Orbits of Product Lie Groups Action

Question

Hi to all,

Let $G$ be a Lie group of linear isometries of $\mathbb{R}^n_{\nu}$ ($\mathbb{R}^n_{\nu}$ is the semi-Euclidean space) and $G_1$ ,$G_2$ two Lie subgroups of $G$. Let $G_1 \times G_2$ as a Lie subgroup of $G$ by identifying it with the inner product of $G_1$ and $G_2$ (suppose $G_1$ , $G_2$ have the required properties to form inner direct product) and let $G_1 \times G_2$ acts on $\mathbb{R}^n_{\nu}$ by the induced action from $G$ . What can we say about the relation between the orbits of $G_1 \times G_2$ and the orbits of $G_1$ and $G_2$? Is a $(G_1 \times G_2)$-orbit isometric (or at least diffeomorphic) to the product of the $G_1$-orbit and $G_2$-orbit?

Best for you.

Answer 1

If all actions are proper, orbits are closed embedded submanifolds. If not, then they are only initial submanifolds (unique injectively immersed submanifolds with a possibly finer topology - think of a 1-param. subgroup of the torus with irrational slope; see thm 6.4 in (here))

Now the $(G\times G_1)$-orbit through $x$ is diffeomorphic to $(G_1\times G_2)/(G_1\times G_2)_x$ (mod out the isotropy group). If $(G_1\times G_2)_x = (G_1)_x\times (G_2)_x$ then the orbit is a product. We always have $(G_1\times G_2)_x \supset (G_1)_x\times (G_2)_x$ so the $(G\times G_1)$-orbit through $x$ is always a submersive quotient of the product of the two orbits.

The source I cited has more information.