What is the definition of an equivariant connection I am looking for a reference that gives a definition of equivariant connection in terms of sheaves. More precisely:
Let $X$ be a complex manifold or an algebraic variety over $\mathbb C$ and $\rho:G\times X\rightarrow X$ an action. An equivariant sheaf is then a sheaf $\mathcal{F}$ on $X$ together with an isomorphism $\Phi:\rho^*\mathcal{F}\rightarrow pr_{2}^*\mathcal{F}$ satisfying a cocycle condition.
A connection for me is a $\mathbb C$ linear morphism
$\nabla:\mathcal{F}\longrightarrow \mathcal{F}\otimes \Omega_X^1$ that satisfies the Leibniz-rule.
What is the right notion of compatibility between these two structures that gives the definition of equivariant connection? Is this written up somewhere? In particular the canonical conection $d$ on $\mathcal{O}_X^n$ should be equivariant. I would also be happy if for whatever reasons things only work on vector bundles. In the best possible case, the reference would also work out how this translates in terms of comodules for quasi-coherent sheaves over affine schemes, but I guess that is easy enough once one has the correct definition.
 A: A good notion seems to be the following:
Recall that for connections, even though they are not $\mathcal{O}_X$-linear, there is a well defined notion of pullback. So we get a map $pr_2^*\nabla:pr_2^*\mathcal{F}\longrightarrow pr_2^*\mathcal{F}\otimes\Omega_{G\times X}^1$. Denote by $pr_2^*\nabla_X$ its composition with the identity tensor the projection $\Omega_{G\times X}^1\cong pr_1^*\Omega_G^1\oplus pr_2^*\Omega_X^1\longrightarrow pr_2^*\Omega^1_X$.
Also, $\Omega_X^1$ carries an equivariant structure as follows:
$\rho^*\Omega_X^1\longrightarrow\Omega_{G\times X}\cong pr_1^*\Omega_G^1\oplus pr_2^*\Omega_X^1\longrightarrow pr_2^*\Omega_X,$
where the first arrow is induced by pullback with $\rho$ and the second is projection. Denote the composition by $\Phi_\Omega$.
For an equivariant sheaf $(\mathcal{F},\Phi_\mathcal{F})$ we get by adjunction a morphism $\Phi_{\mathcal{F}}^{ad}:\mathcal{F}\longrightarrow \rho_*pr_2^*\mathcal{F}$. Now a connection could be coined equivariant if one has the equality (sorry I don't know how to draw commutative diagrams here)
$(\Phi_\mathcal{F}\otimes\Phi_\Omega)^{ad}\circ\nabla=\rho_*(pr_2^*\nabla_X)\circ\Phi_\mathcal{F}^{ad}$
This makes the canonical connection on a trivial bundle (for a possible nontrivial action on generators that does not depend on the $X$ coordinates) and also translates nicely to comodules. I would still be happy to know if this is already written down somewhere.
