Questions tagged [category-o]

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

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10
votes
3answers
418 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 ...
7
votes
0answers
175 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
0answers
206 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
1answer
638 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 ...
6
votes
0answers
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 ...
5
votes
1answer
241 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' ...
5
votes
0answers
124 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 $\...
4
votes
2answers
533 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
2answers
348 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 $$Hom(A, S(B)) \cong Hom(B, A)^*$$ ...
3
votes
1answer
101 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 ...
3
votes
1answer
143 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
1answer
125 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 $\...
2
votes
1answer
205 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 (...
2
votes
0answers
191 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,...