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.