Question on standard filtration Ex 3.7(a) James Humphreys "Representations of Semisimple Lie Algebras in the BGG Category O" Exercise 3.7(a) James Humphreys's "Representations of Semisimple Lie Algebras in the BGG Category O"
Let $V \in \mathcal{O}$ be a module which admits a standard filtration. Suppose that there is a surjective homomorphism $$\phi: V \rightarrow  M(\mu)$$ where $M(\mu)$ is a Verma module. Show that ker$\phi$ also admits a standard filtration.
My attempt: Suppose that $\lambda$ is maximal among the weights of $V$ and let $M(\lambda)$ be the corresponding submodule. 
If $\mu \neq \lambda$, then the composition of the embedding $ M(\lambda) \rightarrow V$ and $ \phi: V \rightarrow  M(\mu)$ yields the composition $$M(\lambda) \rightarrow V \rightarrow M(\mu)$$ which is $0$ by the maximality assumption of $\lambda$. Hence, we have the surjection $$V/M(\lambda) \rightarrow  M(\mu)$$ and since the quotient module $V/M(\lambda)$ also admits a standard filtration (I am guessing we can argue by induction on the length of $V$ but I am a bit confused about this. Can anyone show me this?).
Furthermore, if $\mu = \lambda$, then $\operatorname{ker} \phi \cong V/M(\lambda)$ (is my claim true? why is this so?) $\operatorname{ker} \phi$ admits a standard filtration since $V/M(\lambda)$ has a standard filtration.
 A: This will go much smoother if you say: we can assume that our standard filtration is chosen so that the weights are non-decreasing.  That is, $\{0\}\subset V_1\subset V_2\subset \cdots \subset V$ satisfies $V_{i}/V_{i-1}\cong M(\lambda_i)$ with $\lambda_i> \lambda_j$ implies $i<j$.  (This is because unless $\lambda_i>\lambda_{i+1}$, we have $V_{i+1}/V_{i-1}\cong M(\lambda_i)\oplus M(\lambda_{i+1})$, so we can rearrange the filtration to reorder these weights.)  I assume this is proven somewhere in Humphreys.  Arguments with standard filtrations tend to work much better once you make sure they are in this form.
Let $k$ be maximal such that $V_k$ is killed by the map to $M(\mu)$.  Then, we have a surjective map $V/V_k\to M(\mu)$.  Since the map $V_{k+1}/V_k\to M(\mu)$ isn't zero, we must have $\lambda_{k+1}\leq \mu$.  If $\lambda_{k+1} <\mu$, then the map $V_{k+1}/V_k\to M(\mu)$ isn't surjective, and so $\mu$ must appear as a weight of $M(\lambda_m)$ for $m>k+1$.  This is impossible, since $\lambda_m\geq \mu>\lambda_{k+1}$ implies $m<k+1$.  Thus, we must have $\lambda_{k+1}=\mu$, and $V_{k+1}/V_{k}\to M(\mu)$ is an isomorphism.  
Thus, the quotients $(V_{m}\cap\ker \phi)/(V_{m-1}\cap\ker\phi)\cong V(\lambda_m)$ as long as $m\neq k+1$, and is 0 when $m=k+1$.  Thus, these intersections $V_{m}\cap\ker \phi$ give the desired filtration.
