Recall the following theorem due to Burnside:Let $G$ be a finite group and let $V$ be its irreducible complex representation of dimension greater than 1, then the character
of this representation is $0$ on some element of $G$. Is this statement still correct
if $G$ is any compact Lie group? Thanks.



Is there a generalization of Burnside's theorem for compact Lie groups?
The answer is yes--use the Weyl character formula, for example.
See: [Patrick X. Gallagher, Zeros of group characters. Math. Z. Volume 87 (1965), Number 3.](http://www.springerlink.com/content/l2m14425071477x3/)


