# Understanding category $\mathcal O^\mathfrak p$

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 $\mathcal O_0^\mathfrak p$.

• Hopefully I will have time to give a more full answer later. For now I recommend looking at the short description in my paper with Mazorchuk arxiv.org/abs/1506.07008 as well as the references we give there (this is certainly not the best place to start, but it was one I could find without spending time searching right now). – Tobias Kildetoft Apr 26 '17 at 12:44
• Thank you for your link! I think what I understood when asking agrees with what I understand from §2.4 in your article. I have tried to what I think how I obtain $Z^\mathfrak p M(w)$ from $M(w)$. But still, this doesn't look like the correct modules. – Bubaya Apr 26 '17 at 14:44
• I will need to take a closer look when I am back on a computer, but at first glance these looks ok. The Zuckerman functors are certainly not exact, though whether these are the left- or right exact ones is something I don't know off-hand. I would guess that they are the right exact ones since these are the modules with "important" quotients which should be preserved. I should also note that the derived functors of the Zuckerman functors have been studied and used a fair bit by Mazorchuk among others, so once I can look up references I should be able to find something about this. – Tobias Kildetoft Apr 26 '17 at 17:29
• You can find some stuff about $\mathcal O ^{\mathfrak p}$ in Humphreys: Representations of Semisimple Lie Algebras in the BGG Category $\mathcal O$, chapter 9. – Rafael Mrđen Apr 26 '17 at 21:39
• @rafaelm Thank you. Indeed example 9.5 addresses the Vermas of the 𝔰𝔩₃-case, which seem to be the modules I suggested. But am still uncertain about the projectives. Humphreys states that there is a parabolic Version of BGG reciprocity, which is not the case for the projectives I proposed. What is their correct shape, and how do I obtain it? – Bubaya Apr 27 '17 at 8:56