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.



Irreducible representation of the product of two groups and tensor product
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.


