I am reading the paper "The representation ring of the quantum double of a finite group" by Witherspoon. In Chapter 2 we define a G-equivariant vector bundle on a finite G-set $X$, as a collection of finite dimensional vector spaces $\{U_x\}_{x\in X}$, with a representation of $G$ on $\bigoplus_{x}U_x$ such that $U_x \cdot g=U_{x^g}$.
We then decompose the $\bigoplus_{x}U_x=\bigoplus_{\sigma}U_\sigma$ for each $G$-orbit $\sigma$ on $X$, with $U_\sigma=\bigoplus_{x\in \sigma}U_x$. Now I think I see why these $U_\sigma$ are $kG$-submodules of $\bigoplus_{x}U_x$, since the $$U_\sigma \cdot g=\bigoplus_{x\in \sigma}U_x\cdot g=\bigoplus_{x\in \sigma}U_{x^g}=\bigoplus_{x\in \sigma}U_x=U_\sigma$$ It is claimed these $U_\sigma$ are the indecomposable G-equivariant vector bundles on $X$. But by the definition above, a vector bundle on $X$ is a collection of vector spaces, one for each $x\in X$. But these $U_\sigma$ only have vector spaces defined over the elements in the orbit: $x\in \sigma\subset X$. Is it implicitly assumed we take trivial/zero vector spaces over the remaining points in $X\backslash \sigma$? Otherwise I don't see how these objects can be called vector bundles on $X$.
Many thanks for any help!