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Let $G$ be a discrete finitely generated group which acts properly and freely on a smooth manifold $M$ with compact quotient $M/G$. Is it right to consider any function $f \in C^{\infty}_c(M)$ (with compact support) as smooth section of some finite vector bundle $f' \in C^{\infty}(M/G, E)$ and then as smooth section of corresponding $G$-equivariant bundle $ C^{\infty}_c(M, \hat{E})$?

If it is correct can I define a Fredholm differential operator $D : C^{\infty}_c(M) \rightarrow C^{\infty}_c(M)$ as $G$-invariant operator on $C^{\infty}_c(M, \hat{E})$?

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    $\begingroup$ I am curious to hear how you see a function on $\mathbb{R}$ as a section of a vector bundle on $\mathbb{S}^1$... $\endgroup$
    – abx
    Dec 1, 2019 at 14:03
  • $\begingroup$ Equivariant functions valued in a $G$-module are precisely those which descend to sections of a vector bundle. Arbitrary functions clearly don't. $\endgroup$
    – Ben McKay
    Dec 1, 2019 at 14:13
  • $\begingroup$ I consider functions with compact support, which can be described as functions valued in group ring as follows: $f \rightarrow \sum\limits_{g \in G} f(gx)g$ with property $f(gx) = g^{-1}f(x)$ $\endgroup$
    – Aleksandr Alekseev
    Dec 1, 2019 at 16:24
  • $\begingroup$ @abx: The OP takes the functions to have compact support. $\endgroup$
    – Alex M.
    Dec 7, 2019 at 17:05

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