15
votes
Proofs of Beilinson-Bernstein
In the introduction (section 1.3) of the paper
Ed Frenkel and Dennis Gaitsgory, $D$-modules on the affine Grassmannian and representations of affine Kac-Moody algebras, Duke Math. J. 125(2) (2004) pp ...
- 4,612
14
votes
Proofs of Beilinson-Bernstein
I have sketched several possible proofs in this document:
http://math.mit.edu/~bezrukav/localization_notes.pdf
Please let me know if there are comments or questions.
- 1,526
14
votes
Accepted
Why are VOA characters modular forms (geometrically)?
I'm sure someone here can handwave the intuition behind these statements much better than me. This handwaving version goes like this, one is interested in computing vacuum 1-point functions on a torus,...
- 3,242
12
votes
How should I think about the Grothendieck-Springer alteration?
For Question 1 I agree with dhy that Namikawa's work is the relevant place to start, and from there the booming field of symplectic representation theory, which from one perspective is all about ...
- 24k
12
votes
Accepted
Reference for combinatorics with view towards representation theory/algebraic geometry
M. Haiman "Notes on Macdonald polynomials and the geometry of the Hilbert scheme of points on $\mathbb{P}^2$". By one of the greatest specialists of interactions between combinatorics and ...
- 7,300
11
votes
Examples of function fields Langlands for small genus (<= 2)
If you really want an example of a representation, there's something funny you will find. Any irreducible cuspidal automorphic representation of $GL_n(\mathbb A_F)$ factors as a restricted tensor ...
- 148k
11
votes
Accepted
Chebotarev density theorem and pure weight local systems
As Piotr says, we must assume $U$ normal. The purity assumption is not needed.
There are two steps to this proof
(1) Suppose for any $u \in U$ and $k \in \mathbb{N}$, we have
$Tr(\sigma_u^k,\mathcal{L}...
- 148k
10
votes
Accepted
What kind of algebraic object is $\mathcal{D}_X$? (algebra of diifferential operators). What's special about modules over it?
Proposition 1.2.9 of http://math.columbia.edu/~scautis/dmodules/hottaetal.pdf explains that if $M$ and $N$ are both left $D$-modules and $M'$ and $N'$ are both right $D$-modules then
(a) $M\otimes_{R}...
- 3,545
10
votes
Reference for combinatorics with view towards representation theory/algebraic geometry
I am not that strong on the representation-theory side, but know more about the combinatorics side. If you want to get an overview of the symmetric functions and the associated combinatorics (crystal ...
- 15.8k
9
votes
Accepted
Global Langlands function fields
The abstract of V. Lafforgue's paper https://arxiv.org/abs/1404.6416 says
For any reductive group G over a global function field, we use the cohomology of G-shtukas with multiple modifications and ...
- 148k
9
votes
Statement of local geometric Langlands
There is no precise formulation of local geometric Langlands in the literature, but the rough outline is known and goes back to the papers of Frenkel-Gaitsgory starting with https://arxiv.org/abs/math/...
- 24k
9
votes
Accepted
Branching rule of $S_n$ and Springer theory
This is a nice question. I have never seen this before.
Let us write
$$\mathcal B_u = \{ V_0 \subset V_1 \subset \cdots \subset V_{n-1} \subset V_n = \mathbb C^n : u V_i \subset V_i \}. $$
Let $ \...
- 4,612
9
votes
Accepted
What is the Zhu algebra of a vertex algebra "really"?
This is a question I've asked around periodically and haven't heard a fully satisfying answer for, but I can report what I understood. Let me say off the bat that the nicest statement I'm aware of is ...
- 24k
8
votes
How should I think about the Grothendieck-Springer alteration?
The answer to 1) is that this is a special case of a broader phenomenon for symplectic resolutions (though I think some features are specific to the Grothendieck-Springer case.) For instance you have ...
- 5,958
8
votes
Accepted
Irreducible representations of product of profinite groups
This is not even true for finite groups, in this generality, and not even in characteristic $0$. Consider, for example, the group $Q_8 \times C_3$, where $Q_8$ is the quaternion group and $C_3$ is ...
- 13k
7
votes
Accepted
Component group of stabilizer group of a nilpotent element
No. Example 16 of Sommers - A generalization of the Bala–Carter theorem for nilpotent orbits shows that, if $x$ is a subregular nilpotent element in $\mathsf G_2$, then $\operatorname C(x)$ is $\...
