Let $P$ be a pro-p group. Assume that there is a filtration of $P$ by normal subgroups $P_i$ such that $P_0=P$ and $P_{i+1} < P_i(i\in\mathbb N)$. Let $V$ be an $l$-adic representation of $P$, where the prime $l$ is prime to $p$.

Question: Is there a direct sum decomposition $V=\bigoplus_{i\geq 0,i\in\mathbb Z}V_i$ by $P$-submodules such that the $P_r$-fixed part is given by $V^{P_i}=\bigoplus_{i\leq r} V_i$ ?