

whose image is $\mathcal {L}_{\overline {\mu }} \cap (\tilde {\textsf {t}}^{-1}(\tilde {S}) \times \textsf {t}^{-1}(S))$ . At the same time,

$$ \begin{align*} \dim(\mathcal G \ltimes_S M/G) &= \dim \textsf{t}^{-1}(S) + \dim \overline{\mu}^{-1}(S) - \dim S \\ &= \big(\!\dim \mathcal G + \dim S - \dim X\big) + \big(\!\dim(M/G) + \dim S - \dim X\big) - \dim S \\ &= \dim S + \dim(M/G). \end{align*} $$

On the other hand,

$$ \begin{align*} \dim(\tilde{\textsf{t}}^{-1}(\tilde{S}) \times \textsf{t}^{-1}(S)) &= \dim(\tilde{S}) + \dim(M/G) + \dim(S) + \dim(X) \\ &= \big(\dim(S) + \dim(M/G) - \dim(X)\big) \\ &\quad + \dim(M/G) + \dim(S) + \dim(X) \\ &= 2 \dim(S) + 2 \dim(M/G) \\ &= 2 \dim(\mathcal G \ltimes_S M/G).\\[-34pt] \end{align*} $$

2.7 Quasi-Hamiltonian and quasi-Poisson manifolds

Following Boalch [Reference Boalch10] and a number of his subsequent manuscripts [Reference Boalch and Yamakawa12], we now outline the algebraic and holomorphic approaches to quasi-Hamiltonian and quasi-Poisson geometry in this manuscript. Let $G$ be a Lie group with Lie algebra $\mathfrak {g}$ , exponential map $\mathrm {exp}:\mathfrak {g}\longrightarrow G$ , and fixed $G$ -invariant, non-degenerate, symmetric bilinear form $(\cdot ,\cdot ):\mathfrak {g}\otimes \mathfrak {g}\longrightarrow \mathbb {C}$ . The Cartan 3-form on $G$ is the unique $G$ -invariant element $\eta _G\in \Omega ^3(G)$ satisfying $(\eta _G)_e(x,y,z)=\frac {1}{12}(x,[y,z])$ for all $x,y,z\in \mathfrak {g}$ . Note that this expression defines an element of $\wedge ^3(\mathfrak {g}^*)$ . Using $(\cdot ,\cdot )$ to identify $\mathfrak {g}$ and $\mathfrak {g}^*$ , this is an element $\chi _G\in \wedge ^3\mathfrak {g}$ . One also has $\theta ^L,\theta ^R\in \Omega ^1(G,\mathfrak {g})$ , the left and right-invariant Maurer–Cartan forms on $G$ , respectively.

Suppose that $G$ acts holomorphically on a manifold $M$ . Each $\xi \in \mathfrak {g}$ thereby determines a generating vector field $\xi _M\in H^0(M,TM)$ , that is,

$$ \begin{align*}(\xi_M)(p):=\frac{d}{dt}\bigg\vert_{t=0}\mathrm{exp}(-t\xi)\cdot p\end{align*} $$

for all $p\in M$ . The Lie algebra morphism

$$ \begin{align*}\mathfrak{g}\longrightarrow H^0(M,TM),\quad \xi\mapsto\xi_M\end{align*} $$

extends uniquely to a graded algebra morphism $\wedge ^{\bullet }\mathfrak {g}\longrightarrow H^0(M,\wedge ^{\bullet }(TM))$ . Write $(\chi _G)_M\in H^0(M,\wedge ^3(TM))$ for the image of $\chi _G\in \wedge ^3\mathfrak {g}$ under the latter morphism. One calls $M$ a quasi-Poisson G-manifold [Reference Alekseev, Kosmann-Schwarzbach and Meinrenken1] if it comes equipped with a $G$ -invariant bivector field $\sigma \in H^0(M,\wedge ^2(TM))$ satisfying $[\sigma ,\sigma ]=(\chi _G)_M$ , where $[\sigma ,\sigma ]\in H^0(M,\wedge ^3TM)$ denotes the Schouten bracket of $\sigma $ with itself. Let us write $\overline {M}$ for the quasi-Poisson $G$ -manifold obtained by negating the quasi-Poisson structure on a quasi-Poisson $G$ -manifold $M$ .

Let $M$ be a quasi-Poisson $G$ -manifold. A holomorphic map $\mu :M\longrightarrow G$ is called a moment map if it is equivariant with respect to the conjugation action of $G$ on itself and satisfies a quasi-Poisson counterpart of Hamilton’s equations (see [Reference Alekseev, Kosmann-Schwarzbach and Meinrenken1, Definition 2.2]. One uses the term quasi-Poisson Hamiltonian G-space for a quasi-Poisson $G$ -manifold with a prescribed moment map. Let $(M_1,\sigma _1,\mu _1)$ and $(M_2,\sigma _2,\mu _2)$ be quasi-Poisson Hamiltonian $G$ -spaces. By a map of quasi-Poisson Hamiltonian G-spaces, we mean a holomorphic, $G$ -equivariant map $f:M_1\longrightarrow M_2$ satisfying $\mu _1=\mu _2\circ f$ and $f_*((\sigma _1)_p)=(\sigma _2)_{f(p)}$ for all $p\in M$ . A quasi-Poisson Hamiltonian $G$ -space $(M,\sigma ,\mu )$ is called non-degenerate if $T_pM=\mathrm {im}(\sigma _p^{\vee })+T_p(G\cdot p)$ for all $p\in M$ .

To describe a specialization of the previous paragraph, we again let $G$ act holomorphically on a manifold $M$ . Consider a $G$ -invariant $2$ -form $\omega \in \Omega ^2(M)^G$ and $G$ -equivariant holomorphic map $\mu :M\longrightarrow G$ , where $G$ acts on itself by conjugation. One calls $(M,\omega ,\mu )$ a quasi-Hamiltonian G-space [Reference Alekseev, Malkin and Meinrenken2] if the following conditions are satisfied:

1. (i) $\mathrm {d}\omega =-\mu ^*\eta _G$ ;

2. (ii) $\mathrm {ker}(\omega _p)=\{\xi _M(p):\xi \in \mathrm {ker}(\mathrm {Ad}_{\mu (p)}+\mathrm {id}_{\mathfrak {g}})\}$ for all $p\in M$ ;

3. (iii) $\iota _{\xi _M}\omega =\frac {1}{2}\mu ^*(\theta ^L+\theta ^R,\xi )$ for all $\xi \in \mathfrak {g}$ .

If $(M,\sigma ,\mu )$ is a non-degenerate quasi-Poisson Hamiltonian $G$ -space, then there exists a unique $2$ -form $\omega \in \Omega ^2(M)$ for which $\sigma $ and $\omega $ satisfy [Reference Alekseev, Kosmann-Schwarzbach and Meinrenken1, Equation (32)] and $(M,\omega ,\mu )$ is a quasi-Hamiltonian $G$ -space. Quasi-Hamiltonian $G$ -spaces are thereby in bijective correspondence with non-degenerate quasi-Poisson Hamiltonian $G$ -spaces. At the same time, every quasi-Poisson Hamiltonian $G$ -space partitions into quasi-Hamiltonian leaves [Reference Alekseev, Kosmann-Schwarzbach and Meinrenken1].

2.8 Quasi-symplectic groupoids over quotients of quasi-Hamiltonian manifolds

Suppose that $G$ and $H$ are Lie groups with fixed, non-degenerate, invariant, symmetric bilinear forms on their respective Lie algebras. Let $(M,\omega )$ be a quasi-Hamiltonian $G\times H$ -space with moment map $(\mu ,\nu ):M\longrightarrow G\times H$ . If $G$ acts freely and properly on $M$ , then $M/G$ carries a unique quasi-Poisson $H$ -space structure for which the quotient map $\pi :M\longrightarrow M/G$ is a morphism of quasi-Poisson $H$ -spaces [Reference Bălibanu and Mayrand6, Corollary 4.7]. The moment map on $M/G$ is obtained by descending $\nu $ to a holomorphic map $\overline {\nu }:M/G\longrightarrow H$ .

On the other hand, $M\times \overline {M}$ is a quasi-Hamiltonian $G\times G\times H\times H$ -space. Two applications of fusion [Reference Alekseev, Malkin and Meinrenken2, Theorem 6.1] render $M\times \overline {M}$ a quasi-Hamiltonian $G\times H$ -space. The group $G\times H$ thereby acts diagonally on $M\times M$ , and with moment map

$$ \begin{align*}M\times M\;\substack{\longrightarrow\\[-9pt] \longrightarrow}\; G\times H,\quad (p,q)\mapsto (\mu(p)\mu(q)^{-1},\nu(p)\nu(q)^{-1}).\end{align*} $$

It follows that the quasi-Hamiltonian reduction $(M\times \overline {M})\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{e}}G=(M\times _GM)/G$ is a quasi-Hamiltonian $H$ -space; write $\sigma $ for the underlying $2$ -form on $(M\times _G M)/G$ . One also finds that the Lie groupoid structure on $\mathrm {Pair}(M)$ induces such a structure on $(M\times _G M)/G \;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; M/G$ ; write $\mathrm {Pair}(M)\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{e}}G$ for this new Lie groupoid.

Proof We first check the following compatibility conditions:

1. (1) $\sigma $ is multiplicative;

2. (2) $\mathrm {d}\sigma = \textsf {s}^*\overline {\nu }^*\eta _H - \textsf {t}^*\overline {\nu }^*\eta _H$ ;

3. (3) $\mathrm {d}\overline {\nu }^*\eta _H = 0$ .

Condition (3) is clear.

Let $i : M \times _G M \longrightarrow M \times M$ be the inclusion map and $\pi : M \times _G M \longrightarrow (M \times _G M) / G$ the quotient map. It follows from the definition of the quasi-Hamiltonian 2-form on the fusion product $M \times \overline {M}$ that its pullback by $i^*$ coincides with $i^*(\mathrm {pr}_1^*\omega - \mathrm {pr}_2^*\omega )$ , where $\mathrm {pr}_i : M \times M \to M$ are the projections. This implies that

(2.1) $$ \begin{align} \pi^*\sigma = i^*(\mathrm{pr}_1^*\omega - \mathrm{pr}_2^*\omega). \end{align} $$

Since $\mathrm {pr}_1^*\omega - \mathrm {pr}_2^*\omega $ is multiplicative on $\mathrm {Pair}(M)$ , it follows that $\sigma $ is also multiplicative, verifying (1). To check that (2) holds, note that (2.1) gives

$$ \begin{align*} \pi^*\mathrm{d}\sigma &= i^*(\mathrm{pr}_1^*\mathrm{d}\omega - \mathrm{pr}_2^*\mathrm{d}\omega) \\ &= i^*(\mathrm{pr}_1^*(-\mu^*\eta_G - \nu^*\eta_H) + \mathrm{pr}_2^*(\mu^*\eta_G + \nu^*\eta_H)) \\ &= \pi^*(\textsf{s}^*\overline{\nu}^*\eta_H - \textsf{t}^*\overline{\nu}^*\eta_H), \end{align*} $$

where the last equality follows from the fact that $\mu \circ \mathrm {pr}_1 \circ i = \mu \circ \mathrm {pr}_2 \circ i$ .

