It seems that there are two notions of strongly equivaraint $D_X$- Modules and I would like to know if they are equivalent, or at least how they are related. Let $\rho: G\times X \rightarrow X$ be an action of an algebraic group on a smooth variety over the complex numbers. The first definition goes like this:

An equivariant $D_X$ Module is just a $D_X$ module $M$ together with an isomorphism $$\rho^* M\rightarrow \pi^* M$$ of $D_{G\times X}$ -modules. That isomorphism has to satisfy some cocycle condition.

The other definition is a bit more cumbersome to write down. First it requires just an isomorphism of $O_{G\times X}$ modules, not necessarily of $D_{G\times X}$-modules $$\rho^* M\rightarrow \pi^* M$$ modules, which again satisfies the cocycle conditon. In addition it requires the action map $$D_X\otimes M \rightarrow M$$ to be equivariant. Finally there is another condition to be satisfied: Observe that we get two operations of the liealgera on $M$:

One operation, by directly differentiating the action of $G$ on $M$.

Another operation in the following way: First we differentiate the action of $G$ on $X$, and get a map $$Lie(G)\rightarrow Der_X$$ from the liealgebra into vectorfields on $X$. Because $M$ is a $D_X$ module we can compose this map with the action of vectorfields on $M$ and get our second operation.

We require these operations to coincide.

A more precise definition of the second kind is given here on pages 48-49: http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf

So the question is, are these two notions equivalent?

Edit: If anybody else needs these facts, I found a reference which gives a proof: http://alpha.uhasselt.be/Research/Algebra/Publications/Geq.ps