Let $G$ be a simply connected semi-simple algebraic group over an algebraically closed field of positive characteristic. The Steinberg tensor product theorem gives a tensor product decomposition of an irreducible rational $G$-module $S(\lambda)$ with heighest weight $\lambda$ according to the $p$-adic expansion of $\lambda$.

I am trying to understand the proof of this theorem as given by Cline, Parshall, Scott in Journal of Algebra 63, 264-267 (1980). I have two major problems with this proof:

In theorem 1 where it is proven that any irreducible $\mathcal{U}$-module over the restricted universal enveloping algebra extends uniquely to a rational $G$-module, how do I get this representation $\rho:G \rightarrow PGL(V)$? I was able to see that I have a morphism $G \rightarrow \mathrm{Aut}_{_kAlg}(\mathrm{End}_k(V))$. But what next? It seems to be well-known that the group of algebra-automorphisms of the endomorphism ring is just $PGL(V)$, but why? Moreover, why do I get a morphism of

*algebraic*groups?In the proof of theorem 2, why acts the Lie algebra $L(G)$ trivially on $\mathrm{Hom}_{L(G)}(S _1, S)$? (Here $S$ is some irreducible $G$-module, $S _1$ is an irreducible $L(G)$-submodule of $S$ and $S _1$ also denotes the unique extension to a rational $G$-module). This is not explicitly mentioned, but I think it is used in b). This must have something to do with theorem 1!?

Any hints and ideas are welcome.