Hello,


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.



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> If $v \in M$ is a vector killed by $Lie(G)$, then it is fixed by $G$.
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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 $Hom$).



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> For tori, solvable, nilpotent, semi-simple groups, the expected characterizations of the essential image of $G-mod \to Lie(G)-mod$.
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Thank you,
Sasha



About $G$-modules versus $Lie(G)$-modules for algebraic groups
I'm sure that there are a number of ways of answering this in the affine case; here's one. Let's assume $G$ affine and let's say that $k$ is our algebraically closed field. The following argument may not be the slickest one, but it has the benefit of working in arbitrary characteristic with the enveloping algebra replaced by the hyperalgebra. (In positive characteristic, instead of considering a Lie($G$)-stable vector we'd want to consider a hyperalgebra-stable vector. It is *not* true in positive characteristic that a Lie($G$)-stable vector is $G$-stable).


Now, a $G$-module structure on $M$ is given by a comodule morphism $$c : M \to k[G] \otimes M .$$ Let $U(G)$ denote the enveloping algebra of Lie($G$); equivalently, this is the so-called hyperalgebra of $G$ (hyperalgebra = enveloping algebra when char$(k) = 0$, but not in positive characteristic). We will view $U(G)$ as a subspace of the full linear dual of $k[G]$; namely, $U(G)$ is the subspace of $k[G]^\*$ consisting of elements that vanish on some power of the ideal defining the identity. (You can look in, say, Jantzen's book Representations of Algebraic Groups for more details). Note that $v \in M$ is $G$-stable if and only if $c(v) = 1 \otimes v$.


Now, the action of $U(G)$ on $M$ also comes from the comorphism $c$. Namely, for $v \in M$, if $c(v) = \sum f\_i \otimes v\_i$ then for $X \in U(G)$ we have $$ X.v = \sum X(f\_i) \cdot v\_i ,$$ where $X(f\_i)$ is the dual action of $X$ on $f\_i \in k[G]$. Let $U(G)^+$ denote the augmentation ideal; this is the two-sided ideal of $U(G)$ generated by the Lie algebra inside of $U(G)$. Then $v \in M$ is killed by Lie($G$) if and only if $X.v = 0$ for all $X \in U(G)^+$. (This part only works in characteristic 0; there are slight modifications to be made in positive characteristic). 


So now let's assume that $v \in M$ is killed by Lie$(G)$. Let's write $$ c(v) = \sum f\_i \otimes v\_i . $$ Without loss of generality we may assume that the $v\_i$ are $k$-linearly independent. Since $X.v = 0$ for all $X \in U(G)^+$ this implies $$ \sum X(f\_i) \cdot v\_i = 0 $$ for all $X \in U(G)^+$. Since the $v\_i$ are linearly independent, this means that $X(f\_i) = 0$ for all $i$ and for all $X \in U(G)^+$. Now, any element $f \in k[G]$ that is killed by all elements of $U(G)^+$ must be constant. Hence we have $$ c(v) = \sum 1 \otimes v\_i . $$ But one of the properties of the comodule morphism is that $(\epsilon \otimes Id\_M) \circ c = 1 \otimes Id\_M$, where $\epsilon$ is the augmentation of $k[G]$. Hence $$ \sum 1 \otimes v\_i = 1 \otimes v $$ and $v$ is $G$-stable.


