Let $K=\mathbb F_{p^k}$ a finite field, and $V$ a vector space on $K$. Clearly $PSL(V)=SL(V)/SL(V)\cap Z$ acts on $V$ by the following rule ($Z$ is the subgroup of the scalar functions):
$$v^{(SL(V)\cap Z)\gamma}:=\gamma(v)$$
this action is well defined, and if $W$ is a proper subspace of $V$
$$N_{PSL(V)}(W):=\{x\in PSL(V)\,:\, w^x\in W\quad\forall w\in W\}$$
Such normalizers are the maximal parabolic subgroups. Now I have the following problem that is making me go crazy:
If $T$ is a maximal parabolic subgroup in $PSL(V)$ and $O_p(T)$ is its $p$-radical (i.e. the maximum $p$-supgroup in $T$ that is normal), show that $$C_T(O_p(T))\le O_p(T)$$
The above statement is equivalent to saying that every maximal parabolic subgroup of $PSL(V)$ has characteristic $p$.
If $Q$ is the $p$-radical of $N_{GL(V)}(W)$, up to this moment I've showed only that $C_T(O_p(T))\le C_{PSL(V)}(Q(SL(V)\cap Z)\big/(SL(V)\cap Z))$.
I've found nothing in literature about the above problem.
Many thanks in advance.