We have $\dim (M \times _G M)/G = 2 \dim M/G$ , so it only remains to show that $\ker \sigma _{(p, p)} \cap \ker \mathrm {d}\textsf {s} \cap \ker \mathrm {d}\textsf {t} = 0$ for all $p \in M$ . A general vector in $\ker \mathrm {d}\textsf {s} \cap \ker \mathrm {d}\textsf {t}$ takes the form $(0,(\xi _M)_p)$ for $\xi \in \mathfrak {g}$ , where $\mathrm {d}\mu _p((\xi _M)_p) = 0$ , that is, $\mathrm {Ad}_{\mu (p)^{-1}}\xi - \xi = 0$ . The condition that $(0,(\xi _M)_p) \in \ker \sigma $ then amounts to the condition that $(\xi _M)_p \in \ker \omega _p = \{v_\zeta : \zeta \in \ker (\mathrm {Ad}_{\mu (p)} + 1)\}$ . It follows that also $\mathrm {Ad}_{\mu (p)}(\xi ) + \xi = 0$ , and hence $\xi = 0$ .

3.1 The Moore–Tachikawa conjecture

Let $\mathbf {C}$ be a symmetric monoidal category. A two-dimensional TQFT in $\mathbf {C}$ is a symmetric monoidal functor $\mathbf {Cob}_2\longrightarrow \mathbf {C}$ , where $\mathbf {Cob}_2$ is the category of two-dimensional cobordisms. While one often takes $\mathbf {C}$ to be vector spaces over a fixed field, other cases warrant consideration. This is a context in which to formulate the Moore–Tachikawa conjecture [Reference Moore, Tachikawa, Block, Distler, Donagi and Sharpe40]. Moore and Tachikawa take $\mathbf {C}$ to be $\mathbf {MT}$ , a category of affine symplectic varieties with Hamiltonian actions. Complex semisimple affine algebraic groups constitute the objects of $\mathbf {MT}$ . Morphisms in $\mathrm {Hom}_{\mathbf {MT}}(G,H)$ are certain isomorphism classes of affine Hamiltonian $G\times H$ -varieties. One composes morphisms in $\mathbf {MT}$ via Hamiltonian reduction at level zero: given $[X]\in \mathrm {Hom}_{\mathbf {MT}}(G,H)$ and $[Y]\in \mathrm {Hom}_{\mathbf {MT}}(H,K)$ , we have $[Y]\circ [X]:= [(X\times Y)\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{0}}H]\in \mathrm {Hom}_{\mathbf {MT}}(G,K)$ . Products of objects and morphisms constitute the tensor product of a symmetric monoidal structure on $\mathbf {MT}$ .

Now let $G$ be a connected semisimple affine algebraic group with Lie algebra $\mathfrak {g}$ . Consider the Kostant slice [Reference Kostant33] $\mathrm {Kos}\subseteq \mathfrak {g}$ associated with a principal $\mathfrak {sl}_2$ -triple in $\mathfrak {g}$ [Reference Kostant32], that is, an $\mathfrak {sl}_2$ -triple $(e,h,f)\in \mathfrak {g}^{\times 3}$ with $e,h,f\in \mathfrak {g}_{\mathrm {reg}}$ . It turns out that the affine variety $G\times \mathrm {Kos}$ is symplectic in a natural way, as follows from generalities on Kostant–Whittaker reduction [Reference Kostant34] (see [Reference Bielawski8, Reference Crooks and Röser24] as well). Left multiplication in the first factor defines a Hamiltonian action of $G$ on $G\times \mathrm {Kos}$ . As such, the isomorphism class of $G\times \mathrm {Kos}$ is a morphism from $G$ to the trivial group in $\mathbf {MT}$ . This is a context for the Moore–Tachikawa conjecture; we state it below.

The Moore–Tachikawa conjecture is known to hold in Lie type $A$ , as follows from combining the unpublished work of Ginzburg–Kazhdan [Reference Ginzburg and Kazhdan30] with the results of Braverman–Finkelberg–Nakajima [Reference Braverman, Finkelberg and Nakajima15]. In [Reference Crooks and Mayrand22], we reformulate the Moore–Tachikawa conjecture in the language of shifted symplectic geometry. We also generalize this reformulation to obtain new TQFTs in a $1$ -shifted Weinstein symplectic category. The following are some pertinent details.

3.2 A $1$ -shifted Weinstein symplectic “category”

Recall that two morphisms in a category are composable if and only if the source of one is the target of the other. By relaxing this to a necessary condition for composing morphisms, we arrive at the definition of a “category.” One particularly notable example is Weinstein’s symplectic “category” [Reference Weinstein, Doebner, Andersson and Petry47]. We briefly recall our $1$ -shifted counterpart of this “category” (see [Reference Crooks and Mayrand22, Section 3] for a more comprehensive discussion).

Let $(\mathcal G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; X,\omega ,\phi )$ be a quasi-symplectic groupoid. A 1-shifted Lagrangian on $\mathcal G$ is the data of a Lie groupoid $\mathcal {L}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; Y$ , $2$ -form $\gamma $ on $Y$ , and Lie groupoid morphism $\nu :\mathcal {L}\longrightarrow \mathcal G$ , such that certain compatibility conditions are satisfied (see [Reference Mayrand38, Section 3.1] or [Reference Pantev, Toën, Vaquié and Vezzosi41]). A 1-shifted Lagrangian relation from a quasi-symplectic groupoid $\mathcal G_1$ to a quasi-symplectic groupoid $\mathcal G_2$ is a $1$ -shifted Lagrangian $\mathcal {L}\longrightarrow \mathcal G_1\times \overline {\mathcal G_2}$ on $\mathcal G_1\times \overline {\mathcal G_2}$ . We sometimes adopt the notation

for such a correspondence. Two such correspondences

are declared to be composable if a transversality condition is satisfied (see [Reference Crooks and Mayrand22, Section 2.3] for further details). In this case, the homotopy fiber product [Reference Mayrand38, Section 4.2]

is a $1$ -shifted Lagrangian relation from $\mathcal G_1$ to $\mathcal G_3$ .

In light of the above, we define the 1-shifted Weinstein symplectic “category” $\mathbf {WS}\text{"}_{\mkern -7mu 1}$ as follows. Quasi-symplectic groupoids constitute the objects of $\mathbf {WS}\text{"}_{\mkern -7mu 1}$ . Morphisms from $\mathcal G_1$ and $\mathcal G_2$ are weak equivalence classes [Reference Crooks and Mayrand22, Section 2.4] of $1$ -shifted Lagrangian relations from $\mathcal G_1$ to $\mathcal G_2$ . Given composable $1$ -shifted Lagrangian relations

we define

$$ \begin{align*}[\mathcal{L}_2]\circ[\mathcal{L}_1]:= [\mathcal{L}_1 \times^{\mathrm{h}}_{\mathcal G_2} \mathcal{L}_2]\in\mathrm{Hom}_{\mathbf{WS}"_{\mkern-11mu 1}}(\mathcal G_1,\mathcal G_3).\end{align*} $$

Using a variant of the Wehrheim–Woodward approach [Reference Wehrheim and Woodward45], we complete $\mathbf {WS}"_{\mkern -11mu 1}$ to a genuine category $\mathbf {WS}_1$ . We subsequently show $\mathbf {WS}_1$ to be a symmetric monoidal category (see [Reference Crooks and Mayrand22, Section 3.3]).

3.3 TQFTs valued in $\mathbf {WS}_1$

Recall that if $M$ is a Hamiltonian $\mathcal G_1 \times \overline {\mathcal G_2}$ -space for quasi-symplectic groupoids $\mathcal G_1$ and $\mathcal G_2$ , then the action groupoid $(\mathcal G_1 \times \overline {\mathcal G}_2) \ltimes M$ is a 1-shifted Lagrangian relation from $\mathcal G_1$ to $\mathcal G_2$ via the two projections (see [Reference Calaque, Pantev, Simpson, Toën, Vaquié and Vezzosi17, Example 1.31] or [Reference Mayrand38, Proposition 9.3]). In this way, Hamiltonian $\mathcal G_1 \times \overline {\mathcal G_2}$ -spaces are equivalent to $1$ -shifted Lagrangian structures on $(\mathcal G_1 \times \overline {\mathcal G_2}) \ltimes M$ from $\mathcal G_1$ to $\mathcal G_2$ . Note that this perspective allows one to interpret Hamiltonian $\mathcal G_1 \times \overline {\mathcal G_2}$ -spaces as morphisms in $\mathbf {WS}\text{"}_{\mkern -7mu 1}$ .

Let $\mathcal G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; X$ be a Lie groupoid. A submanifold $S\subseteq X$ is called a global slice if its intersection with each $\mathcal G$ -orbit in $X$ is transverse and a singleton. This slice is called admissible if the isotropy group $\mathcal G_x:= \textsf {s}^{-1}(x)\cap \textsf {t}^{-1}(x)$ is abelian for all $x\in S$ . For $\mathcal G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; X$ , a symplectic groupoid with a global slice $S\subseteq X$ , it follows that $\textsf {t}^{-1}(S) \subseteq \mathcal G$ is a Hamiltonian $\mathcal G$ -space with respect to the action by right multiplication. This leads to the following result from [Reference Crooks and Mayrand22].

As explained in [Reference Crooks and Mayrand22, Section 4.4], the TQFT structure comes from the abelian symplectic groupoid $\mathcal {A} := \mathcal G\big \vert _S$ ; it is Morita equivalent to $\mathcal G$ via the inclusion map $\mathcal {A} \longrightarrow \mathcal G$ . Let us set

$$\begin{align*}\mathcal{A}^{m, n} := \{(a, b) \in \mathcal{A}^{*m} * \mathcal{A}^{*n} : a_1 \ldots a_m = b_1 \ldots b_n\}, \end{align*}$$

for $m,n\in \mathbb {Z}_{\geq 0}$ with $(m,n)\neq (0,0)$ , where $*$ denotes the fiber product over $S$ . We have that $\eta _{\mathcal G,S}(C_{m, n}) = [\mathcal {A}^{m, n}]$ for all $m,n\in \mathbb {Z}_{\geq 0}$ with $(m,n)\neq (0,0)$ , where $C_{m,n}$ is the standard two-dimensional cobordism from $m$ circles to $n$ circles. One views $\mathcal {A}^{m, n}$ as a Lagrangian relation from $\mathcal G^m$ to $\mathcal G^n$ via the maps $\mathcal {A}^{m, n} \longrightarrow \mathcal {A}^m \hookrightarrow \mathcal G^m$ and $\mathcal {A}^{m, n} \longrightarrow \mathcal {A}^n \hookrightarrow \mathcal G^n$ (see [Reference Crooks and Mayrand22, Remark 4.6 and Section 4.4]).

