Let  a compact  topological group  $G$  with invariant  measure $\mu,$ acts on  a simply  connected topological  space $X$ and $\rho$ is  a unitary irreducible   representation of $G$. Let $f:X\to  \mathbb{C}^{n}$ be  a  continuous  function.

>Is it true to say that there  exist $x_{0}\in X$  such that the  following integral is  non invertible:

$$ \int_{G}  \rho(g)f(g.x_{0}))d\mu  $$

This  question is  motivated by
http://mathoverflow.net/questions/176607/integrals-of-representations-over-geodesics#comment458696_176607

and  proposition $3$ of
http://arxiv.org/abs/1110.0091