The characteristic kernel on a regular moment level

Let $v\in T_p{\mu }^{-1}(\alpha )$. By [F1] and [F2], $T_p{\mu }^{-1}(\alpha )=(T_p(G\cdot p){)}^{\omega }$ and $(T_p{\mu }^{-1}(\alpha ){)}^{\omega }=T_p(G\cdot p)$; hence $v$ is in the kernel of $({\iota }^{\ast }\omega {)}_p$ if and only if $v\in T_p(G\cdot p)$, that is, if and only if $v$ lies in the radical of the restriction of $\omega$ to the orbit tangent space $T_p(G\cdot p)$.

For $\xi ,\eta \in \mathfrak{g}$ the value of the orbit restriction is ${\omega }_p({\xi }_M(p),{\eta }_M(p))=\{{\mu }^{\xi },{\mu }^{\eta }\}(p)$: this follows from $X_{{\mu }^{\xi }}=-{\xi }_M$, $X_{{\mu }^{\eta }}=-{\eta }_M$ by [F3] and the definition of the Poisson bracket, or equivalently from ${\omega }_p({\xi }_M,{\eta }_M)=-d{\mu }_p^{\xi }({\eta }_M(p))$.

Let $\xi \in \mathfrak{g}$. By step 1.2 and the bracket identity [F5], ${\omega }_p({\xi }_M(p),{\eta }_M(p))=\{{\mu }^{\xi },{\mu }^{\eta }\}(p)=⟨\alpha ,[\xi ,\eta ]⟩$ for every $\eta \in \mathfrak{g}$.

Consequently ${\xi }_M(p)$ is in the radical of the orbit restriction if and only if $\alpha ([\xi ,\cdot ])=0$, equivalently if and only if $\xi \in {\mathfrak{g}}_{\alpha }$ by [F6].

Therefore the radical of the restriction of $\omega$ to $T_p(G\cdot p)$ equals the image of ${\mathfrak{g}}_{\alpha }$ under the infinitesimal orbit map, namely $T_p(G_{\alpha }\cdot p)$ by [F6].

By step 1.1 the kernel of the restricted form is precisely that radical, so $\ker ({\iota }^{\ast }\omega {)}_p=T_p(G_{\alpha }\cdot p)$.