For every integer $k\geq 1$, recall that $$P_k = \{v\in\mathcal{H}\cap V_\mathbb{Q} \text{ such that } kv\in V_\mathbb{Z}\}.$$ Show that the set $P_k$ is invariant under $\Gamma$.
For every integer $k\geq 1$, recall that
$$P_k = \{v\in\mathcal{H}\cap V_\mathbb{Q} \text{ such that } kv\in V_\mathbb{Z}\}.$$
Show that the set $P_k$ is invariant under $\Gamma$.