Reduction in stages for free proper regular actions

Put $Z_H={\mu }_H^{-1}(0)$ and ${\mathfrak{h}}^0=\{\alpha \in {\mathfrak{g}}^{\ast }:\alpha {\mathrm{\mid }}_{\mathfrak{h}}=0\}$. Normality implies ${\mathrm{Ad}}_g\mathfrak{h}=\mathfrak{h}$ for every $g\in G$, by differentiating conjugation on $H$. Thus equivariance of $\mu$ shows that $Z_H$ is $G$-invariant. Restriction of the moment equations and equivariance to $H$ makes ${\mu }_H$ an equivariant moment map for the restricted symplectic action. The first-stage hypotheses therefore give $M_H=Z_H\mathrm{/}H$, a smooth surjective submersion $\pi :Z_H\to M_H$, and a symplectic form ${\omega }_H$ with ${\pi }^{\ast }{\omega }_H={\iota }_H^{\ast }\omega$. Empty levels are understood as empty manifolds.

Write $c_g$ for conjugation by $g$. Differentiating the identity $q\circ c_g=c_{q(g)}\circ q$ gives $d{q}_e{\mathrm{Ad}}_g={\mathrm{Ad}}_{q(g)}d{q}_e$. In particular, for every $h\in H$, including disconnected components, $q(h)=e$ implies ${\mathrm{Ad}}_h\xi -\xi \in \ker d{q}_e=\mathfrak{h}$. Consequently $H$ acts trivially on ${\mathfrak{h}}^0$. The dual of $d{q}_e$ gives a linear isomorphism $J:(\mathfrak{g}\mathrm{/}\mathfrak{h}{)}^{\ast }\to {\mathfrak{h}}^0$, $J(\beta )(\xi )=\beta (\xi +\mathfrak{h})$, intertwining the two coadjoint actions. For $p\in Z_H$, $\mu (p)\in {\mathfrak{h}}^0$, and equivariance gives $\mu (hp)=\mu (p)$. Therefore $\bar{\mu }(\pi (p))=J^{-1}\mu (p)$ is well defined, with exactly the formula in the statement. This uses normality at the group level, not an assumption that $H$ is connected.

Define $(gH)\cdot \pi (p)=\pi (gp)$. Changing $p$ to $hp$ changes $gp$ by $gh{g}^{-1}\in H$, and changing $g$ to $gh$ has the same effect; hence the action is well defined and inherits the group-action identities. It is smooth: on domains of smooth local sections $s$ of $q$ and $t$ of $\pi$, it is $(a,x)\mapsto \pi (s(a)t(x))$. The moment map is smooth as well, since locally $\bar{\mu }=J^{-1}\mu \circ t$. Equivariance follows from step 2.1: $\bar{\mu }((gH)\cdot \pi (p))=q(g)\cdot \bar{\mu }(\pi (p))$.

For fixed $g$, let $a_g:Z_H\to Z_H$ be its action and let ${\bar{a}}_{q(g)}$ be the induced action on $M_H$. Then $\pi \circ a_g={\bar{a}}_{q(g)}\circ \pi$ and $a_g^{\ast }{\iota }_H^{\ast }\omega ={\iota }_H^{\ast }\omega$, so ${\pi }^{\ast }({\bar{a}}_{q(g)}^{\ast }{\omega }_H-{\omega }_H)=0$. Pullback by a surjective submersion is injective on differential forms: at any target point choose a preimage and lift every tangent argument by the surjective differential. Thus the residual action preserves ${\omega }_H$.

Naturality of exponentials under $q$ and the defining action show that the residual fundamental field of $\xi +\mathfrak{h}$ is $d\pi ({\xi }_M)$ along $Z_H$; the field is tangent there by $G$-invariance. For $v\in T_p{Z}_H$, differentiating ${\bar{\mu }}^{\xi +\mathfrak{h}}\circ \pi ={\mu }^{\xi }{\mathrm{\mid }}_{Z_H}$ gives $d{\bar{\mu }}_{\pi (p)}^{\xi +\mathfrak{h}}(d{\pi }_{p}v)=-{\omega }_p({\xi }_M(p),v)=-{\omega }_H((\xi +\mathfrak{h}{)}_{M_H}(\pi (p)),d{\pi }_{p}v)\mathrm{.}$ Surjectivity of $d{\pi }_p$ proves the moment equation for every tangent vector. Together with steps 3.1 and 4.1 this proves the claimed Hamiltonian residual action and licenses its reduction under the second set of hypotheses.

