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Theorem Let $G$ and $H$ be compact groups. Let $ρ$ be a finite dimensional irreducible continuous representation of $G×H$ over the field of complex numbers. Then $ρ$ is a tensor product of irreducible representations of $G$ and $H$.

In his book L'integration dans les groupes topologiques, Weil proved this theorem under more general conditions. His proof was short and elementary. He used no functional analysis. On the other hand, Pontryagin proved the same theorem using the Peter-Weyl theorem in his famous book.

I was puzzled. Is Weil's proof correct?

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up vote 9 down vote accepted

I don't know either proof, but you don't need any analysis to prove this. Here is a short proof:

Since $\rho$ is finite-dimensional, once restricted to $G = G\times 1,$ we can find an irreducible $G$-subrep'n $\psi$. Consider $Hom_G(\psi,\rho)$; this is a finite-dimensional $H = 1\times H$-representation. Let $\chi$ be an irreducible $H$-subrep'n. Then evaluation gives a non-zero map $\psi \otimes \chi \hookrightarrow \rho,$ which must be an isomorphism, since the source and target are both irreducible.

I would guess that Weil's proof is similar, if it is short and general.

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@Jason: Dear Jason, There is a typo (and thanks for drawing my attention to it). I meant to write "we can find an irreducible $G$-subrep'n $\psi$''. Correction to ensue! Regards, –  Matt E Apr 24 '12 at 2:23
    
Thanks. Let me think about your proof. It may take a while. –  Makoto Kato Apr 24 '12 at 3:55
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Probably better to say "evaluation gives a nonzero map...". It seems to me that you only learn that it is an injection at the same time that you learn that it is an isomorphism. –  David Speyer Apr 24 '12 at 4:28
    
@David: Dear David, A reasonable remark; I'll make an edit! Thanks, and best wishes, –  Matt E Apr 24 '12 at 11:10

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