The morphism $\eta _{\mathcal G,S}(C_{m, n})$ from $\mathcal G^m$ to $\mathcal G^n$ can be described as a Hamiltonian $\mathcal G^m \times \overline {\mathcal G}^n$ -space in the following way. The action groupoid $(\mathcal G \times \mathcal G) \ltimes \mathcal G$ for the Hamiltonian $\mathcal G \times \overline {\mathcal G}$ -space $\mathcal G$ acts as an identity in $\mathbf {WS}_1$ . By composing on both sides with identities

we see that $\eta _{\mathcal G}(C_{m, n})$ can be described as

(3.1)

where

$$\begin{align*}Z := \{(g, h) \in \mathcal G^m \times \mathcal G^n : \textsf{t}(g_1) = \cdots = \textsf{t}(g_m) = \textsf{s}(h_1) = \cdots = \textsf{s}(h_n) \in S\}. \end{align*}$$

Since $\mathcal {A}^{m, n}$ is closed in $\mathcal G^m \times \mathcal G^n$ , it acts freely and properly on $Z$ . It follows that (3.1) is equivalent to

(see [Reference Mayrand38, Theorem 7.2]). On the other hand, a Hamiltonian space is equivalent to a 1-shifted Lagrangian structure on an action groupoid [Reference Mayrand38, Proposition 9.3]. It follows that

$$\begin{align*}\mathcal G^{m, n}_S := Z/\mathcal{A}^{m, n} \end{align*}$$

is a Hamiltonian $\mathcal G^m \times \overline {\mathcal G}^n$ -space with respect to the $2$ -form $\eta $ determined by

(3.2) $$ \begin{align} \pi^*\eta = i^*(\omega, \ldots, \omega), \end{align} $$

where $\pi : Z \to Z / \mathcal {A}^{m, n}$ is the quotient map, $i : Z \hookrightarrow \mathcal G^m \times \mathcal G^n$ is the inclusion map, and $(\omega , \ldots , \omega )$ is the natural 2-form on $\mathcal G^m \times \mathcal G^n$ . We summarize this discussion in the following proposition.

In the case, where $\mathcal G$ is a symplectic groupoid, the Hamiltonian space $Z/\mathcal {A}^{m, n}$ can be seen as a reduction along a submanifold in the sense of Section 2.5. To this end, note that $\mathcal G^{m + n}$ is a Hamiltonian $\overline {\mathcal G}^m \times \mathcal G^n \times \overline {\mathcal G}^n \times \mathcal G^m$ space. One finds that

$$ \begin{align*}S^{m, n} = \{(x, y) \in X^n \times X^m : x_1 = \cdots = x_n = y_1 = \cdots = y_m \in S\}\end{align*} $$

is pre-Poisson with stabilizer subgroupoid $\mathcal {A}^{m, n}$ . It follows from (3.2) that $\mathcal G^{m, n}_S$ is the reduction of $\mathcal G^{m + n}$ along $S^{m,n}$ with respect to $\mathcal {A}^{m, n} \longrightarrow S^{m, n}$ .

3.4 Connections to the Moore–Tachikawa conjecture and multiplicative counterparts

Let $G$ be a connected semisimple affine algebraic group with Lie algebra $\mathfrak {g}$ . Use the Killing form to freely identify $\mathfrak {g}^*$ and $\mathfrak {g}$ as Poisson varieties. At the same time, let $(e,h,f)\in \mathfrak {g}^{\times 3}$ be a principal $\mathfrak {sl}_2$ -triple. The affine subvariety $\mathrm {Kos}:= e+\mathfrak {g}_f\subseteq \mathfrak {g}$ is an admissible global slice to the restricted cotangent groupoid $(T^*G)_{\text {reg}}:= (T^*G)\big \vert _{\mathfrak {g}_{\text {reg}}}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\;\mathfrak {g}_{\mathrm {reg}}$ . Theorem 3.2 then yields an explicit TQFT $\eta _{\mathcal G,S}:\mathbf {Cob}_2\longrightarrow \mathbf {WS}_1$ , where $\mathcal G=(T^*G)_{\text {reg}}$ and $S=\mathrm {Kos}$ . Note that Proposition 3.3 gives explicit Hamiltonian spaces in the image of $\eta _{\mathcal G,S}$ ; these are called the open Moore–Tachikawa varieties. In [Reference Crooks and Mayrand22], we “affinize” $\eta _{\mathcal G,S}$ to construct the Moore–Tachikawa TQFT in a category of affine Hamiltonian schemes.

There is a multiplicative counterpart to the above. A first step is to replace $T^*G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\;\mathfrak {g}$ with the double $\mathrm {D}(G)=G\times G\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; G$ , a quasi-symplectic groupoid integrating the Cartan–Dirac structure on $G$ [Reference Alekseev, Malkin and Meinrenken2]. One then replaces $\mathrm {Kos}\subseteq \mathfrak {g}$ with a Steinberg slice $\mathrm {Ste}\subseteq G$ [Reference Steinberg43]. If $G$ is simply-connected, it turns out that $\mathrm {Ste}$ is an admissible global slice to the quasi-symplectic groupoid $\mathrm {D}(G)_{\text {reg}}:= \mathrm {D}(G)\big \vert _{G_{\text {reg}}}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; G_{\text {reg}}$ . By [Reference Bălibanu and Mayrand6, Example 2.30], this groupoid $\mathcal G$ and slice $S$ satisfy the hypotheses of Theorem 3.2 and Proposition 3.3. The quasi-Hamiltonian spaces $\eta _{\mathcal G,S}(C_{m,n})$ are studied in [Reference Bălibanu and Mayrand6], as multiplicative analogs of the open Moore–Tachikawa varieties. We refer the reader to Sections 8 and 9 for precise descriptions of the objects in this paragraph.

4.1 Parabolic subgroups and subalgebras

Given a parabolic subgroup $P\subseteq G$ (resp. parabolic subalgebra $\mathfrak {p}\subseteq \mathfrak {g}$ ), write $U(P)\subseteq P$ (resp. $\mathfrak {u}(\mathfrak {p})\subseteq \mathfrak {p}$ ) for the unipotent radical of $P$ (resp. nilpotent radical of $\mathfrak {p}$ ). The quotients $L(P):= P/U(P)$ and $\mathfrak {l}(\mathfrak {p}):= \mathfrak {p}/\mathfrak {u}(\mathfrak {p})$ will be called the universal Levi factors of $P$ and $\mathfrak {p}$ , respectively. If $\mathfrak {p}$ is the Lie algebra of $P$ , then $\mathfrak {u}(\mathfrak {p})$ and $\mathfrak {l}(\mathfrak {p})$ are the Lie algebras of $U(P)$ and $L(P)$ , respectively.

4.2 Regular elements in reductive Lie algebras

Let $K$ be a connected reductive affine algebraic group with Lie algebra $\mathfrak {k}$ . Write $K_x\subseteq K$ and $\mathfrak {k}_x\subseteq \mathfrak {k}$ for the $K$ and $\mathfrak {k}$ -centralizers of $x\in \mathfrak {k}$ under the adjoint representation, respectively. Consider the regular locus

$$ \begin{align*}\mathfrak{k}_{\text{reg}}:=\{x\in\mathfrak{k}:\dim\mathfrak{k}_x=\mathrm{rank}\hspace{2pt}\mathfrak{k}\}.\end{align*} $$

If we identify $\mathfrak {k}$ with $\mathfrak {k}^*$ via a non-degenerate, $K$ -invariant, symmetric bilinear form on the former, then $\mathfrak {k}_{\text {reg}}$ becomes the regular locus of the Poisson variety $\mathfrak {k}^*$ . We are principally interested in the regular locus of $\mathfrak {k}=\mathfrak {l}(\mathfrak {p})$ for a parabolic subalgebra $\mathfrak {p}\subseteq \mathfrak {g}$ .

4.3 Regular elements in Levi factors

Let $P\subseteq G$ be a parabolic subgroup with Lie algebra $\mathfrak {p}\subseteq \mathfrak {g}$ . Observe that $U(P)$ acts trivially on the $P$ -module $\mathfrak {l}(\mathfrak {p})$ . One thereby obtains an action of the reductive group $L(P)$ on $\mathfrak {l}(\mathfrak {p})$ ; it is the adjoint representation of $L(P)$ . Write $P_{[x]}\subseteq P$ and $L(P)_{[x]}\subseteq L(P)$ for the stabilizers of $[x]\in \mathfrak {l}(\mathfrak {p})$ under the actions of $P$ and $L(P)$ , respectively. It is straightforward to verify that $L(P)_{[x]}=P_{[x]}/U(P)$ for all $[x]\in \mathfrak {l}(\mathfrak {p})$ .

Proof We begin by proving (i). To this end, observe that $x+\mathfrak {u}(\mathfrak {p})$ is stable under the adjoint action of $P_{[x]}$ on $\mathfrak {g}$ . It therefore suffices to prove that the $P_{[x]}$ -orbit of any point in $(x+\mathfrak {u}(\mathfrak {p}))\cap \mathfrak {g}_{\mathrm {reg}}$ is open and dense in $x+\mathfrak {u}(\mathfrak {p})$ . We also recognize that $P_{[x]}$ -orbits are open in their closures, and that $x+\mathfrak {u}(\mathfrak {p})$ is irreducible. This further reduces us to proving that the dimension of the $P_{[x]}$ -orbit of any point in $(x+\mathfrak {u}(\mathfrak {p}))\cap \mathfrak {g}_{\mathrm {reg}}$ equals $\dim (x+\mathfrak {u}(\mathfrak {p}))$ .

Suppose that $y\in (x+\mathfrak {u}(\mathfrak {p}))\cap \mathfrak {g}_{\mathrm {reg}}$ . It follows that

(4.1) $$ \begin{align}\begin{aligned}\dim (P_{[x]}\cdot y) & = \dim P_{[x]}-\dim(P_{[x]}\cap G_y) \\ & \geq \dim P_{[x]}-\dim G_y \\ & =\dim P_{[x]}-\ell\\ &=\dim L(P)_{[x]}+\dim U(P)-\ell \\ & \geq \dim U(P) \\ & =\dim(x+\mathfrak{u}(\mathfrak{p})),\end{aligned}\end{align} $$

where the fifth line follows from the fact that

$$ \begin{align*}\dim L(P)_{[x]}\geq\mathrm{rank}\hspace{1pt}L(P)=\mathrm{rank}\hspace{1pt}G=\ell.\end{align*} $$

Since $P_{[x]}\cdot y\subseteq x+\mathfrak {u}(\mathfrak {p})$ , the two inequalities in (4.1) must be equalities. We conclude that $\dim (P_{[x]}\cdot y)=\dim (x+\mathfrak {u}(\mathfrak {p}))$ and $\dim L(P)_{[x]}=\ell $ . This verifies (i), as well as the forward implication in (ii).

