About G-modules with good filtrations

Question

Let $k$ be an algebraically closed field of positive characteristic, and let $G$ be a reductive algebraic group over $k$ (for instance a classical group).

Let $V$ be a (rational) $G$-module. We say that $V$ admits a good filtration [see for instance Jantzen, Representations of Algebraic Groups, §4.19] if there exists an ascending chain of $G$-modules $0=V_0 \subset V_1 \subset V_2 \subset ... \subset V$ such that $V=\cup V_i$ and, for each $i \geq 1$, the quotient $V_i/V_{i-1}$ is isomorphic to some $H^0(\lambda) \otimes N_i$, where $N_i$ is a (non necessary finite dimensional) vector space on which $G$ acts trivially, and $H^0(\lambda)$ is the usual induced module $ind_B^G(k_{-\lambda})$, $B \subset G$ being a Borel subgroup, $\lambda$ a dominant weight, and $k_{-\lambda}$ the one dimensional representation of $B$ corresponding to $-\lambda$.

My questions are the following:

1) If $M_1$ and $M_2$ are two $G$-submodules of $M$ such that $M_1,M_2$, and $M$ admit a good filtration, does $M_1 \cap M_2$ also admit a good filtration?

2) If $M$ is a $G$-module with a good filtration and if $N \subset M$ is any $G$-submodule, does there exist a unique minimal $G$-submodule $N'$ such that $N \subset N' \subset M$ and $N'$ admits a good filtration?

Answer 1

The answer to both questions is no.

For 1) let $V=H^0(\lambda)$ with $\lambda$ dominant, $U\leq V$ any submodule that doesn't have a good filtration and $\Delta(U)=\{(u,u): u\in U\}\leq V\oplus V$. Take $M_1=V\oplus0\leq (V\oplus V)/\Delta(U)$, $M_2=0\oplus V\leq(V\oplus V)/\Delta(U)$, and $M$ an injective hull of $(V\oplus V)/\Delta(U)$. Then $M_1\cap M_2\cong U$ doesn't have a good filtration.

For 2) take the same example, with $N=M_1\cap M_2$. Then $M_1$ and $M_2$ are both minimal submodules of $M$ with good filtrations that have $N$ as a submodule.