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:


1. 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?
2. 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.



Proof of Steinberg's tensor product theorem
The 1980 CPS paper is short but not easy to read without enough background.
They gave the first conceptual alternative to Steinberg's somewhat opaque
and computational proof of the tensor product theorem in 1963 (which built
on the 1950s work of Curtis on "restricted" Lie algebra representations
coming from the algebraic group plus the older work of Steinberg's teacher
Richard Brauer on rank 1). Steinberg relied quite a bit on working with
covering groups and projective representations.


CPS already realized the importance of
getting beyond the Lie algebra by using Frobenius kernels. The best modern
source is the large but well-organized 1987 book by J.C. Jantzen,
Representations of Algebraic Groups (expanded AMS edition in 2003). Here
the foundations are worked out thoroughly and the CPS proof is given an
efficient treatment in part II, 3.16-3.17. While CPS had in mind the
analogy with Clifford theory for finite groups, Jantzen gives a self-contained
treatment avoiding use of projective representations or Skolem-Noether.


Apart from sources, the essential goal is to single out the finitely many
"restricted" simple modules for the Lie algebra among the infinitely many
simple (rational) modules for the ambient algebraic group, then realize the
latter modules as twisted tensor products of the former. This requires a notion of
Frobenius morphism for each power of the prime, Frobenius kernels being
infinitesimal group schemes. The Lie algebra just plays the role of
first Frobenius kernel (a normal subgroup scheme), so the Clifford theory
analogue developed by Ballard and CPS makes sense here. 


