In my dissertation I proved a certain theorem(s) concerning the representation theory of a direct product G x H of algebraic groups over a field, given those of G and H. But I would wager 100:1 that this theorem is well known, and as such I am looking for a primary reference for it. I have combed (my personal) literature for it, e.g. Jantzen, Humphreys, etc., but cannot find it.


The theorem(s) can be found in section 2 of the following pdf file:


<http://homepages.utoledo.edu/mcrumle/references%20needed.pdf>


Please ignore the other sections of the file as they pertain to questions having little relevance here.


Thanks in advance for any help.


$\underline{\text{Question, version 2.0}}$


Sorry again for the confusion, my notation appears to have been sloppy. And sorry that I haven't posted the entire thing on this page, but I really think it's more efficient (for everyone) that I post it here:


<http://homepages.utoledo.edu/mcrumle/Reference%20Needed%20-%20Direct%20Products.pdf>


This theorem I believe is much more basic than the gracious people who bothered to try to answer it (in my garbled terminology) have made it out to be. I am talking about arbitrary representations (not irreducible, etc.) over arbitrary fields (not algebraically closed, etc.) for arbitrary algebraic groups (not reductive, etc.). It is simply the statement that any $G \times H$-module structure on a $k$-vector space $V$ can be factored (as a product of matrices, or linear maps if you like) uniquely into a product of commuting representations for $G$ and $H$ on that same vector space. In section 2 there are no tensor products or direct sums of vector spaces taking place; the vector space $V$ is fixed throughout, and all three of the groups $G$, $H$, and $G \times H$ are acting on it.


The corollaries that follow in section 3 merely re-affirm the suspicion that, for fixed $G \times H$-representations on the fixed vector spaces $V$ and $W$, things like "tensor product", "direct sum", and "morphism" of $G \times H$-modules should behave as expected with respect to their respective $G$ and $H$ modules.


Thank you once again for any help. 



Reference needed for representation theory of direct products of algebraic groups over a field (of arbitrary characteristic)
I don't think the statement that you linked in your revised question needs a reference, or even an explicit proof. The fact that comodule structures can be pushed forward along coalgebra homomorphisms is sufficiently clear that you should be able to get away with stating a precise claim. Similarly, I think the statement that the structure of a $G \times H$-module is equivalent to commuting $G$-module and $H$-module structures is also straightforward enough that you can state it without proof or reference. (Others may disagree, but I think I'm on the majority side in this particular case.)


A common way to deal with straightforward facts that are often used later in a paper is to write the claim as a lemma, followed by:



> 
> **Proof:** Omitted.
> 
> 
> 


If the proof is not entirely obvious, it may help to write a sentence or two on how to get through the tricky part.


