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.
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.