In representation theory of linear algebraic groups, we consider the "irreducible" and "completely reducible" types of representations $(V, \rho)$,
- a $G$-representation is irreducible if {0},V are the only G-stable subspaces
- a $G$-representation is completely reducible if it is a direct sum of G-stable irreducible subspaces.
We have a charakterization of the diagonal matrices in $GL_n(k)$ where every representation is completely reducible "into characters" (sloppy expression).
What do we know for the general case? I see that for the "classical" groups ($GL_n(k)$, $SL_n(k)$, $O_n$, $SO_n$ and finite groups) every representation is completely reducible. Okay, but is it known, in which way it is - i.e. which direct sum of $V_i$'s?
And what about the non-linear-reductive groups? I guess there are groups that have neither an irreducible representation nor a are linear-reductive - is something known about them?
I hope I am not asking to much, and I am rather just hoping for a reference like: "Did you forget about theorem X", which answers this part of your question? And I hope this is a valid stack-exchange question. Thanks a lot for your advices.
(G-stable is meant with respect to the action of the l.a.group $G$ on $V$. $k$ is an algebraically closed field, $\text{char} k =0$, in general for simplicity. A l.a.group is called linear-reductive if every G-representation is completely reducible) Addition: V is a finite-dimensional vector space over the field k, and for the G-representation, $\rho: G \to GL(V)$ has to be a homomorphism of linear algebraic groups.