This is from Anton Deitmar's "Principles of Harmonic Analysis":
A vector $\xi ∈ V_{\pi}$ is called square integrable if $D_{\xi} = V_{\pi}$, i.e., if $W_{\xi}(\eta): x \mapsto \langle \eta,\pi(x)\xi\rangle$ is square integrable on $G$ for every $\eta ∈ V_{\pi}$. We put $D_{\pi}$ = {$\xi \in V_{\pi} : \xi$ is square integrable}. It is easily checked that $D_{\pi}$ is a $\pi(G)$-invariant linear subspace of $V_{\pi}$.
I am struggling to figure out why it is clear that $D_{\pi}$ is $\pi(G)$-invariant and any help would be much appreciated.