# Questions tagged [category-o]

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

17
questions

**1**

vote

**1**answer

131 views

### 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}=\...

**5**

votes

**1**answer

237 views

### Questions to the proof of Lemma 9.3 in Humphreys “Representations of Semisimple Lie algebras in the BGG Category $\mathcal{O}$”

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ with root system $\Phi$, Weyl group $W$ and Cartan decomposition $\mathfrak{g}=\mathfrak{h}\oplus \bigoplus_{\alpha \in \Phi} \mathfrak{...

**4**

votes

**0**answers

51 views

### 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 ...

**3**

votes

**2**answers

217 views

### Morphism of Verma modules

I'm trying to understand morphism of Verma modules and consider the following example.
PART 1:
Consider $\mathfrak{g}$=$\mathfrak{gl}_3$ over $\mathbb{C}$ with positive roots \begin{equation*}\Phi_+=\{...

**2**

votes

**1**answer

129 views

### Checking axiom of Category $\mathcal{O}$

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $G$ be a split connected reductive algebraic group over $K$ with Borel $B$. We have the associated Lie algebras $\mathfrak{g}=$Lie$(G)$ and $\...

**3**

votes

**1**answer

106 views

### BGG Category $\mathcal{O}$ is not closed under extension

What is the reason for the BBG category $\mathcal{O}$ failing to be closed under extensions i.e which of the 3 axioms of $\mathcal{O}$ does not hold under taking extensions?
Is there a prototype of ...

**4**

votes

**2**answers

537 views

### What are the “tensor-closed” object of the BGG category $\mathcal{O}$ of a semisimple Lie algebra $\mathfrak{g}$?

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra and we can consider its BGG category $\mathcal{O}$. It is well-known that $\mathcal{O}$ is not closed under tensor product, i....

**10**

votes

**3**answers

424 views

### Is the tensor product of two infinite dimensional objects in the BGG category $\mathcal{O}$ of a semisimple Lie algebra always not in $\mathcal{O}$?

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra (according to a comment of Victor Ostrik, we need to further require that $\mathfrak{g}$ is simple) and we can consider its ...

**5**

votes

**0**answers

130 views

### 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 $\...

**2**

votes

**0**answers

197 views

### 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,...

**7**

votes

**0**answers

177 views

### 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 ...

**7**

votes

**0**answers

208 views

### 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}...

**6**

votes

**1**answer

641 views

### Socle of tilting modules in the BGG category $\mathcal{O}$ over a semisimple Lie algebra

Suppose that $\mathfrak{g}$ is a finite dimensional, complex, semisimple Lie algebra. Let $\mathcal{O}$ be the BGG category over $\mathfrak{g}$.
Tilting module theory play an important role in the ...

**4**

votes

**2**answers

353 views

### Serre functor of a subcategory (in particular parabolic category O)

For an additive category $\mathcal C$ there is the notion of a Serre functor on $\mathcal C$, i.e. a an autoequivalence $S$ of $C$ such that there exist isomorphisms
$$Hom(A, S(B)) \cong Hom(B, A)^*$$
...

**6**

votes

**0**answers

295 views

### 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 ...

**2**

votes

**1**answer

206 views

### Springer Action on Centre of Parabolic Category O (after Brundan)

I recently learned of a result of Brundan describing the centre of the regular block of parabolic category $\mathcal{O}$ for $\mathfrak{gl}_{n}$ as the cohomology of a corresponding Springer fibre (...

**5**

votes

**1**answer

243 views

### Endomorphisms in Category O and Schubert Classes

Let $\mathfrak{g}\supset \mathfrak{b}\supset \mathfrak{h}$ be a complex semisimple Lie algebra, with choice of Borel and Cartan subalgebras, $W$ the Weyl group.
W. Soergel's 'Endomorphismensatz' ...