The space that you seek is the two-sided ideal in $\mathbb C[G]$ generated by the character of $\pi$ (see details below).
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 $$ \epsilon_\pi = \sum_{g\in G} \frac{d_\pi}{|G|}\chi_\pi(g^{-1})g, $$ where $\chi_\pi$ is the character of $\pi$. See for example Theorem 1.7.9 in my book, the relevant part of which is also available online here (see Theorem 7.9).