I would like to know clear references about the following facts:
Let $G$ be a connected algebraic group (over alg. closed field in char. 0), $Lie(G)$ its Lie algebra, $M$ a $G$-module. I don't assume that $G$ is affine, but if there is a nice simple reference with $G$ affine, then I'll like it too.
If $v \in M$ is a vector killed by $Lie(G)$, then it is fixed by $G$.
This will establish that $G-mod \to Lie(G)-mod$ is fully faithful .(by the usual method of interpreting a morphism as an element of the inner $Hom$).
For tori, solvable, nilpotent, semi-simple groups, the expected characterizations of the essential image of $G-mod \to Lie(G)-mod$.
Thank you, Sasha