- 12.9k
6
votes
Accepted
Equivariant quantum cohomology of conical symplectic resolutions
First of all, I don't understand why you say that there are no holomorphic curves - a typical example of such a space is $T^*{\mathbb C}{\mathbb P}^1$ and it perfectly has non-trivial holomoprhic ...
- 7,037
6
votes
A combinatorial expression of Hall-Littlewood polynomials
Here is a simple observation (which you probably already know). If you send $t\to0$, then $b_\lambda$ in your formula disappears, and it remains to minimize $\Delta$. The minimum is then achieved on ...
- 61
6
votes
Accepted
Coulomb branch varieties and symplectic singularities
There is a very recent paper by G. Bellamy that shows that every Coulomb branch is a symplectic singularity.
The paper is very nice. You can find it here
- 3,318
6
votes
Reductive group with simply connected derived group has all root groups $\mathrm{SL}_2$
$\newcommand\Z{\mathbb Z}$Point 1 has already been answered by @spin, and by @nfdc23 at Centralizers of subtori in reductive groups, derived subgroups. It is also Proposition 2.1(i) of Kac and ...
- 12.9k
6
votes
Accepted
Geometric explanation for the $\widehat{\mathfrak{sl}}_2$ free field realisation (alias $L_1(\mathfrak{sl}_2)\to V(\sqrt{2}\mathbf{Z})$)
As Pavel Safranov commented, this is done in the paper arxiv.org/abs/0710.5247 by Zhu. I've skimmed the paper and will sketch how I think it works.
Write $\mathcal{L}_G$ for the determinant line ...
- 5,701
6
votes
Frobenius reciprocity for Deligne-Lusztig induction/restriction
Deligne-Lusztig induction is not a functor between module categories (at the level of characters it sends characters to virtual characters, and not in general to actual characters).
What is true is ...
- 35.2k
6
votes
Accepted
Operations on filtered / graded vector spaces via $\mathbb{A}^1 / \mathbb{G}_m$
For question 1, by the Rees construction the corresponding quasi-coherent sheaf is the graded $k[t]$-vector space $V\otimes k[t]$. This is simply the quasi-coherent sheaf $V\otimes_k\mathcal O_{\...
- 3,054
5
votes
Kostant's $G$-invariant part in the sym power ring of adjoint representation?
As Kostant himself says there, p. 330 (with $\mathfrak g$ complex reductive, $G$ the adjoint group, $J$ for your $I$):
Here the structure of $I$ is given by a theorem of Chevalley. This asserts ...
- 31.5k
5
votes
Intuitive reason that the regular representation is a uniform function
I think the two equalities are the same, since $\operatorname{reg}_T$ equals $\sum_\theta \theta$; so I'll focus on the equality
$$\operatorname{reg}_G = \frac{1}{\lvert G^F\rvert_p} \sum_{T, \theta} \...
- 12.9k
5
votes
Accepted
Intuitive reason that the regular representation is a uniform function
This is an interesting question, but the kind of geometric or structural intuition your are looking for may not exist. To put it another way, the reason behind the fact in the OP is a non-trivial ...
- 3,813
5
votes
Accepted
Interesting properties of "coadjoint" orbits inside $V\in \operatorname{Rep}G$
Generally they can be classified when the action of $G$ on $V$ is visible, which by definition means there are finitely many orbits in the nullcone. Irreducible visible pairs $(G, V)$ were classified ...
- 3,230
5
votes
What's an example of an $n$-step nilpotent Lie algebra whose centre is not $g^{(n)}$?
Any direct sum of a $n$-step nilpotent Lie algebra and a nonzero abelian Lie algebra satisfies the required property.
Acknowledgement: After posting this answer, I realized that this class of examples ...
- 5,878
4
votes
Accepted
Is $\mathbb{P}^1$ the only smooth projective curve with a locally split tangent lie algebroid?
If $f: X \to \mathbb A^n$ is an etale map, then we can pull back vector fields on $\mathbb A^n$ to vector fields on $X$. This pullback operation is a lie algebroid homomorphism. Hence if we pull back ...
- 148k
4
votes
Interactions (functors) between equivariant sheaves for different groups?
In light of Marc Hoyois's observation that $Sh_G(X)$ is the same as presheaves on some easy category, I will answer the more general question:
Given a functor $f \colon \mathcal{C} \to \mathcal{D}$,...
- 5,898
Only top scored, non community-wiki answers of a minimum length are eligible