Is the Moment Map for a Product Group Action a Morse Function?

[Ssnow]

It is know that let $M$ a compact symplectic manifold with $G=T^{d_{T}}$ a torus of dimension $d_{T}$ acting on $M$ in Hamiltonian fashion with Moment map $\Phi:M\rightarrow \mathfrak{t}^{*}$, then $\Phi^{\xi}=\langle \Phi(m),\xi\rangle$ is a Morse function in each of its component (for $\xi\in\mathfrak{t}$). What I want to discuss here is what happen if we consider now the product group $P=G\times T^{d_{T}}$ where $G$ is a Lie group of dimension $d_{G}$, assuming that $P$ acts in Hamiltonian way and that at each point of $M$ the moment map of $G$ is $\Phi_{G}(m)=0\in \mathfrak{g}^{*}$. It is $\Phi_{P}$ a Morse function?

 

[jvicens]

Hello,

Thank you for bringing up this interesting topic. The Morse function property of the moment map is an important aspect of Hamiltonian actions on symplectic manifolds. In the case of a torus action on a symplectic manifold, the moment map is indeed a Morse function on each component, as you mentioned. This is because the torus action is free and the fixed points are isolated, allowing for a clean decomposition into components.

In the case of a product group action, such as the one you propose with $P=G\times T^{d_{T}}$, the moment map may not necessarily be a Morse function. This is because the fixed points of the action will not be isolated, but rather form submanifolds of the symplectic manifold. In this case, the moment map will not have a clean decomposition into components and may not satisfy the Morse function property.

However, there are cases where the moment map for a product group action can still be a Morse function. For example, if the Lie group $G$ is a compact torus, then the fixed points of the action will still be isolated and the moment map will be a Morse function on each component. Additionally, if the Lie group $G$ is a compact semisimple group, then the fixed points will again be isolated and the moment map will be a Morse function on each component.

In general, the Morse function property of the moment map for a product group action will depend on the specific structure of the Lie groups involved, as well as the symplectic manifold on which they act. I hope this helps to shed some light on the topic and I look forward to further discussion on this topic.