# Questions tagged [category-o]

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

24
questions

5
votes

0
answers

104
views

### On Soergel's results concerning projectives modules in category $\mathcal{O}$

I am looking for a translated proof of two results of Soergel usually referred to as endomorphismensatz and struktursatz.
Both of those results were shown in the paper
Soergel, W. (1990). Kategorie đť’Ş...

6
votes

1
answer

348
views

### An alternative form of the Kazhdan-Lusztig conjecture

Fix a complex semisimple Lie algebra $\mathfrak{g}$. Denote by $W$ the corresponding Weyl group, with length function $\ell$ and Bruhat order $\leq$. Let $\lambda$ be an integral anti-dominant weight. ...

7
votes

0
answers

196
views

### Twisted D-module structure on pushfoward of structure sheaf of Bruhat cell

Apologies for the basic question, but my experience on math.stackexchange tells me that this will go unanswered there.
Background: In the principal block, the dual Verma modules (with highest weight $...

10
votes

1
answer

353
views

### Basic algebra of $\mathcal{O}_0(\mathfrak{sl}_n(\mathbb{C}))$ â€” Reference request

It is well known that the principal block $\mathcal{O}_0$ of the BGG category $\mathcal{O}$ of a semisimple Lie algebra is equivalent to the category of finitely generated modules over a certain ...

4
votes

1
answer

191
views

### Spectral sequence from standard/Verma filtration/flag to compute Lie algebra cohomology of tensor product with respect to $\mathfrak{n}$

I'm not sure this question fully qualifies as a research-level math question, but from my (limited) past experience on stackexchanged I feared this question might not get an answer there.
Setting: the ...

1
vote

0
answers

113
views

### Kazhdan-Lusztig Conjecture over non-algebraically closed field

Let $G$ be a split connected semi-simple (or reductive) algebraic group over a (non-archimedean) field $k$ of characteristic zero. Denote by $\mathfrak{g}=\mathrm{Lie}(G)$ the semi-simple (or ...

1
vote

2
answers

219
views

### Computing kernel in the category $\mathcal{O}$

Let $\mathfrak{g}=\mathfrak{gl}_3$ over $\mathbb{C}$ with positive roots
\begin{equation*}
\Phi_+=\{\alpha_1=(1,-1,0),\alpha_2=(1,0,-1),\alpha_3=(0,1,-1)\}.
\end{equation*}
Consider the morphism
\...

4
votes

0
answers

122
views

### Questions to the proof of Proposition 9.3 in Humphreys â€śRepresentations of Semisimple Lie algebras in the BGG Category $\mathcal{O}$"

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ with Cartan subalgebra $\mathfrak{h}$, root system $\Phi \subset \mathfrak{h}^*$ and Weyl group $W$. Fix a set of positive roots $\Phi^+...

5
votes

1
answer

435
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

79
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

338
views

### Morphism of Verma modules

$\DeclareMathOperator\Hom{Hom}$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 ...

2
votes

1
answer

138
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 $\...

4
votes

1
answer

244
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 ...

10
votes

3
answers

592
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

209
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 $\...

3
votes

0
answers

337
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 ĂĽber den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), 421-445,...

7
votes

0
answers

203
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 ...

8
votes

0
answers

243
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

753
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

734
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....

4
votes

2
answers

437
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
$$Hom(A, S(B)) \cong Hom(B, A)^*$$
...

7
votes

0
answers

308
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 ...

5
votes

1
answer

299
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' ...

2
votes

1
answer

242
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 (...