To prove the backward implication in (ii), choose a Cartan subalgebra $\mathfrak {h}\subseteq \mathfrak {g}$ and Borel subalgebra $\mathfrak {b}\subseteq \mathfrak {g}$ satisfying $\mathfrak {h}\subseteq \mathfrak {b}\subseteq \mathfrak {p}$ . These choices yield sets of roots $\Phi \subseteq \mathfrak {h}^*$ , positive roots $\Phi ^{+}\subseteq \Phi $ , and simple roots $\Delta \subseteq \Phi ^{+}$ . There is a unique subset $\Gamma \subseteq \Delta $ satisfying $\mathfrak {p}=\mathfrak {l}\oplus \mathfrak {u}(\mathfrak {p})$ , where

$$ \begin{align*}\mathfrak{l}:=\bigg(\bigoplus_{\alpha\in\mathrm{span}_{\mathbb{Z}}(\Gamma)\cap\Phi^+}\mathfrak{g}_{-\alpha}\bigg)\oplus\mathfrak{h}\oplus\bigg(\bigoplus_{\alpha\in\mathrm{span}_{\mathbb{Z}}(\Gamma)\cap\Phi^+}\mathfrak{g}_{\alpha}\bigg).\end{align*} $$

It suffices to prove that $x+\mathfrak {u}(\mathfrak {p})\cap \mathfrak {g}_{\mathrm {reg}}\neq \emptyset $ for all $x\in \mathfrak {l}_{\text {reg}}$ .

Suppose that $x\in \mathfrak {l}_{\text {reg}}$ . Choose $e_{\alpha }\in \mathfrak {g}_{\alpha }\setminus \{0\}$ for each $\alpha \in \Delta $ , and consider the elements

$$ \begin{align*}e_{\Gamma}:=\sum_{\alpha\in\Gamma}e_{\alpha}\quad\text{and}\quad e_{\Delta}:=\sum_{\alpha\in\Delta}e_{\alpha}.\end{align*} $$

At the same time, let $\mathfrak {b}_{-}\subseteq \mathfrak {g}$ denote the opposite Borel subalgebra with respect to $\mathfrak {h}$ and $\mathfrak {b}$ . The regularity of $x$ in $\mathfrak {l}$ allows one to find $g\in L$ satisfying $\mathrm {Ad}_g(x)\in e_{\Gamma }+\mathfrak {b}_{-}$ [Reference Kostant33, Theorem 8], where $L\subseteq G$ is a Levi subgroup integrating $\mathfrak {l}$ . Setting $y:= \mathrm {Ad}_g(x)$ and noting that $\mathrm {Ad}_g(\mathfrak {u}(\mathfrak {p}))=\mathfrak {u}(\mathfrak {p})$ , it suffices to prove that $(y+\mathfrak {u}(\mathfrak {p}))\cap \mathfrak {g}_{\mathrm {reg}}\neq \emptyset $ . On the other hand, note that $(e_{\Delta }-e_{\Gamma })\in \mathfrak {u}(\mathfrak {p})$ and $y+(e_{\Delta }-e_{\Gamma })\in e_{\Delta }+\mathfrak {b}_{-}$ . The former implies that $y+(e_{\Delta }-e_{\Gamma })\in y+\mathfrak {u}(\mathfrak {p})$ , and the latter tells us that $y+(e_{\Delta }-e_{\Gamma })\in \mathfrak {g}_{\mathrm {reg}}$ [Reference Kostant33, Lemma 10]. We conclude that $(y+\mathfrak {u}(\mathfrak {p}))\cap \mathfrak {g}_{\mathrm {reg}}\neq \emptyset $ . This completes the proof of (ii). Part (iii) follows immediately from (ii).

4.4 Residual actions on associated bundles

Consider a closed subgroup $H\subseteq G$ and finite-dimensional $H$ -module $V$ . Let $G\times H$ act on $G\times V$ by

$$ \begin{align*}(k,h)\cdot(g,v):=(kgh^{-1},h\cdot v),\quad (k,h)\in G\times H,\text{ }(g,v)\in G\times V.\end{align*} $$

The action of $H=\{e\}\times H\subseteq G\times H$ admits a geometric quotient

$$ \begin{align*}G\times_H V:= (G\times V)/H,\end{align*} $$

to which the action of $G=G\times \{e\}\subseteq G\times H$ descends. This quotient variety forms a $G$ -equivariant vector bundle over $G/H$ , with bundle projection

$$ \begin{align*}\theta_{V}:G\times_H V\longrightarrow G/H,\quad [g:v]\mapsto [g].\end{align*} $$

Now suppose that $K\subseteq H$ is a closed, normal subgroup. We may form the geometric quotient variety $G\times _K V$ , as above. Note that the action of $G\times H$ on $G\times V$ descends to an action of $G\times (H/K)$ on $G\times _K V$ . The natural map $G\times _K V\longrightarrow G\times _H V$ is then a geometric quotient of $G\times _K V$ by $H/K = \{e\}\times (H/K)\subseteq G\times (H/K)$ .

4.5 The Poisson Hamiltonian $G$ -variety $G\times _P\mathfrak {p}$

Consider a parabolic subgroup $P\subseteq G$ with Lie algebra $\mathfrak {p}\subseteq \mathfrak {g}$ . Let $\pi :G\longrightarrow G/U(P)$ denote the quotient morphism. The differential $\mathrm {d}\pi _e:\mathfrak {g}\longrightarrow T_{[e]}(G/U(P))$ is surjective with kernel $\mathfrak {u}(\mathfrak {p})$ , and so descends to a $U(P)$ -module isomorphism

$$ \begin{align*}\mathfrak{g}/\mathfrak{u}(\mathfrak{p})\overset{\cong}\longrightarrow T_{[e]}(G/U(P)),\quad [\xi]\mapsto \overline{\xi}:=\mathrm{d}\pi_e(\xi).\end{align*} $$

By inverting the induced isomorphism of dual vector spaces, one obtains a $U(P)$ -module isomorphism

$$ \begin{align*}\alpha:(\mathfrak{g}/\mathfrak{u}(\mathfrak{p}))^*\overset{\cong}\longrightarrow T_{[e]}^*(G/U(P)).\end{align*} $$

Note that

$$ \begin{align*}\alpha(\phi)(\overline{\xi})=\phi([\xi])\end{align*} $$

for all $\phi \in (\mathfrak {g}/\mathfrak {u}(\mathfrak {p}))^*$ and $\xi \in \mathfrak {g}$ , and that $\alpha $ induces an isomorphism of $G$ -equivariant vector bundles over $G/U(P)$ . This isomorphism is given by

(4.2) $$ \begin{align}\varphi:G\times_{U(P)}(\mathfrak{g}/\mathfrak{u}(\mathfrak{p}))^*\overset{\cong}\longrightarrow T^*(G/U(P)),\quad [g:\phi]\mapsto ([g],\alpha(\phi)\circ ((\mathrm{d}L_g)_{[e]})^{-1}),\end{align} $$

where $L_g:G/U(P)\longrightarrow G/U(P)$ is left multiplication by $g\in G$ .

Consider the action of $G\times G$ on $G$ defined by

$$ \begin{align*}(h,k)\cdot g:= hgk^{-1},\quad (h,k)\in G\times G,\text{ }g\in G.\end{align*} $$

There is an induced action of $G\times L(P)$ on $G/U(P)$ . The group $G\times L(P)$ thereby acts on the cotangent bundle $T^*(G/U(P))$ . On the other hand, consider the $P$ -modules $(\mathfrak {g}/\mathfrak {u}(\mathfrak {p}))^*$ and $\mathfrak {p}$ . The discussion in Section 4.4 yields induced actions of $G\times P$ on $G\times (\mathfrak {g}/\mathfrak {u}(\mathfrak {p}))^*$ and $G\times \mathfrak {p}$ . The same discussion implies that these induced actions descend to ones of $G\times L(P)$ on $G\times _{U(P)}(\mathfrak {g}/\mathfrak {u}(\mathfrak {p}))^*$ and $G\times _{U(P)}\mathfrak {p}$ .

Proof It suffices to prove that $\varphi $ is $L(P)$ -equivariant. An examination of (4.2) reveals that this is so if and only if

$$ \begin{align*}\alpha(p\cdot\phi)\circ((\mathrm{d}L_{gp^{-1}})_{[e]})^{-1}=\alpha(\phi)\circ((\mathrm{d}L_g)_{[e]})^{-1}\circ((\mathrm{d}R_{p^{-1}})_{[e]})^{-1}\end{align*} $$

as elements of $T_{[gp^{-1}]}^*(G/U(P))$ for all $p\in P$ and $[g:\phi ]\in G\times _{U(P)}(\mathfrak {g}/\mathfrak {u}(\mathfrak {p}))^*$ , where $R_{p^{-1}}:G/U(P)\longrightarrow G/U(P)$ is the result of letting right multiplication by $p^{-1}$ descend from a map $G\longrightarrow G$ to a map $G/U(P)\longrightarrow G/U(P)$ . To this end, note that each element of $T_{[gp^{-1}]}(G/U(P))$ is given by $(\mathrm {d}L_{gp^{-1}})_{[e]}(\overline {\xi })$ for some $\xi \in \mathfrak {g}$ . We are therefore reduced to proving that

(4.3) $$ \begin{align} & (\alpha(p\cdot\phi)\circ((\mathrm{d}L_{gp^{-1}})_{[e]})^{-1})((\mathrm{d}L_{gp^{-1}})_{[e]}(\overline{\xi})) \nonumber\\ &\quad =(\alpha(\phi)\circ((\mathrm{d}L_g)_{[e]})^{-1}\circ((\mathrm{d}R_{p^{-1}})_{[e]})^{-1})((\mathrm{d}L_{gp^{-1}})_{[e]})(\overline{\xi}))\end{align} $$

for all $p\in P$ , $[g:\phi ]\in G\times _{U(P)}(\mathfrak {g}/\mathfrak {u}(\mathfrak {p}))^*$ , and $\xi \in \mathfrak {g}$ . Our proof will consist of showing that each side of (4.3) is equal to $\phi ([\mathrm {Ad}_{p^{-1}}(\xi )])$ .

Observe that left-hand side of (4.3) is

$$ \begin{align*}\alpha(p\cdot\phi)(\overline{\xi})=(p\cdot\phi)([\xi])=\phi([\mathrm{Ad}_{p^{-1}}(\xi)]).\end{align*} $$

The right-hand side of (4.3) can be simplified by observing that

$$ \begin{align*} ((\mathrm{d}L_g)_{[e]})^{-1}\circ ((\mathrm{d}R_{p^{-1}})_{[e]})^{-1}\circ (\mathrm{d}L_{gp^{-1}})_{[e]} &=\mathrm{d}(L_{g^{-1}}\circ R_p\circ L_{gp^{-1}})_{[e]} \\ &=\mathrm{d}(L_{p^{-1}}\circ R_p)_{[e]}=\overline{\mathrm{Ad}_{p^{-1}}},\end{align*} $$

where $\overline {\mathrm {Ad}_{p^{-1}}}:T_{[e]}(G/U(P))\longrightarrow T_{[e]}(G/U(P))$ is the result of letting $\mathrm {Ad}_{p^{-1}}:\mathfrak {g}\longrightarrow \mathfrak {g}$ descend to an automorphism of $T_{[e]}(G/U(P))$ . The right-hand side is therefore given by

$$ \begin{align*}\alpha(\phi)(\overline{\mathrm{Ad}_{p^{-1}}}(\overline{\xi}))=\alpha(\phi)(\overline{\mathrm{Ad}_{p^{-1}}(\xi)})=\phi([\mathrm{Ad}_{p^{-1}}(\xi)]),\end{align*} $$

completing the proof.

