I am trying to get a “picture” of $\mathcal O_0^\mathfrak p$, so maybe you could correct what I think to have understood:
Let's consider $\mathfrak{sl}_3$, call the simple reflections $s, t$ and the longest element $w$. Let's consider the parabolic subalgebra given by $I=\{t\}$. The coset representatives of minimal length are $W^\mathfrak p=\{e, s, st\}$. The simple modules in $\mathcal O_0^\mathfrak p$ are those simples with highest weight $w.0$ such that $tw > w$; these are $L(e), L(s)$ and $L(st)$. What are Verma modules and projective covers in $\mathcal O_0^\mathfrak p$? These are the maximal quotients of the ordinary Verma modules and projective covers that contain only the composition factors listed above.
By taking the “forbidden” composition factors and modding out the submodule they generate (marked like that), I think I should get:
M(e): L(e) L(s)L(t)L(st)L(ts)L(w)M(s): L(s) L(st)L(ts)L(w)M(st): L(st)L(w)
and for the remaining projective covers, the following composition factors remain:
P(s): L(s)
L(st) L(e)
P(st): L(st)
L(s)
Are these indeed the images of these modules in $\mathcal O_0^\mathfrak p$? I am astonished that neither the inclusions of Verma modules nor the inclusions of projectives remain inclusions. Furthermore, I doubt that $P^\mathfrak p(s)$ has a non-simple socle.
You see, I haven't got a good picture of what's happening here, so I appreciate any hints for introductory texts on how to work with $\mathcal O_0^\mathfrak p$.