Put $Z={\mu }^{-1}(0)$ and $W={\bar{\mu }}^{-1}(0)$. The equality $J\bar{\mu }\pi =\mu {\mathrm{\mid }}_{Z_H}$ implies $Z={\pi }^{-1}(W)$, and the restricted map $P=\pi {\mathrm{\mid }}_Z:Z\to W$ is surjective with fibres exactly the $H$-orbits. The levels are embedded by [F5]; their inclusions give the usual induced smooth structures, so $P$ is smooth. More explicitly, a map into an embedded submanifold is smooth when its ambient composite is smooth, by the coordinates in which the submanifold is a coordinate plane. For $u\in T_{w}W=\ker d{\bar{\mu }}_w$, choose $p\in Z$ over $w$ and lift $u$ to $v\in T_p{Z}_H$ using $d{\pi }_p$. Differentiating the displayed identity yields $d{\mu }_{p}v=Jd{\bar{\mu }}_{w}u=0$. Hence $v\in T_{p}Z$ by [F5], proving $P$ is a submersion. It therefore has smooth local sections through every point by [F4].

Let $R:W\to W\mathrm{/}(G\mathrm{/}H)$ and $Q:Z\to Z\mathrm{/}G$ be the quotient maps. Both are smooth surjective submersions by the assumed free proper actions and [F4]. Define $\varphi :W\mathrm{/}(G\mathrm{/}H)\to Z\mathrm{/}G$ by $\varphi (R(P(p)))=Q(p)$. This is well defined and bijective: two points of $Z$ have $P$-images in the same $G\mathrm{/}H$-orbit exactly when one differs from a $G$-translate of the other by an element of $H$, which is exactly equality of their $G$-orbits. To verify the nontrivial direction explicitly, if $P(p^{\mathrm{\prime }})=(gH)P(p)$, then $P(p^{\mathrm{\prime }})=P(gp)$, so $p^{\mathrm{\prime }}=hgp$ for some $h\in H$. Smoothness follows locally by choosing sections $r$ of $R$ and $s$ of $P$, shrinking domains so that $s\circ r$ is defined: $\varphi =Q\circ s\circ r$. Conversely, for a local section $t$ of $Q$, the inverse is $R\circ P\circ t$, hence smooth. Thus this is a diffeomorphism, proved directly without a double-quotient theorem.

Write $\mathrm{\Omega }$ for the two-stage reduced form on $W\mathrm{/}(G\mathrm{/}H)$, ${\omega }_G$ for the one-stage form on $Z\mathrm{/}G$, $j:W\hookrightarrow M_H$ and $i:Z\hookrightarrow M$. Their defining identities give $R^{\ast }\mathrm{\Omega }=j^{\ast }{\omega }_H$, $Q^{\ast }{\omega }_G=i^{\ast }\omega$, and $P^{\ast }{j}^{\ast }{\omega }_H=i^{\ast }\omega$ by restricting ${\pi }^{\ast }{\omega }_H={\iota }_H^{\ast }\omega$ to $Z$. Therefore $(R\circ P{)}^{\ast }{\varphi }^{\ast }{\omega }_G=Q^{\ast }{\omega }_G=i^{\ast }\omega =(R\circ P{)}^{\ast }\mathrm{\Omega }\mathrm{.}$ The composite $R\circ P$ is a surjective submersion, so the injectivity argument in step 4.1 gives ${\varphi }^{\ast }{\omega }_G=\mathrm{\Omega }$.

If $Z$ is empty, then $W$ is empty and both final reductions are empty, with the unique empty symplectomorphism. The first-stage construction still applies even when $Z_H$ is nonempty. The extreme cases $H=\{e\}$ and $H=G$ give the identity reduction at one of the stages. A subgroup with zero Lie algebra can be nontrivial and discrete; no identification $Z_H\mathrm{/}H=Z_H$ is asserted in that case, and the group-level argument in step 2.1 applies unchanged. Countable choice is inherited as stated in [A1]; the local sections used above are pointwise local constructions, not a selected global section. This completes all claims.