Let $D$ be a differential operator acting between the spaces of smooth sections of two vector bundles $E,F$ over *compact* manifold $M$. If $M$ is not simply connected one can construct the universal covering $\widetilde{M}$ of $M$. If we do so, we have an action of $\pi_1(M)$ on $\widetilde{M}$.

I would like to understand in detail how we can

lift$D$ to this universal covering in such a way that this operator will be equivariant.

If it is discussed in detail somewhere, I'll be satisfied with giving me some references (my quess is the following: first we have a covering map $p: \widetilde{M} \to M$ and then we can pull back the bundles $E,F$ to bundles $p^*(E),p^*(F)$: the action of $\pi_1(M)$ on $\widetilde{M}$ gives us the action on sections $(s \vartriangleleft g)(x)=s(g \vartriangleright x)$: but when we come to differential operator I don't know how to proceed further).

And once we have the lift $\widetilde{D}$ and we know that it is equivariant I read that one can associate to it a class in $K_0(C^*_r(\pi_1(M)))$. This is the 'higher index' of $D$.

I would like also to understand the main ingredients of this construction: how this higher index is defined, why it lives in $K$-theory of the reduced $C^*$-algebra of the group, do I need the equivariant $K$-theory to perform this construction?

Forgive me if my question is too broad: I'm starting my attempt to understand index theorem (the original one, due to Atiyah-Singer) but before this I would like to have some general picture about various generalisations (at least to know the formulation of theorems and understand constructions involved).