The space that you seek is the two-sided ideal in $\mathbb C[G]$ generated by the character of $\pi$.

This follows from the explicit Wedderburn decomposition of $\mathbb C[G]$. If we write
$$
\mathbb C[G] = \bigoplus_{\pi \in \hat G} M_{d_\pi}(\mathbb C),
$$
then the identity element of the $\pi$th summand is 
$$
\epsilon_\pi = \frac{d_\pi}{|G|}\chi_\pi,
$$
where $\chi_\pi$ is the character of $\pi$. See for example Theorem 1.7.9 in [my book][1], the relevant part of which is also available online [here][2] (see Theorem 7.9).


  [1]: http://www.imsc.res.in/~amri/rtcv/
  [2]: http://www.cimpa-icpam.org/IMG/pdf/RepresentationTh.pdf