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Questions tagged [geometric-representation-theory]

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Smoothness of some varieties related to the Slodowy slice

Let $G$ be a complex algebraic group with simple Lie algebra $\mathfrak{g} = \operatorname{Lie} G$. Let $\mathcal{B}$ be the flag variety consisting of all the Borel subalgebras of $\mathfrak{g}$. Let ...
Haris's user avatar
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4 votes
0 answers
138 views

What advantages do perverse sheaves provide over D-modules? (or vice versa)

My question is as in the title: What advantages do perverse sheaves provide over D-modules? (or vice versa) As a specific example: could something like the modular generalized Springer correspondence ...
Andrea B.'s user avatar
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3 votes
1 answer
128 views

Factoring out an element of a root subgroup to make a conjugation integral

Fix a nonarchimedean local field $L/\mathbb{Q}_p$ with ring of integers $\mathcal{O}$, uniformizer $\varpi$, and residue field $k$. If I take a matrix $$\begin{pmatrix} a & \varpi b \\ c & d \...
Ashwin Iyengar's user avatar
1 vote
0 answers
64 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_{...
Filip's user avatar
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2 votes
0 answers
117 views

Flag variety type Beilinson resolution

The Beilinson resolution is a locally free sheaves resolution for sheaf $\Delta_*\mathcal{O}_{\mathbb{P}}$,where $\Delta: \mathbb{P}\to \mathbb{P}\times\mathbb{P}$ is the diagonal embedding of ...
fool rabbit's user avatar
3 votes
0 answers
99 views

Different definitions of the thick affine flag variety

I have seen several different definitions of the so called "thick" affine flag variety associated to an affine Lie algebra, and I am having trouble seeing why they are the same. Some ...
Qixian Zhao's user avatar
3 votes
1 answer
305 views

Representation theory and topology of Teichmüller space

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\char{char}$I am reading a note on Teichmüller space, and I come across a somewhat algebraic problem in the picture below,...
Kenny S's user avatar
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1 vote
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153 views

Definition of nearby cycle over an affine line

In some famous papers like Gaitsgory's "Construction of central elements in the affine Hecke algebra via nearby cycles" and Beilinson-Bernstein's "A proof of Jantzen conjectures", ...
Allen Lee's user avatar
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126 views

What is Pic of the torus global affine Grassmannian?

Let $T$ be a torus and $X$ a proper smooth curve over characteristic $0$ algebraically closed field $k$. What is $\text{Pic}(\text{Gr}_{T,X^n})$? Here $\text{Gr}_{T,X^n}$ is the BD Grassmannian over $...
Pulcinella's user avatar
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8 votes
1 answer
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Two versions of parabolic category O

Let $ \mathfrak{g} $ be a semisimple Lie algebra with corresponding complex semisimple group $ G$. Let $ P \subset G $ be a parabolic subgroup. Let $ W^P $ be the set of shortest coset ...
Joel Kamnitzer's user avatar
2 votes
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Multiplicities of components of a Springer fibre

Given a Springer fibre of type A, are multiplicities of its irreducible components known in general, or at least in the special cases of two-row/hook types? By multiplicities I mean considering a ...
Filip's user avatar
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1 answer
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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 ...
enihcamemit's user avatar
2 votes
0 answers
113 views

Lie Algebra representations outside of generalized central characters

For a simple Lie algebra $\mathfrak{g}$, we can view its category of representations as fibered over $\operatorname{Spec}Z(\mathfrak{g})$ (a representation will lie over a point if the center's action ...
E. KOW's user avatar
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additive vs multiplicative quiver/hypertoric varieties - properties

It is a standard fact that a smooth Nakajima quiver variety / hypertoric variety $X$ has the following properties: It is holomorphic symplectic $(X,\omega)$, in fact 1'. hyperkahler It has a ...
Filip's user avatar
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2 votes
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172 views

Are parabolic Springer fibers equal dimensional?

Let $G$ be a simple algrbraic group ( of type BCDEFG ) over the complex number $\mathbb{C}$, let $P$ be a parabolic subgroup of $G$, suppose we have a resolution of singularities $\mu: T^*(G/P)\to \...
fool rabbit's user avatar
2 votes
1 answer
205 views

Component group of stabilizer group of a nilpotent element

Let $G$ be a semisimple complex algebraic group, and $\mathfrak{g}$ be its Lie algebra. $G$ acts on $\mathfrak{g}$ by adjoint action. Let $x$ be an nilpotent element in $\mathfrak{g}$, and $G(x)\...
fool rabbit's user avatar
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 $\...
user avatar
6 votes
0 answers
186 views

Bundles equivariant with respect to a transitive Lie algebra action

Suppose an algebraic group $G$ transitively acts on a variety $X$. Then it is well known that $G$-equivariant vector bundles on $X$ are in correspondence with representations of the stabilizer of a ...
Ilya Dumanski's user avatar
12 votes
1 answer
582 views

What is the Zhu algebra of a vertex algebra "really"?

