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?



About G-modules with good filtrations
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.


