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Let $G_1, G_2$ be two lie groups, $V$ be a finite dimensional (continuous) irreducible complex representation of $G_1 \times G_2$, must $V \cong V_1 \otimes V_2$ for some irreducible representation $V_i$ of $G_i$?

If $G_i$ are compact, this is true by Peter-Weyl theorem.

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    $\begingroup$ if reps are over R, answer is no, even for cpt G. if reps are over C, reduce to f.d. assoc. alg. case where ans. is yes. $\endgroup$ – Peter McNamara Sep 1 '18 at 0:15
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If the field is $\mathbb C$, There are many ways of seeing this. For $i=1,2$ we may replace $G_i$ by its Zariski closure in $GL(V_i)$ without changing the hypotheses or the conclusion. But if an algebraic subgroup $G\subset GL(V)$ is irreducible, then it is reductive (the unipotent radical will have a fixed space which is G invariant and hence zero).

We now use the fact that the reductive group $G_i$ has a Zariski dense compact group $K_i$; we may thus replace $G_i$ by the compact $K_i$ where you have accepted the result.

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