Given any vertex algebra $V$, you can give a particular quotient $\DeclareMathOperator\Zhu{Zhu}\Zhu V=V/\cdots$ an algebra structure using (a small amount of) the vertex algebra structure. As far as ...
Pulcinella's user avatar
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3 votes
0 answers
332 views

Understanding the proof of the Springer correspondence

Let $G$ be a connected reductive group over an algebraically closed field $k$ with Weyl group $W$. Let $$ \mathcal{S} = R\pi_*\mathbb{Q}_\ell[\dim \mathcal{N}] $$ be the Springer sheaf, where $\...
zygomatic's user avatar
5 votes
0 answers
198 views

Relations between Whittaker functions/W algebras and Stokes data/resurgence

Skippable background: A Whittaker function is more or less a function on a flag manifold which is twisted-invariant for the action of a unipotent subgroup. E.g. consider functions $f$ on $\mathbf{P}^1$...
Pulcinella's user avatar
  • 5,565
8 votes
1 answer
283 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 ...
Pulcinella's user avatar
  • 5,565
7 votes
0 answers
207 views

Twisted D-module structure on pushfoward of structure sheaf of Bruhat cell

Apologies for the basic question, but my experience on math.stackexchange tells me that this will go unanswered there. Background: In the principal block, the dual Verma modules (with highest weight $...
Fan Zhou's user avatar
  • 301
5 votes
1 answer
232 views

Exterior products of irreducible representations of sl_2(C)

It is well-known that $\mathfrak{sl}_2(\mathbb{C})$ admits exactly one irreducible representation $V_n$ of dimension $n+1$ for all $n\geq 0$. It is explicitly given by the action on homogeneous ...
AThomas's user avatar
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11 votes
0 answers
1k views

Roadmap to geometric Langlands for a mathematical physics student

I am a student of both mathematics and physics, who has recently been studying string theory. My mathematics background is mostly differential geometry (principal bundles, Lie groups, etc.), although ...
Daniel Waters's user avatar
2 votes
0 answers
111 views

Levi quotients of parahorics in loop group

I am looking for some references on Levi quotients of parahorics in $LG = G(\mathbb{C}((t)))$, $G$ being an algebraic group with Weyl group $W$. I have read that parahoric subgroups of $LG$ are in ...
user492133's user avatar
4 votes
0 answers
137 views

Verlinde formula via chiral/vertex algebras

The Verlinde formula gives an explicit formula for any finite dimensional simple Lie algebra $\mathfrak{g}$ an explicit formula for the dimension of conformal blocks for the associated WZW conformal ...
Pulcinella's user avatar
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5 votes
0 answers
120 views

Is there a derived version of affine Schur-Weyl duality?

One version of affine Schur-Weyl duality states that there is a fully faithful functor from representation of $A_r$ affine Hecke algebra to the representation of $A_n$ affine Lie group assuming $r<...
Xu Kai's user avatar
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1 vote
1 answer
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Dimension of maximal split subtorus and fixed point subspace of Lie algebra

Let $F = \mathbb{C}((t))$. Let $G$ be a complex semisimple algebraic group. Then conjugacy classes of maximal tori in $G(F)$ are in bijection with conjugacy classes in $W$, the Weyl group of $G$ with ...
user492130's user avatar
4 votes
1 answer
315 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 $\...
lw h's user avatar
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0 answers
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The closure of the orbits of $\mathcal{F} \times \mathcal{F}$

Let $V$ be a finite-dimensional vector space over an algebraically closed field $F$, and $\dim V=d$. Let $G=GL(V)$, $P$ is the parabolic subgroup of $G$. Let $\mathcal{F}$ be the set of all partial ...
zhichengzhang's user avatar
1 vote
0 answers
188 views

Visualizing the affine Bruhat decomposition for $\operatorname{SL}_2$

$ \newcommand\Fl{\mathcal{F}\!\ell} \newcommand\numC{\mathbb{C}} \newcommand\numZ{\mathbb{Z}} \newcommand\ringO{\mathbb{O}} \newcommand\ringK{\mathbb{K}} \newcommand\power{\...
Gaussler's user avatar
  • 295
0 votes
0 answers
169 views

Relationship between vector bundles and modules

THE GROTHENDIECK RING IN GEOMETRY AND TOPOLOGY - M.F. ATIYAH §1. The Grothendieck ring in homotopy theory I am going to be talking about vector bundles, i.e. fibre bundles with fibre a vector space ...
Abel 's user avatar
  • 51
4 votes
0 answers
161 views

Frobenius kernel by Greenberg functor

I was not able to find much on representation theory (with a geometric perspective) of algebraic groups over local artinian rings. In particular, studying on the classic book of Jantzen, I was asking ...
Mattia's user avatar
  • 41
2 votes
0 answers
166 views

