Questions tagged [category-o]

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

Filter by
Sorted by
Tagged with
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 𝒪...
alerouxlapierre's user avatar
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. ...
Estwald's user avatar
  • 1,027
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 $...
Fan Zhou's user avatar
  • 181
10 votes
1 answer
353 views

Basic algebra of $\mathcal{O}_0(\mathfrak{sl}_n(\mathbb{C}))$ — 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 ...
Rida Saabna's user avatar
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 ...
Fan Zhou's user avatar
  • 181
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 ...
KKD's user avatar
  • 463
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 \...
KKD's user avatar
  • 463
4 votes
0 answers
122 views

Questions to the proof of Proposition 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 Cartan subalgebra $\mathfrak{h}$, root system $\Phi \subset \mathfrak{h}^*$ and Weyl group $W$. Fix a set of positive roots $\Phi^+...
KKD's user avatar
  • 463
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{...
KKD's user avatar
  • 463
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 ...
KKD's user avatar
  • 463
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 ...
KKD's user avatar
  • 463
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 $\...
KKD's user avatar
  • 463
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 ...
KKD's user avatar
  • 463
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 ...
Zhaoting Wei's user avatar
  • 8,647
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 $\...
SamJeralds's user avatar
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 über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), 421-445,...
userabc's user avatar
  • 667
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 ...
Rafael Mrden's user avatar
  • 1,358
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}...
Julian Kuelshammer's user avatar
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 ...
Steven's user avatar
  • 159
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....
Zhaoting Wei's user avatar
  • 8,647
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 $$Hom(A, S(B)) \cong Hom(B, A)^*$$ ...
Clemens Koppensteiner's user avatar
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 ...
Hu Jun's user avatar
  • 71
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' ...
George Melvin's user avatar
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 (...
George Melvin's user avatar