# Set of singular points for momentum map (with coisotropic action)

Let $G$ be a Lie-group acting on a connected symplectic manifold $M'$ in a hamiltonian way, with an $\operatorname{Ad}^*_G$-equivariant momentum map. Assuming $G$ acts properly on $M'$, we can restrict ourself to the open and dense subset $M$ of regular points. So $M$ is still connected. Since $M$ is a $G$-invariant subset and open, $G$ acts also hamiltonian on $M$ with $\Phi \colon M \to \mathfrak{g}^*$.

Assuming $\Phi(M) = W$ is a manifold, $\Phi$ being a surjective submersion on $W$ and $G$ acting properly on $W$.

Since $G$ acts also properly on $W$, we have a principle orbit-type in $W$. Denote by $W_{reg}$ the open and dense set of points $y\in W$, such that $G \cdot y$ is a principal orbit.

Now using Sard, we know, that there is an open and dense subset $W_0 \subset W$, such that $$y \in W_0 \ \Leftrightarrow \ y \text{ is a regular value and regular point }$$

What can we now say about the preimage $\Phi^{-1}(W_0)$?

1) Could it be, that it is open and dense?

2) if not, is it, if $G$ is compact?

3) what if the $G$-orbits in $M$ are coisotropic?

4) what if $G$ is compact and the $G$-orbits in $M$ are coisotropic?

Maybe one example: If $\mathcal{O}$ is some coadjoint $U(n)$-orbit and $\pi \colon \mathcal{O} \to \mathfrak{u}(n-1)^*$ is the projection, induced by the embedding $U(n-1) \subset U(n)$, then $\pi$ is the momentum map for the coadjoint $U(n-1)$ action on $\mathcal{O}$. Then $\mathcal{O}_{reg}$ is exactly the preimage of $W_0$.

## 1 Answer

The equivariant Darboux theorem is valid whenever the action is proper (see for example Heinzner, Loose: A global slice theorem for proper Hamiltonian actions. Manuscripta Math. 98 (1999), 295–305). This implies that locally a Hamiltonian proper action is real analytic. In particular, the moment map is (locally) real analytic. Therefore the set of regular points is dense since its complement is locally a real analytic closed subset.

• Thank you very much for your response. So, by saying that the equivariant Darboux theorem implies, that locally a Hamiltonian proper action is real analytic, do you mean, that it is a straight corollary of this theorem, or that it follows from (for instance) the coisotropic embedding? – Olorin May 19 '16 at 15:38
• The equivariant Darboux theorem implies that locally everything looks like a vector bundle over the orbit with all the standard structures which are all real analytic. – Friedrich Knop May 19 '16 at 16:26