Invariant elements of homogeneous vector bundles over homogeous spaces

Question

For a compact homogeneous space $K/L$. Consider a homogeneous vector bundle, which is to say, a vector bundle of the form $$ \mathcal{V} \simeq K \times_{\rho} \Lambda, $$ where $\rho$ is a representation of $L$ on $\Lambda$. It seems clear that $^{G}\Gamma(\mathcal{V})$, the left $G$-invariant sections of $\Gamma(\mathcal{V})$, admit the following linear isomorphism $$ ^{G}\Gamma(\mathcal{V}) \simeq ^H \Lambda, $$ but I am finding it difficult to formulate a precise proof - the main problem being my poor understanding of how the Peter-Weyl theorem relates to smooth functions, as opposed to continuous functions.

Answer 1

This does not depend on compactness of the homogeneous space and the proof does not need the Peter-Weyl theorem. You can directly establish a bijective correspondence between the two spaces. In one direction, the map is just given by evaluation in $eH$. In the other direction you take an $H$-invariant element $y\in\Lambda$ and show that defining $\sigma(gH)$ to be the class of $(g,y)$ leads to a well defined $G$-invariant section, which is evidently smooth.