# Questions tagged [category-o]

Questions related to the Bernstein-Gelfand-Gelfand category O and generalizations

7
questions with no upvoted or accepted answers

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### Correspondence between Verma module morphisms and invariant differential operators - is it exact?

For a (complex, connected, simply connected, simple) Lie group $G$, a parabolic subgroup $P \subseteq G$, and a $\mathfrak g$-integral $\mathfrak p$-dominant weight $\lambda$, we can construct ...

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### Induction from the Borel subalgebra to BGG category $\mathcal{O}$

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with a choice of Cartan subalgebra $\mathfrak{h}$, Borel $\mathfrak{b}$, and nilpotent radical $\mathfrak{n}$. Let $\mathcal{O}...

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### A question on the resolution of parabolic Verma module $M_I(\lambda)$ in BGG category O

I am reading Humphrey's book "Representations of semisimple Lie algebras in the BGG Category O" on Page 189, Proposition 9.6, where he remarked that "Note that if we had developed the full BGG ...

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### Motivation and Difference of Category O Definition for Kac-Moody Algebras

My first encounter with Category $\mathcal{O}$ was (perhaps unusually) learning about Kac-Moody algebras from Kac's book. Kac takes the following definition:
The Category $\mathcal{O}$ has objects $\...

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### Technics for computing weights of kernel, image or cokernel in category $\mathcal{O}$

As in the title I looking for technics to compute weights of kernels, images or cokernels in category $\mathcal{O}$ besides checking everything directly by hand.
To be more concrete, consider the ...

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### Translation of Soergel's 1990 paper on category O

Is there any English translation for the folowing paper of Soergel?
Kategorie $\mathcal{O}$, perverse Garben, und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), 421-445,...

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### Sufficent condition for being an object of $\mathcal{O}^{\mathfrak{p}}$

Let $G$ be a Lie group with simple roots $\Delta$ and Weyl group $W$. For $I \subset \Delta$ let $P=P_I \subset G$ be the corresponding (standard) parabolic subgroup. Denote by $\mathfrak{g}=\...