Let $\langle \cdot ,\cdot \rangle :\mathfrak {g}\otimes \mathfrak {g}\longrightarrow \mathbb {C}$ be the Killing form. One knows that $\mathfrak {p}$ is the annihilator of $\mathfrak {u}(\mathfrak {p})$ in $\mathfrak {g}$ with respect to this form. We thereby obtain a $P$ -module isomorphism $\mathfrak {p}\longrightarrow (\mathfrak {g}/\mathfrak {u}(\mathfrak {p}))^*$ , whose composition with $\alpha :(\mathfrak {g}/\mathfrak {u}(\mathfrak {p}))^*\overset {\cong }\longrightarrow T^*_{[e]}(G/U(P))$ is

(4.4) $$ \begin{align}\beta:\mathfrak{p}\overset{\cong}\longrightarrow T^*_{[e]}(G/U(P)),\quad\beta(x)(\overline{\xi})=\langle x,\xi\rangle,\quad x\in\mathfrak{p},\text{ }\xi\in\mathfrak{g}.\end{align} $$

This combines with (4.2) and Proposition 4.3 to give a $G\times L(P)$ -equivariant isomorphism

(4.5) $$ \begin{align}\psi_{\mathfrak{p}}:G\times_{U(P)}\mathfrak{p}\overset{\cong}\longrightarrow T^*(G/U(P)),\quad [g:x]\mapsto([g],\beta(x)\circ((\mathrm{d}L_g)_{[e]})^{-1})\end{align} $$

of vector bundles over $G/U(P)$ . On the other hand, the $G\times L(P)$ -action on $T^*(G/U(P))$ is Hamiltonian with respect to the canonical symplectic structure on the latter. We may therefore equip $G\times _{U(P)}\mathfrak {p}$ with the symplectic Hamiltonian $G\times L(P)$ -variety structure for which $\psi _{\mathfrak {p}}$ is an isomorphism of symplectic Hamiltonian $G\times L(P)$ -varieties. A straightforward exercise shows

$$ \begin{align*}G\times_{U(P)}\mathfrak{p}\longrightarrow\mathfrak{g}\times\mathfrak{l}(\mathfrak{p}),\quad [g:x]\mapsto (\mathrm{Ad}_g(x),-[x])\end{align*} $$

to be a moment map, where the Killing form is used to identify $\mathfrak {g}^*$ (resp. $\mathfrak {l}(\mathfrak {p})^*$ ) with $\mathfrak {g}$ (resp. $\mathfrak {l}(\mathfrak {p})$ ). It follows that

$$ \begin{align*}G\times_P\mathfrak{p}=(G\times_{U(P)}\mathfrak{p})/L(P)\end{align*} $$

is a Poisson Hamiltonian $G$ -space with moment map

$$ \begin{align*}\mu_{\mathfrak{p}}:G\times_P\mathfrak{p}\longrightarrow\mathfrak{g},\quad [g:x]\mapsto\mathrm{Ad}_g(x).\end{align*} $$

5.1 The partial Grothendieck–Springer resolution $\mu _{\mathcal {C}}:\mathfrak {g}_{\mathcal {C}}\longrightarrow \mathfrak {g}$

The adjoint representation induces an action of $G$ on the set of parabolic subalgebras of $\mathfrak {g}$ . An orbit of this action will be called a conjugacy class of parabolic subalgebras. If $\mathcal {C}$ is one such conjugacy class, then the nonnegative integer $d_{\mathcal {C}}:= \dim \mathfrak {p}$ is independent of $\mathfrak {p}\in \mathcal {C}$ . It is straightforward to verify that $\mathcal {C}$ constitutes a closed $G$ -orbit in the Grassmannian $\mathrm {Gr}(d_{\mathcal {C}},\mathfrak {g})$ of $d_{\mathcal {C}}$ -dimensional subspaces of $\mathfrak {g}$ . It follows that $\mathcal {C}$ is a smooth projective variety carrying an algebraic $G$ -action.

Fix a conjugacy class $\mathcal {C}$ of parabolic subalgebras of $\mathfrak {g}$ . Let $\pi _{\mathcal {C}}:\mathfrak {g}_{\mathcal {C}}\longrightarrow \mathcal {C}$ be the $G$ -equivariant vector bundle obtained by pulling the tautological bundle on $\mathrm {Gr}(d_{\mathcal {C}},\mathfrak {g})$ back along the inclusion $\mathcal {C}\subseteq \mathrm {Gr}(d_{\mathcal {C}},\mathfrak {g})$ . It follows that

$$ \begin{align*}\mathfrak{g}_{\mathcal{C}}=\{(\mathfrak{p},x)\in\mathcal{C}\times\mathfrak{g}:x\in\mathfrak{p}\},\end{align*} $$

and that $\pi _{\mathcal {C}}(\mathfrak {p},x)=\mathfrak {p}$ for all $(\mathfrak {p},x)\in \mathfrak {g}_{\mathcal {C}}$ . One also has the $G$ -equivariant morphism

$$ \begin{align*}\mu_{\mathcal{C}}:\mathfrak{g}_{\mathcal{C}}\longrightarrow\mathfrak{g},\quad(\mathfrak{p},x)\mapsto x.\end{align*} $$

5.2 The Poisson geometry of $\mathfrak {g}_{\mathcal {C}}$

Let $\mathcal {C}$ be a conjugacy class of parabolic subalgebras of $\mathfrak {g}$ . Suppose that $\mathfrak {p}\in \mathcal {C}$ , and let $P\subseteq G$ be the parabolic subgroup integrating $\mathfrak {p}$ . One has the $G$ -equivariant variety isomorphisms

$$ \begin{align*} &\phi:G/P\overset{\cong}\longrightarrow\mathcal{C},\quad [g]\mapsto\mathrm{Ad}_g(\mathfrak{p})\quad\text{and}\quad \\ &\delta_{\mathfrak{p}}:G\times_P\mathfrak{p}\overset{\cong}\longrightarrow\mathfrak{g}_{\mathcal{C}},\quad [g:x]\mapsto(\mathrm{Ad}_g(\mathfrak{p}),\mathrm{Ad}_g(x)),\end{align*} $$

and commutative diagram

Proof As $\dim \mathfrak {g}_{\mathcal {C}}=\dim \mathfrak {g}$ , it suffices to prove that $(\mathrm {d}\mu _{\mathcal {C}})_{(\mathfrak {p},x)}$ is surjective if and only if $\mathfrak {g}_x\cap \mathfrak {u}(\mathfrak {p})=\{0\}$ . Consider the quotient morphism $\pi :G\times \mathfrak {p}\longrightarrow G\times _P\mathfrak {p}$ . The composite map $\delta _{\mathfrak {p}}\circ \pi :G\times \mathfrak {p}\longrightarrow \mathfrak {g}_{\mathcal {C}}$ is a submersion that sends $(e,x)$ to $(\mathfrak {p},x)$ . It therefore suffices to prove that

$$ \begin{align*}\mathrm{d}(\mu_{\mathcal{C}}\circ\delta_{\mathfrak{p}}\circ\pi)_{(e,x)}:\mathfrak{g}\oplus\mathfrak{p}\longrightarrow\mathfrak{g}\end{align*} $$

is surjective if and only if $\mathfrak {g}_x\cap \mathfrak {u}(\mathfrak {p})=\{0\}$ . On the other hand, a straightforward calculation shows $\mu _{\mathcal {C}}\circ \delta _{\mathfrak {p}}\circ \pi :G\times \mathfrak {p}\longrightarrow \mathfrak {g}$ to be given by

$$ \begin{align*}(\mu_{\mathcal{C}}\circ\delta_{\mathfrak{p}}\circ\pi)(g,y)=\mathrm{Ad}_g(y)\end{align*} $$

for all $(g,y)\in G\times \mathfrak {p}$ . It follows that

$$ \begin{align*}\mathrm{d}(\mu_{\mathcal{C}}\circ\delta_{\mathfrak{p}}\circ\pi)_{(e,x)}(\xi,\eta)=[\xi,x]+\eta\end{align*} $$

for all $(\xi ,\eta )\in \mathfrak {g}\oplus \mathfrak {p}$ . The image of $\mathrm {d}(\mu _{\mathcal {C}}\circ \delta _{\mathfrak {p}}\circ \pi )_{(e,x)}$ is therefore $[\mathfrak {g},x]+\mathfrak {p}$ . We conclude that $\mathrm {d}(\mu _{\mathcal {C}}\circ \delta _{\mathfrak {p}}\circ \pi )_{(e,x)}$ is surjective if and only if $([\mathfrak {g},x]+\mathfrak {p})^{\perp }=\{0\}$ , or equivalently $\mathfrak {g}_x\cap \mathfrak {u}(\mathfrak {p})=\{0\}$ .

Recall the Poisson Hamiltonian $G$ -variety structure on $G\times _P\mathfrak {p}$ from Section 4.5. In this context, we have the following result.

Proof Part (ii) is a straightforward computation, while (iii) is an immediate consequence of (i) and (ii). It therefore remains only to prove (i) and (iv). To prove (i), note that the uniqueness assertion follows immediately from the fact that $\delta _{\mathfrak {p}}$ is a bijection for all $\mathfrak {p}\in \mathcal {C}$ . The same fact also allows one to choose $\mathfrak {p}\in \mathcal {C}$ , and then endow $\mathfrak {g}_{\mathcal {C}}$ with the Poisson variety structure for which $\delta _{\mathfrak {p}}$ is an isomorphism of Poisson varieties. Our task is to prove that this Poisson structure does not depend on $\mathfrak {p}$ . This is the task of showing

$$ \begin{align*}\delta_{\mathfrak{p}_2}^{-1}\circ\delta_{\mathfrak{p}_1}:G\times_{P_1}\mathfrak{p}_1\longrightarrow G\times_{P_2}\mathfrak{p}_2\end{align*} $$

to be a Poisson variety isomorphism for all $\mathfrak {p}_1,\mathfrak {p}_2\in \mathcal {C}$ with corresponding parabolic subgroups $P_1,P_2\subseteq G$ , respectively.

Suppose that $\mathfrak {p}_1,\mathfrak {p}_2\in \mathcal {C}$ , and choose $h\in G$ satisfying $\mathrm {Ad}_h(\mathfrak {p}_1)=\mathfrak {p}_2$ . A straightforward exercise reveals that

$$ \begin{align*}(\delta_{\mathfrak{p}_2}^{-1}\circ\delta_{\mathfrak{p}_1})([g:x])=[gh^{-1}:\mathrm{Ad}_h(x)]\end{align*} $$

for all $[g:x]\in G\times _{P_1}\mathfrak {p}_1$ . Let us also observe that

$$ \begin{align*}\eta:G\times_{U(P_1)}\mathfrak{p}_1\longrightarrow G\times_{U(P_2)}\mathfrak{p}_2,\quad [g:x]\mapsto [gh^{-1}:\mathrm{Ad}_h(x)],\quad [g:x]\in G\times_{U(P_1)}P_1\end{align*} $$

is well-defined and makes

commute. It therefore suffices to prove that $\eta $ is an isomorphism of symplectic varieties.

