All Questions
12 questions
0
votes
1
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205
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What's an example of an $n$-step nilpotent Lie algebra whose centre is not $g^{(n)}$?
Let $\mathfrak g$ be a Lie algebra. $\mathfrak g^{(1)}=\mathfrak g$ and $\mathfrak g^{(n+1)}=[\mathfrak g,\mathfrak g^{(n)}]=\mathbb R$-span$\{[X,Y]:X\in\mathfrak g,Y\in\mathfrak g^{(n)}\}$. The ...
- 275
3
votes
0
answers
58
views
Locally finite positive energy modules generated by singular vectors at positive levels?
This is question is about whether or not certain modules for an affine Lie algebra are generated by their singular vectors. I begin with some background.
Backround on affine Lie algebras. Let $\...
user108998
4
votes
1
answer
348
views
Verma modules and Borel–Weil
Let $\mathfrak{g}$ be a semisimple Lie algebra and fix a root system. Let $\mathfrak{b}:=\mathfrak{h}\oplus\bigoplus_{\alpha\in R^+}\mathfrak{g}_\alpha$. The complex irreducible representation of $\...
- 181
4
votes
1
answer
189
views
Equivalence of categories of modules over $U(\mathfrak{g})$ on which $Z(\mathfrak{g})$ acts by central characters
$\newcommand{\g}{\mathfrak{g}}$Setting: $\mathfrak{g}$ is a semisimple complex Lie algebra. Here $\chi_\lambda$ denotes the central character corresponding to the action of $Z(\g)$ on a highest weight ...
- 311
1
vote
1
answer
324
views
Nakajima quiver varieties for ADE quiver with one dimensional framing
Let $Q$ be a quiver of type $ADE$, $I$ is the set of vertices of $Q$. Let $\mathfrak{M}({\mathbf{v}},{\mathbf{w}})$ be a Nakajima quiver variety for such quiver (here ${\mathbf{v}}=(v_i)_{i \in I}$ is ...
- 163
3
votes
0
answers
126
views
Nakajima reflection functors and the flavour/framing group action
Nakajima has constructed so-called reflection functors that are isomorphisms between different quiver varieties that have the same framing $\mathbf{w}:$
$$\Phi_{\sigma}:\mathfrak{M}_\zeta(Q,\mathbf{v}...
- 1,677
2
votes
1
answer
144
views
Characterisation of even nilpotent elements in $\mathfrak{sl}_n$
Is there a ''nice'' classification of even nilpotent elements in $\mathfrak{sl}_n,$ using the correspondence between nilpotent elements and partitions of n? By an even element, I mean an element $e$, ...
- 1,677
5
votes
2
answers
491
views
Kostant's $G$-invariant part in the sym power ring of adjoint representation?
Let $g$ be a Lie algebra, say $sl_n(\mathbb C)$. It is considered as the adjoint representation of $G=SL_n(\mathbb C)$.
A famous theorem of Kostant from "Lie Group Representations on Polynomial ...
- 3,804
40
votes
1
answer
4k
views
Roadmap to Geometric Representation Theory (leading to Langlands)?
I believe there has been at least one question similar to this one and yet I still think this particular question deserves to have a thread of its own.
I'm becoming increasingly fascinated by stuff ...
- 7,789
8
votes
2
answers
743
views
Confusion about Subcategories of Category $\mathcal{O}$
So, in learning about category $\mathcal{O}$ representations of a semisimple Lie algebra $\mathfrak{g}$, I've come across two natural kinds of subcategories, and I think I'm confused about their ...
- 183
7
votes
0
answers
167
views
How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in the framework of D-modules?
Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be ...
- 9,019
2
votes
2
answers
959
views
Around the socle filtration of a Verma module
Work in the context of a finite dimensional simple Lie algebra over $\mathbb{C}$.
Write $W$ for the Weyl group and $\leq$ for the Bruhat order. For $w\in W$ let $\Delta_w$ denote the Verma module of ...
- 1,595