Counting points of parabolic Springer fibers

Let $G$ be a reductive group over an (algebraically closed ) field. To each parabolic subgroup $P \subseteq G$ and $x \in G$ we can consider two types of partial Springer fibers associated to it : $$1)...
Tommaso Scognamiglio's user avatar
4 votes
1 answer
179 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 ...
Fan Zhou's user avatar
  • 301
5 votes
0 answers
148 views

Concrete computation of pullback of the $D$-module $\mathbb{C}[t,t^{-1}]$

I have some problems in calculating some example explicitly. Consider $$ \{0\} \overset{i}{\rightarrow} \mathbb{C} \overset{j}{\leftarrow} \mathbb{C}^*.$$ Then $Rj_+\mathbb{C}[t,t^{-1}] = \mathbb{C}[t,...
XT Chen's user avatar
  • 1,094
6 votes
1 answer
510 views

Chebotarev density theorem and pure weight local systems

How do we deduce the following statement from the Chebotarev density theorem? The statement is from Ngo's Fundamental Lemma paper. Let $U$ be a scheme of finite type over $\mathbb{F}_q$. Let $\...
userabc's user avatar
  • 677
5 votes
1 answer
158 views

Interesting properties of "coadjoint" orbits inside $V\in \operatorname{Rep}G$

Let $G$ be a reductive group over $\mathbf{C}$. It acts on the dual of its Lie algebra $\mathfrak{g}^*$ by conjugation. One can describe the orbits of $\mathfrak{g}^*$ explicitly (e.g. using Jordan ...
Pulcinella's user avatar
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20 votes
0 answers
545 views

Your favourite alternative proof of Borel–Weil–Bott

There is a really nice proof of Borel–Weil–Bott, essentially using parabolic induction (see Proof of Borel-Weil-Bott Theorem, Lurie - A proof of the Borel–Weil–Bott theorem or Demazure - A very simple ...
5 votes
1 answer
260 views

Two identities involving Ext functors in the context of D-modules

I have several questions regarding proposition 2.3 in "Cherednik and Hecke algebras of varieties with a finite group action", by Pavel Etingof. Let $X$ be a complex affine algebraic variety ...
Fernando Peña Vázquez's user avatar
7 votes
0 answers
147 views

comparison of polynomial loop group and smooth loop group

I have a question about Section 8.6 of Pressley-Segal's Loop groups book. Let $G$ be a compact, connected Lie group. Proposition 8.6.6 concerns the comparison of homotopy type between its polynomial ...
onefishtwofish's user avatar
4 votes
1 answer
245 views

A problem on Kazhdan–Lusztig theorem

I am reading Chriss, Ginzburg's book Representation theory and complex geometry. In theorem 7.2.16 it says that the convolution action of the Steinberg variety $St=\tilde{\mathcal{N}}\times_\mathfrak{...
Yang's user avatar
  • 71
2 votes
1 answer
187 views

Geometric meaning of inducing a representation from a parabolic subgroup of a Weyl group

What is the geometric meaning of inducing a representation from a parabolic subgroup of a Weyl group? Could Springer theory of Weyl group representations be used to obtain such a geometric meaning?
IntegrableSystemsEnthusiast's user avatar
2 votes
0 answers
203 views

Error in Proposition 8.7.1 of Pressley–Segal

Let $G$ be a connected, compact Lie group, $T$ a maximal torus. Let $LG$ be the group of smooth (or polynomial) loops and $X=LG/T$ the affine flag variety ($T$ acts say by right multiplication). In ...
onefishtwofish's user avatar
15 votes
1 answer
524 views

Branching rule of $S_n$ and Springer theory

Let $u\in\mathrm{GL}_n$ be a unipotent element, let $\mathcal{B}_u$ be the variety of Borel subgroups containing $u$, and let $d=\dim \mathcal{B}_u$. Then Springer theory tells us that $H^{2d}(\...
user148212's user avatar
  • 1,606
11 votes
0 answers
660 views

Sign error in Chriss-Ginzburg?

On page 118, Theorem 2.7.26 (iii) in Chriss-Ginzburg "Representation Theory and Complex Geometry" there is a formula for the convolution of the classes of conormal bundles of $Y_{12}\subset ...
Anton Mellit's user avatar
  • 3,592
11 votes
0 answers
367 views

Perverse sheaves and representation theory

At my university we are having a working group on perverse sheaves, with the aim of applying them to representation theory (Lusztig canonical bases for quivers/quantum groups etc). We are still ...
Tommaso Scognamiglio's user avatar
1 vote
1 answer
296 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 ...
Asav's user avatar
  • 163
11 votes
2 answers
906 views

Reference for combinatorics with view towards representation theory/algebraic geometry

I'm making this post to ask for a reference about combinatorics: I'm a PhD student in representation theory/algebraic geometry. My background is mostly in algebra and geometry (and also mostly ...
Tommaso Scognamiglio's user avatar