All Questions
Tagged with geometric-representation-theory lie-algebras
18 questions
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
8
votes
1
answer
295
views
What are twisted Verma modules? Basic properties?
Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbf{C}$, $\lambda$ be a weight in the nonnegative Weyl chamber and $w_1,w_2\in W$. Then the twisted Verma modules are defined e.g. in Andersen and ...
- 5,701
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
6
votes
0
answers
173
views
Equivalence between $\mathcal{D}_\lambda$ modules and $\mathcal{D}_{0}$ modules
Fix $G$ a finite dimensional reductive group and $\lambda$ a weight. Apparently the category of $\mathcal{D}_\lambda$ modules on $G/B$ is equivalent to the category of $\mathcal{D}_0$ modules on $G'/B'...
- 5,701
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
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
3
votes
3
answers
264
views
Nontrivial Poisson relations for affine Poisson algebras
Let $A$ be a polynomial algebra over a field of characteristic $0$ in the variables $x_1,\dots,x_n$. Consider polynomials $f_1,\dots,f_m\in A$ and let $I$ be the ideal they generate in $A$. Moreover, ...
- 359
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
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 $\mathbf{w}:$
$$\Phi_{\sigma}:\mathfrak{M}_\zeta(Q,\mathbf{v}...
- 1,677
3
votes
0
answers
176
views
Double loop groups and cohomology
Let $G$ be a connected reductive group over $\mathbb{C}$ of Lie algebra $\mathfrak{g}$.
What is the value of $H^{3}(\mathfrak{g}((t))((s)),\mathbb{C})$?
- 3,472
2
votes
2
answers
959
views
Around the socle filtration of a Verma module
Work in the context of a finite dimensional simple Lie algebra over $\mathbb{C}$.
Write $W$ for the Weyl group and $\leq$ for the Bruhat order. For $w\in W$ let $\Delta_w$ denote the Verma module of ...
- 1,595
2
votes
1
answer
144
views
Characterisation of even nilpotent elements in $\mathfrak{sl}_n$
Is there a ''nice'' classification of even nilpotent elements in $\mathfrak{sl}_n,$ using the correspondence between nilpotent elements and partitions of n? By an even element, I mean an element $e$, ...
- 1,677
1
vote
1
answer
324
views
Nakajima quiver varieties for ADE quiver with one dimensional framing
Let $Q$ be a quiver of type $ADE$, $I$ is the set of vertices of $Q$. Let $\mathfrak{M}({\mathbf{v}},{\mathbf{w}})$ be a Nakajima quiver variety for such quiver (here ${\mathbf{v}}=(v_i)_{i \in I}$ is ...
- 163
1
vote
0
answers
79
views
Extension of a type A Springer fibre
Given a decomposition $p=(p_1,\dots,p_n)$ of $n$, one can associate its corresponding
partial flag variety $$\mathcal{B}_p=\{F=(0=F_0\subset F_1\subset \dots \subset F_n=\mathbb{C}^n) \mid \dim F_i/F_{...
- 1,677
0
votes
1
answer
205
views
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
-1
votes
1
answer
301
views
Is $\mathbb{P}^1$ the only smooth projective curve with a locally split tangent lie algebroid?
Let $C$ be a smooth projective curve over an algebraically closed field $k$.
The tangent lie algeborid $\mathcal{T}_C$ of $C$ is just sheaf of vector fields on $C$ equipped with the usual lie ...
- 7,789