Consider the variety isomorphism

$$ \begin{align*}G/U(P_1)\overset{\cong}\longrightarrow G/U(P_2),\quad [g]\mapsto [gh^{-1}],\quad [g]\in G/U(P)_1,\end{align*} $$

and the symplectic variety isomorphism

$$ \begin{align*}\zeta:T^*(G/U(P_1))\overset{\cong}\longrightarrow T^*(G/U(P_2))\end{align*} $$

that it induces. It is straightforward to check that

commutes, where $\psi _{\mathfrak {p}_1}$ and $\psi _{\mathfrak {p}_2}$ are defined in (4.5). Since $\zeta $ , $\psi _{\mathfrak {p}_1}$ , and $\psi _{\mathfrak {p}_2}$ are symplectic variety isomorphisms, the same must be true of $\eta $ . This proves (i).

We now prove (iv). A first step is to invoke Proposition 5.1; it implies that the differential of $\mu _{\mathcal {C}}$ is an isomorphism at each point in $\mu _{\mathcal {C}}^{-1}(\mathfrak {g}_{\text {reg}}\cap \mathfrak {g}_{\text {ss}})\subseteq \mathfrak {g}_{\mathcal {C}}$ , where $\mathfrak {g}_{\text {ss}}\subseteq \mathfrak {g}$ is the locus of semisimple elements. Since moment maps are Poisson morphisms (resp. $G$ -equivariant morphisms), requiring $\mu _{\mathcal {C}}$ to be a moment map determines the Poisson bivector field (resp. generating vector fields of the $G$ -action) on $\mu _{\mathcal {C}}^{-1}(\mathfrak {g}_{\text {reg}}\cap \mathfrak {g}_{\text {ss}})\subseteq \mathfrak {g}_{\mathcal {C}}$ . We also note that $\mu _{\mathcal {C}}^{-1}(\mathfrak {g}_{\text {reg}}\cap \mathfrak {g}_{\text {ss}})$ is a non-empty open subset of the irreducible variety $\mathfrak {g}_{\mathcal {C}}$ . It follows that forcing $\mu _{\mathcal {C}}$ to be a moment map determines the Poisson bivector field on $\mathfrak {g}_{\mathcal {C}}$ , as well as the generating vector fields of the $G$ -action on $\mathfrak {g}_{\mathcal {C}}$ . As $G$ is connected, these generating vector fields determine the $G$ -action on $\mathfrak {g}_{\mathcal {C}}$ . The previous two sentences imply (iv).

Recall that the regular locus $X_{\text {reg}}$ of a Poisson manifold $X$ is the union of its top-dimensional symplectic leaves. The rank of a Poisson manifold is the supremum of its symplectic leaf dimensions. Returning to the notation of this section, we have the following result.

Proof We first prove (i). Suppose that $(\mathfrak {p},x)\in \mathfrak {g}_{\mathcal {C}}$ , and let $P\subseteq G$ be the parabolic subgroup integrating $\mathfrak {p}$ . Recall that the canonical isomorphism $\delta _{\mathfrak {p}}:G\times _P\mathfrak {p}\longrightarrow \mathfrak {g}_{\mathcal {C}}$ is one of Hamiltonian $G$ -spaces. Since $\delta ([e:x])=(\mathfrak {p},x)$ , it suffices to prove that $[e:x]\in (G\times _P\mathfrak {p})_{\text {reg}}$ if and only if $[x]\in \mathfrak {l}(\mathfrak {p})_{\text {reg}}$ .

Recall that $G\times _{U(P)}\mathfrak {p}$ is a symplectic Hamiltonian ( $G\times L(P)$ )-space with moment map

$$ \begin{align*}(\nu_1,\nu_2):G\times_{U(P)}\mathfrak{p}\longrightarrow\mathfrak{g}\times\mathfrak{l}(\mathfrak{p}),\quad [g:y]\mapsto (\mathrm{Ad}_g(y),-[y]).\end{align*} $$

In light of Section 4.5, the symplectic leaves of $G\times _P\mathfrak {p}$ are the connected components of the Hamiltonian reductions of $G\times _{U(P)}\mathfrak {p}$ by $L(P)$ . It follows that the symplectic leaves of $G\times _P\mathfrak {p}$ are given by

$$ \begin{align*}\nu_2^{-1}([y])/L(P)_{[y]}=G\times_{P_{[y]}}((-y)+\mathfrak{u}(\mathfrak{p}))\subseteq G\times_P\mathfrak{p},\end{align*} $$

as $[y]$ ranges over $\mathfrak {l}(\mathfrak {p})$ . We conclude that $G\times _{P_{[x]}}(x+\mathfrak {u}(\mathfrak {p}))$ is the symplectic leaf of $G\times _P\mathfrak {p}$ through $[e:x]$ . This implies that $[e:x]\in (G\times _P\mathfrak {p})_{\text {reg}}$ if and only if $[x]$ achieves the minimal $P$ -centralizer dimension of vectors in $\mathfrak {l}(\mathfrak {p})$ . On the other hand, it is clear that $\dim P_{[x]}=\dim L(P)_{[x]}+\dim U(P)$ . One concludes that $\dim P_{[x]}$ is minimal among the $P$ -centralizer dimensions of vectors in $\mathfrak {l}(\mathfrak {p})$ if and only if $[x]\in \mathfrak {l}(\mathfrak {p})_{\text {reg}}$ . The proof of (i) is therefore complete.

To prove (ii), suppose that $(\mathfrak {p},x)\in (\mathfrak {g}_{\mathcal {C}})_{\text {reg}}$ . It suffices to prove that the symplectic leaf of $\mathfrak {g}_{\mathcal {C}}$ through $(\mathfrak {p},x)$ has dimension $\dim \mathfrak {g}-\ell $ . In light of the previous paragraph, this is the task of showing the dimension of $G\times _{P_{[x]}}(x+\mathfrak {u}(\mathfrak {p}))$ to be $\dim \mathfrak {g}-\ell $ . We have

$$ \begin{align*}\dim (G\times_{P_{[x]}}(x+\mathfrak{u}(\mathfrak{p}))) &= \dim G+\dim U(P)-\dim P_{[x]}\\ &=\dim G-\dim L(P)_{[x]}\\ &=\dim\mathfrak{g}-\ell, \end{align*} $$

where the last line follows from (i).

Let $\mathcal {B}$ denote the conjugacy class of Borel subalgebras of $\mathfrak {g}$ , that is, the full flag variety of $G$ . Note that $\mathfrak {g}_{\mathcal {B}}=\widetilde {\mathfrak {g}}$ is the full Grothendieck–Springer resolution of $\mathfrak {g}$ [Reference Evens and Lu26]. The following well-known result is a special case of the previous result.

6.1 The symplectic groupoid $(T^*G)_{\mathcal {C}}$

Let $\mathfrak {p}\subseteq \mathfrak {g}$ be a parabolic subalgebra integrating to a parabolic subgroup $P\subseteq G$ . Recall the discussion of the symplectic variety $G\times _{U(P)}\mathfrak {p}$ and Poisson variety $G\times _P\mathfrak {p}=(G\times _{U(P)}\mathfrak {p})/L(P)$ in Section 4.5. In light of Proposition 2.1, we may form the symplectic groupoid

$$ \begin{align*}(T^*G)_{\mathfrak{p}}:=\mathrm{Pair}(G\times_{U(P)}\mathfrak{p})\mkern-4mu\mathbin{/\mkern-5mu/}_{\mkern-4mu{0}}L(P)\;\substack{\longrightarrow\\[-9pt] \longrightarrow}\; G\times_P\mathfrak{p}.\end{align*} $$

Write $\textsf {s}_{\mathfrak {p}}:(T^*G)_{\mathfrak {p}}\longrightarrow \mathfrak {g}_{\mathcal {C}}$ (resp. $\textsf {t}_{\mathfrak {p}}:(T^*G)_{\mathfrak {p}}\longrightarrow \mathfrak {g}_{\mathcal {C}}$ ) for the result of composing the source (resp. target) of $(T^*G)_{\mathfrak {p}}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; G\times _P\mathfrak {p}$ with the Poisson isomorphism $\delta _{\mathfrak {p}}:G\times _P\mathfrak {p}\longrightarrow \mathfrak {g}_{\mathcal {C}}$ . It follows that $(T^*G)_{\mathfrak {p}}$ is a symplectic groupoid over $\mathfrak {g}_{\mathcal {C}}$ with source $\textsf {s}_{\mathfrak {p}}$ and target $\textsf {t}_{\mathfrak {p}}$ ; groupoid multiplication and inversion are induced from those of the pair groupoid of $G\times _{U(P)}\mathfrak {p}$ , and composing $\delta _{\mathfrak {p}}^{-1}:\mathfrak {g}_{\mathcal {C}}\longrightarrow G\times _P\mathfrak {p}$ with the unit bisection of $(T^*G)_{\mathfrak {p}}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; G\times _{P}\mathfrak {p}$ gives the unit bisection of $(T^*G)_{\mathfrak {p}}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\;\mathfrak {g}_{\mathcal {C}}$ . Straightforward computations also reveal that

$$ \begin{align*} & \textsf{s}_{\mathfrak{p}}([[g_1:x_1]:[g_2:x_2]])=(\mathrm{Ad}_{g_1}(\mathfrak{p}),\mathrm{Ad}_{g_1}(x_1))\quad\text{and}\quad \\ &\textsf{t}_{\mathfrak{p}}([[g_1:x_1]:[g_2:x_2]])=(\mathrm{Ad}_{g_2}(\mathfrak{p}),\mathrm{Ad}_{g_2}(x_2))\end{align*} $$

for all $[[g_1:x_1]:[g_2:x_2]]\in (T^*G)_{\mathfrak {p}}$ .

Proposition 6.1 Let $\mathcal {C}$ be a conjugacy class of parabolic subalgebras of $\mathfrak {g}$ . If $\mathfrak {p}_1,\mathfrak {p}_2\in \mathcal {C}$ , then there is a canonical $G$ -equivariant isomorphism

(6.1)

of symplectic groupoids.

