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 yesuse the Weyl character formula, for example. See: Patrick X. Gallagher, Zeros of group characters. Math. Z. Volume 87 (1965), Number 3. 


$\mathbb{C}$
) of a connected semisimple compact Lie group. Why would an answer be interesting? There are similarities with the finite group situation, including orthogonality relations, but also some big differences. As far as I can see, the proof for finite groups doesn't readily translate to compact Lie groups. The answer could well be yes (or no), but does it have implications? – Jim Humphreys May 28 '10 at 22:21