Let $G$ be a finite cyclic group and $X$ a smooth manifold equipped with a trivial $G$-action.
It is known that we can decompose every $G$-equivariant vector bundle with respect to the action:
Every smooth $G$-equivariant vector bundle $V \to X$ can be decomposed as the Whitney sum $V=\bigoplus_{\chi}V_\chi$ of $G$-equivariant vector bundles.
Here $\chi$ runs the character group $X(G):=\mathrm{Hom}(G,\mathbb C^\ast)$ and the $G$-action on $V_\chi$ is given by $g \cdot v = \chi(g)v$ ($g \in G$, $v \in V_\chi$).
I understand the case when $X$ is a point. This is just the eigenspace decomposition.
Question: Why do the eigenspaces vary smoothly so that $V_\chi$ gives a smooth subbundle of $V$?
Notes:
- I do not mention above whether the vector bundle is real or complex. Maybe we should assume it to be complex.
- If you know a reference containing the proof, please let me know.