Let  a compact  topological group  $G$  with invariant  measure $\mu,$ acts on  a simply  connected  compact topological  space $X$ and $\rho$ is  a  $n$-dimensional unitary   representation of $G$. Let $f:X\to  M_{n}(\mathbb{C})$ be  a  continuous  function.  Under what of the  following two  conditions we can say:

>There  exist $x_{0}\in X$  such that the  following integral is  non invertible:

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


**Condition 1:** $G$ is  abelian and $\rho$ is  a  non trivial representation.

**Condition 2:** $\rho$ is an irreducible  representation. 

This  question is  motivated by this [post](http://mathoverflow.net/questions/176607/integrals-of-representations-over-geodesics#comment458696_176607)
and  proposition $3$ of this  [note](http://arxiv.org/abs/1110.0091).