Proof Choose $h\in G$ satisfying $\mathrm {Ad}_h(\mathfrak {p}_1)=\mathfrak {p}_2$ . Let $P_1,P_2\subseteq G$ be the parabolic subgroups integrating $\mathfrak {p}_1,\mathfrak {p}_2$ , respectively. It follows that $hP_1h^{-1}=P_2$ , and that conjugation by $h$ descends to an algebraic group isomorphism $\phi :L(P_1)\overset {\cong }\longrightarrow L(P_2)$ . Let us also recall that $G\times L(P_i)$ acts on $G/U(P_i)$ for $i=1,2$ . In this context, we observe that the isomorphism

$$ \begin{align*}\zeta:G/U(P_1)\overset{\cong}\longrightarrow G/U(P_2),\quad [g]\mapsto[gh^{-1}]\end{align*} $$

satisfies the equivariance condition

$$ \begin{align*}\zeta((g,[p])\cdot[k])=(g,\phi([p]))\cdot\zeta([k])\end{align*} $$

for all $(g,[p])\in G\times L(P_1)$ and $[k]\in G/U(P_1)$ . On the other hand, the proof of Proposition 5.2 explains that

$$ \begin{align*}\eta:G\times_{U(P_1)}\mathfrak{p}_1\longrightarrow G\times_{U(P_2)}\mathfrak{p}_2,\quad [g:x]\mapsto [gh^{-1}:\mathrm{Ad}_h(x)],\quad [g:x]\in G\times_{U(P_1)}P_1\end{align*} $$

is the isomorphism of cotangent bundles induced by $\zeta $ . We conclude that $\eta $ is an isomorphism from the Hamiltonian $(G\times L(P_1))$ -variety $G\times _{U(P_1)}\mathfrak {p}_1$ to the Hamiltonian $(G\times L(P_2))$ -variety $G\times _{U(P_2)}\mathfrak {p}_2$ , where $L(P_1)$ and $L(P_2)$ are identified via $\phi $ . This makes it clear that

$$ \begin{align*}\Psi_{(\mathfrak{p}_1,\mathfrak{p}_2)}:(T^*G)_{\mathfrak{p}_1}\longrightarrow (T^*G)_{\mathfrak{p}_2},\quad [\alpha_1:\alpha_2]\mapsto [\eta(\alpha_1):\eta(\alpha_2)]\end{align*} $$

is a well-defined, $G$ -equivariant symplectic variety isomorphism. It is also clear that (6.1) commutes if this isomorphism is taken as the top horizontal arrow, and straightforward to verify that $\Psi _{(\mathfrak {p}_1,\mathfrak {p}_2)}$ is an isomorphism of symplectic groupoids.

It remains only to prove that $\Psi _{(\mathfrak {p}_1,\mathfrak {p}_2)}$ does not depend on the choice of $h\in G$ satisfying $\mathrm {Ad}_h(\mathfrak {p}_1)=\mathfrak {p}_2$ . To this end, let $h,k\in G$ be such that $\mathrm {Ad}_h(\mathfrak {p}_1)=\mathfrak {p}_2$ and $\mathrm {Ad}_k(\mathfrak {p}_1)=\mathfrak {p}_2$ . It follows that $\mathrm {Ad}_{hk^{-1}}(\mathfrak {p}_2)=\mathfrak {p}_2$ , or equivalently that $hk^{-1}\in P_2$ . We also have

$$ \begin{align*}(gh^{-1},\mathrm{Ad}_h(x))=(gk^{-1}(hk^{-1})^{-1},\mathrm{Ad}_{hk^{-1}}\mathrm{Ad}_k(x))\end{align*} $$

for all $(g,x)\in G\times \mathfrak {p}_1$ . In particular, $(gh^{-1},\mathrm {Ad}_h(x))$ and $(gk^{-1},\mathrm {Ad}_k(x))$ belong to the same $L(P_2)$ -orbit in $G\times _{U(P_2)}\mathfrak {p}_2$ for all $[g:x]\in G\times _{U(P_1)}\mathfrak {p}_1$ . This fact forces $\Psi _{(\mathfrak {p}_1,\mathfrak {p}_2)}$ to be independent of the choice mentioned in the first sentence of this paragraph.

We are now equipped to prove Main Theorem 1. For the sake of convenience, we restate it below.

Let $\mathcal {C}$ be a conjugacy class of parabolic subalgebras of $\mathfrak {g}$ . Consider the set

$$ \begin{align*}(T^*G)_{\mathcal{C}}:=\bigg(\bigsqcup_{\mathfrak{p}\in\mathcal{C}}(T^*G)_{\mathfrak{p}}\bigg)\Bigm/\sim,\end{align*} $$

where $\sim $ is the equivalence relation defined by

$$ \begin{align*}(\alpha_1\in(T^*G)_{\mathfrak{p}_1})\sim(\alpha_2\in(T^*G)_{\mathfrak{p}_2})\Longleftrightarrow\alpha_2=\Psi_{(\mathfrak{p}_1,\mathfrak{p}_2)}(\alpha_1).\end{align*} $$

Main Theorem 1 now follows from our next result.

Corollary 6.2 The set $(T^*G)_{\mathcal {C}}$ has a unique $G$ -equivariant symplectic groupoid structure such that

is a $G$ -equivariant symplectic groupoid isomorphism for all $\mathfrak {p}\in \mathcal {C}$ .

Given $(\mathfrak {p},x)\in \mathfrak {g}_{\mathcal {C}}$ , the isomorphism $(T^*G)_{\mathcal {C}}\cong (T^*G)_{\mathfrak {p}}$ restricts to an isomorphism

(6.2) $$ \begin{align}((T^*G)_{\mathcal{C}})_{(\mathfrak{p},x)} & \cong ((T^*G)_{\mathfrak{p}})_{[e:x]} \\ & = \left\{[[g:y]:[h:z]]\in (G\times_{U(P)}\mathfrak{p})\times_{\mathfrak{l}(\mathfrak{p})}(G\times_{U(P)}\mathfrak{p})/L(P):[g:y] \right. \notag \\ & \left. =[e:x]=[h:z]\text{ in }G\times_P\mathfrak{p}\right\}\notag\end{align} $$

between the isotropy groups of $(\mathfrak {p},x)$ and $[e:x]\in G\times _P\mathfrak {p}$ . We use this isomorphism to freely identify the two isotropy groups in our next proposition.

Proposition 6.3 If $(\mathfrak {p},x)\in \mathfrak {g}_{\mathcal {C}}$ , then

$$ \begin{align*} &L(P)_{[x]}\longrightarrow ((T^*G)_{\mathcal{C}})_{(\mathfrak{p},x)}=\left(\left(G\times_{U(P)}\mathfrak{p})\times_{\mathfrak{l}(\mathfrak{p})}(G\times_{U(P)}\mathfrak{p})\right)/L(P)\right)_{[e:x]},\quad \\ & [p]\mapsto [[p:\mathrm{Ad}_{p^{-1}}(x)]:[e:x]]\end{align*} $$

is a well-defined isomorphism of algebraic groups.

Proof A straightforward exercise reveals that

(6.3) $$ \begin{align} P_{[x]}\longrightarrow \left(\left(G\times_{U(P)}\mathfrak{p})\times_{\mathfrak{l}(\mathfrak{p})}(G\times_{U(P)}\mathfrak{p})\right)/L(P)\right)_{[e:x]},\quad\!\! p\mapsto [[p:\mathrm{Ad}_{p^{-1}}(x)]:[e:x]]\end{align} $$

defines an algebraic group morphism. It therefore suffices to prove that (6.3) is surjective with kernel $U(P)$ . To establish surjectivity, suppose that $[[g:y]:[h:z]]\in \left (\left (G\times _{U(P)}\mathfrak {p})\times _{\mathfrak {l}(\mathfrak {p})}(G\times _{U(P)}\mathfrak {p})\right )/L(P)\right )_{[e:x]}$ . It follows that $g,h\in P$ , $y=\mathrm {Ad}_{g^{-1}}(x)$ , and $z=\mathrm {Ad}_{h^{-1}}(x)$ . The condition $[y]=[z]\in \mathfrak {l}(\mathfrak {p})$ then implies that $gh^{-1}\in P_{[x]}$ , so that

$$ \begin{align*}[[g:y]:[h:z]]=[[g:\mathrm{Ad}_{g^{-1}}(x)]:[h:\mathrm{Ad}_{h^{-1}}(x)]=[[gh^{-1}:\mathrm{Ad}_{(gh^{-1})^{-1}}(x)]:[e:x]]\end{align*} $$

must be in the image of (6.3).

It remains to prove that $U(P)$ is the kernel of (6.3). To this end, note that $p\in P_{[x]}$ belongs to this kernel if and only if $[[p:\mathrm {Ad}_{p^{-1}}(x)]:[e:x]]=[[e:x]:[e:x]]$ . This is equivalent to the existence of $q\in P$ such that

$$ \begin{align*}[[p:\mathrm{Ad}_{p^{-1}}(x)]:[e:x]]=[[q:\mathrm{Ad}_{q^{-1}}(x)]:[q:\mathrm{Ad}_{q^{-1}}(x)]].\end{align*} $$

We may rephrase this as the condition that

$$ \begin{align*}(p,\mathrm{Ad}_{p^{-1}}(x))=(qu,\mathrm{Ad}_{(qu)^{-1}}(x))\quad\text{and}\quad (e,x)=(qv,\mathrm{Ad}_{(qv)^{-1}}(x))\end{align*} $$

for some $q\in P$ and $u,v\in U(P)$ . Straightforward manipulations and substitutions show that this holds if and only if

$$ \begin{align*}(p,\mathrm{Ad}_{p^{-1}}(x))=(v^{-1}u,\mathrm{Ad}_{(v^{-1}u)^{-1}}(x))\end{align*} $$

for some $u,v\in U(P)$ , or equivalently $p\in U(P)$ . The kernel of (6.3) is therefore equal to $P_{[x]}\cap U(P)$ . Since $U(P)$ acts trivially on the $P$ -module $\mathfrak {l}(\mathfrak {p})$ , the kernel must be $U(P)$ .

6.2 Global slices to $(T^*G)_{\mathcal {C}}$

Let $\mathcal {C}$ be a conjugacy class of parabolic subalgebras of $\mathfrak {g}$ . Fix a principal $\mathfrak {sl}_2$ -triple $(e,h,f)\in \mathfrak {g}^{\times 3}$ , and consider the associated Kostant slice

$$ \begin{align*}\mathrm{Kos}:= e+\mathfrak{g}_f\subseteq\mathfrak{g}.\end{align*} $$

One knows that $\mathrm {Kos}$ is a fundamental domain for the adjoint action of $G$ on $\mathfrak {g}_{\mathrm {reg}}$ [Reference Kostant33]. It is also known to be a Poisson transversal for the Poisson structure on $\mathfrak {g}=\mathfrak {g}^*$ , where the Killing form is used to identify $\mathfrak {g}$ and $\mathfrak {g}^*$ (see [Reference Kostant32, Reference Kostant33] for the required ingredients, and [Reference Gan and Ginzburg28] for the case of arbitrary Slodowy slices). We conclude that

$$ \begin{align*}\mathrm{Kos}_{\mathcal{C}}:=\mu_{\mathcal{C}}^{-1}(\mathrm{Kos})\subseteq\mathfrak{g}_{\mathcal{C}}\end{align*} $$

is a Poisson transversal. On the other hand, let $((T^*G)_{\mathcal {C}})_{\text {reg}}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\;(\mathfrak {g}_{\mathcal {C}})_{\text {reg}}$ denote the pullback of $(T^*G)_{\mathcal {C}}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\;\mathfrak {g}_{\mathcal {C}}$ to $(\mathfrak {g}_{\mathcal {C}})_{\text {reg}}\subseteq \mathfrak {g}_{\mathcal {C}}$ . We relate the preceding discussion to the notion of an admissible global slice, as defined in Section 3.3.

