A compact-group moment map can be averaged to an equivariant one when the affine obstruction vanishes

The integrand $(g,p)\mapsto g^{-1}\cdot \mu (g\cdot p)$ is smooth, being a composition of the smooth action, the smooth coadjoint action and $\mu$; since $G$ is compact, integrating the finitely many components of this ${\mathfrak{g}}^{\ast }$-valued function against the normalized Haar measure defines a smooth map $\bar{\mu }:M\to {\mathfrak{g}}^{\ast }$. [A2, F1, F2].

Equivariance: for $h\in G$ and $p\in M$, make the right-translation substitution $k=gh$, so $g=k{h}^{-1}$ and $g^{-1}=h{k}^{-1}$. Right invariance of Haar then gives $\bar{\mu }(h\cdot p)={\int }_{G}h\cdot (k^{-1}\cdot \mu (k\cdot p))d{\mu }_G(k)=h\cdot \bar{\mu }(p)\mathrm{.}$ No equivariance of the original infinitesimal moment map is used. [step 1.1, F1, F2].

Component equations: for fixed $g\in G$ and $\xi \in \mathfrak{g}$, the function $p\mapsto ⟨g^{-1}\cdot \mu (g\cdot p),\xi ⟩={\mu }^{{\mathrm{Ad}}_g\xi }(g\cdot p)$ has differential $d{\mu }_{g\cdot p}^{{\mathrm{Ad}}_g\xi }(d(a_g{)}_{p}v)=-{\omega }_{g\cdot p}(({\mathrm{Ad}}_g\xi {)}_M(g\cdot p),d(a_g{)}_{p}v)=-{\omega }_p({\xi }_M(p),v)$ by [F2] and [F3], independently of $g$. Integrating over $G$ gives $d{\bar{\mu }}_p^{\xi }(v)=-{\omega }_p({\xi }_M(p),v)$, the component moment equation for $\bar{\mu }$. [step 1.1, F2, F3].

By step 2.2 the averaged map satisfies the component moment equations, and by step 2.1 it is coadjoint equivariant; hence $\bar{\mu }$ is an equivariant moment map for the action. [step 2.1, step 2.2].

For each $\xi$, step 2.2 and [F2] show that ${\bar{\mu }}^{\xi }$ and ${\mu }^{\xi }$ are Hamiltonians for the same vector field, so [F4] and connectedness make their difference constant. Linearity in $\xi$ therefore gives a constant covector $\delta :=\bar{\mu }-\mu \in {\mathfrak{g}}^{\ast }$. Since $\bar{\mu }$ is equivariant, its bracket defect vanishes; expanding its defect and using that constants are Poisson-central gives $0=c(\xi ,\eta )-\delta ([\xi ,\eta ])$. Thus the constant cocycle $c$ of [F5] is the coboundary represented by $\delta$. In general $\delta$ need not be coadjoint-fixed, because $\mu$ need not be equivariant. [step 2.2, step 3.1, F2, F4, F5].

The construction began from the supplied linear family of component Hamiltonians; no step here produces such a family when the closed one-forms $-{\iota }_{{\xi }_M}\omega$ have no primitives, so averaging trivializes only the affine obstruction of a supplied infinitesimal moment map. [step 4.1, A2, A1].