8
votes
Socle of tilting modules in the BGG category $\mathcal{O}$ over a semisimple Lie algebra
EDIT: Following a conversation with Ivan Losev, the situation is clearer now. Consider the principal block of $\mathcal{O}$. Recall two facts:
1) the socle of any Verma module is $L_{w_0}$,
2) ...
- 5,867
5
votes
Is the tensor product of two infinite dimensional objects in the BGG category $\mathcal{O}$ of a semisimple Lie algebra always not in $\mathcal{O}$?
The answer to the question here is yes. (More generallly, if $M$ and $N$ lie in $\mathcal(O)$ and are both infinite dimensional, then $M \otimes N \notin \mathcal{O}$.)
The proof is eaiest to ...
- 52.4k
4
votes
Accepted
Spectral sequence from standard/Verma filtration/flag to compute Lie algebra cohomology of tensor product with respect to $\mathfrak{n}$
I think the issue here is that the subquotients in the standard filtration have weights (your $\lambda_i$'s) which are ordered the other way. To be clear, if the weights of $\nu_0$, ..., $\nu_n$ of $L$...
- 482
4
votes
Accepted
BGG Category $\mathcal{O}$ is not closed under extension
You can usually extend two modules from $\mathcal{O}$ by a module which is not semisimple for the Cartan subalgebra (i.e. fails to be a weight module). See Exercise 3.1. in [J. E. Humphreys, ...
- 1,368
3
votes
Accepted
An alternative form of the Kazhdan-Lusztig conjecture
Let $G$ be the simply-connected Lie group with Lie algebra $\mathfrak g$, let $B\subset G$ be the Borel subgroup, and let $X=G/B$ be the flag variety. There are equivalences $$\mathrm{Mod}_f(\mathfrak ...
- 1,897
3
votes
Accepted
Basic algebra of $\mathcal{O}_0(\mathfrak{sl}_n(\mathbb{C}))$ — Reference request
You can find quiver and relations (not sure if they are admissible always) here: http://www.math.uni-bonn.de/ag/stroppel/Quivers.pdf
In particular the explicit algebra is only fully understoof for $...
- 26.1k
3
votes
Is the tensor product of two infinite dimensional objects in the BGG category $\mathcal{O}$ of a semisimple Lie algebra always not in $\mathcal{O}$?
Edit: As @ZhaotingWei points out this is wrong. In fact, in my example we seem to have $M\simeq \bigoplus_{n=4}^\infty \Delta(-n\rho)$ which is certainly not in $\mathcal{O}$.
Inspired by @...
- 116
2
votes
Is the tensor product of two infinite dimensional objects in the BGG category $\mathcal{O}$ of a semisimple Lie algebra always not in $\mathcal{O}$?
I will answer the question in the affirmative for $\mathfrak{sl}_n$, and in general provide a property of the Weyl group that would imply an affirmative answer in general. This property is true in ...
- 8,726
2
votes
Accepted
Computing kernel in the category $\mathcal{O}$
Here is a less direct, but shorter, proof using some non-trivial machinery. Denote by $s$, $t$ the simple reflections, and $M_w := M(w \cdot 0)$ where $w \in W$, and $\cdot$ is the "shifted" ...
- 1,368
2
votes
Accepted
Morphism of Verma modules
PART 1:
The element $u$ must have weight $-\alpha_2$, since $\mu = \lambda - \alpha_2.$
In $U(\mathfrak{n^-})$ there are only two linearly independent elements that have such weight (assuming PBW ...
- 8,157
1
vote
Accepted
Checking axiom of Category $\mathcal{O}$
The module $U(\mathfrak{g})\otimes_{U(\mathfrak{b})} N$ is locally $U(\mathfrak{b})$-finite and there is a surjective $U(\mathfrak{g})$-homomorphism $\varphi\colon U(\mathfrak{g})\otimes_{U(\mathfrak{...
- 8,157
1
vote
Computing kernel in the category $\mathcal{O}$
So far I only found the following messy proof. In case I did a mistake or you know a more elegant way it would be a pleasure to know. It seems to me that can be generalised to any "BGG square&...
- 463
1
vote
Accepted
Questions to the proof of Lemma 9.3 in Humphreys "Representations of Semisimple Lie algebras in the BGG Category $\mathcal{O}$"
Thanks to the outstanding help of LSpice I present a version of more detailed proof of the two parts above. Do not hesitate to point out mistakes.
"$(1) \Rightarrow (2)$": Fix $\alpha \in I$ ...
- 463
1
vote
Morphism of Verma modules
For part 2:
With
\begin{equation*}
y_{\alpha_1}=\begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix},y_{\alpha_2}=\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 &...
- 463
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