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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.

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Well, that depends. How are you given $V$? – Qiaochu Yuan Jul 12 '12 at 19:20
    
By $G$-representation I mean, that $\rho: G \to GL(V)$ is a homomorphism of linear algebraic groups where V is a finite-dimensional $k$ vector space – Suedklee Jul 12 '12 at 19:53
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I know that. I can't tell you anything about how to decompose $V$ until you tell me how you found it. Depending on how you found it this could be an open problem in general even for $\text{GL}_n$. – Qiaochu Yuan Jul 12 '12 at 22:18
    
What do you mean by "how you found it". Sounds like for each given $G$ one would have to guess a representation? Do I have to choose something? – Suedklee Jul 13 '12 at 6:57
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You have to choose a representation. Perhaps I am misunderstanding your question. I read your question as "how do I know what irreducible representations a given representation $V$ is a direct sum of?" My answer is "it depends on how you are given $V$." Does this make sense? – Qiaochu Yuan Jul 13 '12 at 7:16

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