It is well-known that if a Lie group $G$ acts on a symplectic manifold $(W,\omega)$ preserving $\omega$, then we can easily find a $G$-invariant almost complex structure $J$ by choosing first a $G$-invariant Riemannian metric -- that always exists by averaging an arbitrary one.

Then there is a unique $A \in \Gamma(\mathrm{End}(TW))$ that satisfies $\langle v,w\rangle = \omega(v,Aw)$ for all $v,w \in TW$. Both the metric and $\omega$ are $G$-invariant thus $A$ also must be, and then we basically only have to normalize $A$ to obtain the desired almost complex structure (in this case an $\omega$-compatible almost complex structure).

A similar method should work for taming almost complex structures since these are also in bijection to a convex domain of a vector space.

What about a general almost complex structures? If $W$ is a smooth manifold that admits almost complex structures, and if $G$ is a connected compact Lie group acting smoothly on $W$, is there any other method to obtain an almost complex structure on $W$ that is $G$-invariant?

Or are there any (easy) counter examples?


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