Proof We first show $\mathrm {Kos}_{\mathcal {C}}$ to be a global slice to $((T^*G)_{\mathcal {C}})_{\text {reg}}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\;(\mathfrak {g}_{\mathcal {C}})_{\text {reg}}$ . To this end, note that the orbits of $((T^*G)_{\mathcal {C}})_{\text {reg}}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\;(\mathfrak {g}_{\mathcal {C}})_{\text {reg}}$ are precisely the top-dimensional symplectic leaves of $\mathfrak {g}_{\mathcal {C}}$ . This combines with Proposition 5.3 to reduce us to proving the following: $\mathrm {Kos}_{\mathcal {C}}$ has a non-empty intersection with the symplectic leaf of $\mathfrak {g}_{\mathcal {C}}$ through $(\mathfrak {p},x)$ if and only if $[x]\in \mathfrak {l}(\mathfrak {p})_{\text {reg}}$ , in which case the intersection is a singleton. By Lemma 4.1 and Proposition 5.2, this is equivalent to proving the following: $\mu _{\mathfrak {p}}^{-1}(\mathrm {Kos})$ has a non-empty intersection with the symplectic leaf of $G\times _P\mathfrak {p}$ through $[e:x]$ if and only if $(x+\mathfrak {u}(\mathfrak {p}))\cap \mathfrak {g}_{\mathrm {reg}}\neq \emptyset $ , in which case the intersection is a singleton. This leaf is $G\times _{P_{[x]}}(x+\mathfrak {u}(\mathfrak {p}))$ , as established in the proof of Proposition 5.3. The “if and only if” assertion now follows immediately from the fact that $\mathrm {Kos}$ is a fundamental domain for the adjoint action of $G$ on $\mathfrak {g}_{\mathrm {reg}}$ .

To complete our proof that $\mathrm {Kos}_{\mathcal {C}}$ is a global slice, let $(\mathfrak {p},x)\in \mathfrak {g}_{\mathcal {C}}$ be such that $(x+\mathfrak {u}(\mathfrak {p}))\cap \mathfrak {g}_{\mathrm {reg}}\neq \emptyset $ . Our task is to show that any two elements of $G\times _{P_{[x]}}(x+\mathfrak {u}(\mathfrak {p}))\cap \mu _{\mathfrak {p}}^{-1}(\mathrm {Kos})$ must coincide. To this end, suppose that $[g,y],[h:z]\in G\times _P(x+\mathfrak {u}(\mathfrak {p}))\cap \mu _{\mathfrak {p}}^{-1}(\mathrm {Kos})$ . It follows that $\mathrm {Ad}_g(y),\mathrm {Ad}_h(z)\in \mathrm {Kos}$ , implying that $y,z\in \mathfrak {g}_{\mathrm {reg}}$ . By Lemma 4.1(i), $z=\mathrm {Ad}_p(y)$ for some $p\in P$ . One consequence is that $y$ and $z$ must be conjugate to the same element of $\mathrm {Kos}$ , that is, $\mathrm {Ad}_g(y)=\mathrm {Ad}_h(z)=\mathrm {Ad}_{hp}(y)$ . We conclude that $g^{-1}hp\in G_y$ . Since Lemma 4.2 now implies that $g^{-1}hp\in P$ , we have

$$ \begin{align*}[h:z]=[h:\mathrm{Ad}_p(y)]=[hp:y]=[hp(g^{-1}hp)^{-1}:\mathrm{Ad}_{g^{-1}hp}(y)]=[g:y].\end{align*} $$

This establishes that $\mathrm {Kos}_{\mathcal {C}}$ is a global slice to $((T^*G)_{\mathcal {C}})_{\text {reg}}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\;(\mathfrak {g}_{\mathcal {C}})_{\text {reg}}$ .

It remains only to establish that the global slice $\mathrm {Kos}_{\mathcal {C}}$ is admissible. Given $(\mathfrak {p},x)\in \mathrm {Kos}_{\mathcal {C}}$ , the previous two paragraphs imply that $[x]\in \mathfrak {l}(\mathfrak {p})_{\text {reg}}$ . We conclude that $L(P)_{[x]}$ is abelian [Reference Kostant33, Proposition 14]. By Proposition 6.3, the isotropy group $((T^*G)_{\mathcal {C}})_{(\mathfrak {p},x)}$ is also abelian. It follows that $\mathrm {Kos}_{\mathcal {C}}$ is admissible.

We have the following specialization to the case of the flag variety $\mathcal {B}$ of $\mathfrak {g}$ . It is presumably well known to experts. Note that $\mathfrak {g}_{\mathcal {B}}=\widetilde {\mathfrak {g}}$ is the full Grothendieck–Springer resolution in this case.

6.3 Grothendieck–Springer alterations of the Moore–Tachikawa TQFT

Recall that Main Theorem 2 is the following assertion.

The proof is now straightforward. Let $\mathcal {C}$ be a conjugacy class of parabolic subalgebras of $\mathfrak {g}$ . As $(\mathfrak {g}_{\mathcal {C}})_{\text {reg}}$ is Poisson, Theorem 6.4 implies that $\mathcal G=((T^*G)_{\mathcal {C}})_{\text {reg}}$ and $S=\mathrm {Kos}_{\mathcal {C}}$ satisfy the hypotheses of Theorem 3.2. Write $\eta _{\mathcal {C}}:\mathbf {Cob}_2\longrightarrow \mathbf {WS}_1$ for the resulting TQFT; it satisfies Main Theorem 2. Proposition 3.3 gives an explicit variety $(T^*G)_{\mathcal {C}}^{m,n}:= \eta _{\mathcal {C}}(C_{m,n})$ for $(m,n)\neq (0,0)$ . The following two special cases warrant further discussion:

1. (i) If $\mathcal {C}=\{\mathfrak {g}\}$ , then $\eta _{\mathcal {C}}$ is the open Moore–Tachikawa TQFT constructed in [Reference Crooks and Mayrand22]. The varieties

   $$ \begin{align*}(T^*G)^{m,n}:= (T^*G)_{\mathcal{C}}^{m,n}\end{align*} $$
   are sometimes called open Moore–Tachikawa varieties, and feature in several recent works [Reference Bielawski9, Reference Crooks and Mayrand21, Reference Crooks and Mayrand22, Reference Ginzburg and Kazhdan30]. One may affinize these varieties to obtain a scheme-theoretic version of the Moore–Tachikawa TQFT [Reference Crooks and Mayrand22, Reference Ginzburg and Kazhdan30].

2. (ii) Suppose that $\mathcal {C}=\mathcal {B}$ is the conjugacy class of Borel subalgebras. By Corollary 6.5, $\mathrm {Kos}_{\mathcal {C}}=\mathrm {Kos}_{\mathcal {B}}$ is an admissible global slice to the entire symplectic groupoid $(T^*G)_{\mathcal {C}}=(T^*G)_{\mathcal {B}}$ .

Fix a pair of nonnegative integers $(m,n)\neq (0,0)$ . One might seek relationships among the varieties $(T^*G)_{\mathcal {C}}^{m,n}$ , as $\mathcal {C}$ ranges over the conjugacy classes of parabolic subalgebras of $\mathfrak {g}$ . Section 7 addresses this issue.

7.1 A useful result

Let $\mathcal {C}$ be a conjugacy class of parabolic subalgebras of $\mathfrak {g}$ . Suppose that $\mathfrak {p}\in \mathcal {C}$ , and let $P\subseteq G$ be the parabolic subgroup integrating $\mathfrak {p}$ . Recall from Section 4.5 that $G\times _{U(P)}\mathfrak {p}$ is a symplectic Hamiltonian $G\times L(P)$ -variety. Let $\phi _{\mathfrak {p}}:G\times _{U(P)}\mathfrak {p}\longrightarrow \mathfrak {g}$ denote the moment map for the Hamiltonian action of $G=G\times \{e\}\subseteq G\times L(P)$ . On the other hand, recall that $G\times _P\mathfrak {p}=(G\times _{U(P)}\mathfrak {p})/L(P)$ is a Poisson Hamiltonian $G$ -variety with moment map $\mu _{\mathfrak {p}}$ . One has $\mathcal {L}_{\mu _{\mathfrak {p}}}=\mathcal {L}_{\overline {\phi _{\mathfrak {p}}}}$ in the notation of Section 2.6. Note that $\mathcal {L}_{\mu _{\mathfrak {p}}}$ is a Lagrangian subgroupoid of $(\mathrm {Pair}(G\times _{U(P)}\mathfrak {p})\mkern -4mu\mathbin {/\mkern -5mu/}_{\mkern -4mu{0}}L(P))\times \overline {T^*G}=(T^*G)_{\mathfrak {p}}\times \overline {T^*G}$ , and that Lemma 2.4 gives information about Lagrangian intersections with $\mathcal {L}_{\mu _{\mathfrak {p}}}$ .

Use the isomorphism in Corollary 6.2 to identify $(T^*G)_{\mathcal {C}}$ with $(T^*G)_{\mathfrak {p}}$ , and Proposition 5.2(i) to identify $\mathfrak {g}_{\mathcal {C}}$ with $G\times _P\mathfrak {p}$ . Lemma 2.4 then implies the following about $\mathcal {L}_{\mu _{\mathcal {C}}}$ .

7.2 Proof of Main Theorem 3

To prepare for what follows, let $X$ and $Y$ be symplectic varieties. Our convention is to define a Lagrangian relation from $X$ to $Y$ to be an immersed Lagrangian submanifold $L\subseteq X\times \overline {Y}$ . We sometimes adopt the notation $L:X\Longrightarrow Y$ for a Lagrangian relation $L$ from $X$ to $Y$ . If $L:X\Longrightarrow Y$ and $M:Y\Longrightarrow Z$ are Lagrangian relations between symplectic varieties $X$ , $Y$ , and $Z$ , then $L$ and $M$ may be composed as set-theoretic relations. Let $\Delta (Y)\subseteq Y\times Y$ denote the diagonal copy of $Y$ . We call $M$ and $N$ composable if $L\times M$ are $X\times \Delta (Y)\times Z$ are transverse in $X\times Y\times Y\times Z$ , and the projection map $X\times Y\times Y\times Z\longrightarrow X\times Z$ restricts to an immersion $(L\times M)\cap (X\times \Delta (Y)\times Z)\longrightarrow X\times Z$ . The image of this projection is then Lagrangian relation $M\circ L:X\Longrightarrow Z$ [Reference Bates and Weinstein7, Proposition 5.28]. We call $L$ and $M$ strongly composable if the map $(L\times M)\cap (X\times \Delta (Y)\times Z)\longrightarrow X\times Z$ is an embedding of manifolds. In this case, $M\circ L$ is a Lagrangian submanifold of $X\times \overline {Y}$ .