Let $G$ be a compact Lie group, $V$ a finite-dimensional orthogonal $G$-representation. Consider the unit sphere in $V$, i.e. the subspace of $V$ consisting of vectors of length $1$. Since $V$ is orthogonal, this space inherits a $G$-action.
When $G$ is finite, I can write down a $G$ CW complex structure on such a sphere in $V$. My question is: what happens if $G$ is a compact Lie group? Does a sphere in $V$ still admit a $G$ CW complex structure?
You can use Morse theory to show that every smooth compact $G$-manifold is given by some equivariant handelbody decomposition (note that in this case the handles are disc bundles over $G/H$, not necessarily $G/H \times D^n$!) and give those handles a $G$-triangulation. This does mean that a $G$-CW complex structure on a $G$-manifold is usually much larger than the natural object arising from Morse theory. In principle things should be easier for representation spheres, but I don't know any place where an elementary proof is written down.