# Questions tagged [geometric-representation-theory]

The geometric-representation-theory tag has no usage guidance.

183
questions

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### 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 ...

11
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1
answer

293
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### 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 ...

3
votes

0
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157
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### 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 $\...

4
votes

0
answers

155
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### 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 $\mathbf{P}^1$...

8
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1
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### 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 ...

7
votes

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### 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 $...

5
votes

1
answer

165
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### 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 ...

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514
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### 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 ...

2
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67
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### 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 ...

4
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118
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### 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 ...

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### 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<...

1
vote

1
answer

63
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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 ...

4
votes

1
answer

233
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### 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 $\...

0
votes

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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 ...

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116
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### 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{\...

0
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0
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96
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### 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 ...

4
votes

0
answers

121
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### 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 ...

2
votes

0
answers

153
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### 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)...

4
votes

1
answer

149
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### 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 ...

5
votes

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answers

136
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### 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,...

6
votes

1
answer

432
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### 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 $\...

5
votes

1
answer

139
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### 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 ...

16
votes

0
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399
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### 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

235
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### 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 ...

7
votes

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114
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 ...

4
votes

1
answer

232
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{...

2
votes

1
answer

141
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?

2
votes

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answers

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### 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 ...

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votes

1
answer

381
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### 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}(\...

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496
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### 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 ...

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### 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 ...

1
vote

1
answer

221
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### 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 ...

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2
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### 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 ...

3
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100
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### Poincare polynomials for Borel Moore homology and fibrations

For an algebraic variety $X$ over $\mathbb{C}$, we denote $H_k(X)$ as its Borel-Moore homology of degree $k$. Let us define the Poincare polynomial associated it by
$$P(X)=\sum_{k\in \mathbb{N}}dim ...

4
votes

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answers

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### Stratified fibration property of the "Ran" affine Grassmannian

Let us consider the so-called Ran Grassmannian $Gr_{Ran}$, i.e. the geometric object defined e.g. in [Zhu, An Introduction to the affine Grassmannian and the Geometric Satake equivalence, Definition 3....

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241
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### The geometric "hands-on" vs. algebraic approach to nearby cycles

Feel free to skip to the question below; the following is just context and discussion:
An interesting, but seemingly less used result in the theory of nearby cycles of constructible sheaves is (...

8
votes

1
answer

580
views

### Why are VOA characters modular forms (geometrically)?

In Zhu's seminal paper, he proves (5.3.2) that if $V$ is a vertex algebra the character of all of its modules are modular forms! (This is not literally true- there are conditions).
I have always found ...

5
votes

1
answer

720
views

### Statement of local geometric Langlands

A precise statement of the global geometric Langlands conjecture is well-known.
However, I am unable to find a statement of the local Langlands conjecture. Does anyone have a modern statement or a ...

1
vote

1
answer

169
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### Irreducible real representations of $\mathrm{SL}(2,\mathbb{R})$

I am looking for a classification of irreducible real representations of $\mathrm{SL}(2,\mathbb{R})$ of finite dimension (in the following by "representation" I mean a representation of ...

1
vote

0
answers

79
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### Characteristic functions of character sheaves on tori

I am currently reading a set of lecture notes by V. Ostrik and G. Williamson, Character sheaves, tensor categories and non-Abelian Fourier transform. In Theorem 1.1, they make the assertion that the ...

8
votes

3
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470
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### Intuitive reason that the regular representation is a uniform function

Corollary 12.14 of Digne-Michel's book Representations of finite groups of Lie type gives various decompositions of the regular representation $\operatorname{reg}_G$ in terms of the Deligne-Lusztig ...

6
votes

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### Irreducible representations of product of profinite groups

It is a standard fact in the representation theory of finite groups that for $G,H$ finite groups, all of the irreducible representations of $G \times H$ are the external tensor product of irreps of $G$...

3
votes

2
answers

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### Frobenius reciprocity for Deligne-Lusztig induction/restriction

I am currently trying to understand the properties of Deligne-Lusztig induction, following Carter's Finite groups of Lie type and Digne-Michel's Representations of finite groups of Lie type. I am ...

10
votes

1
answer

432
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### Is the affine closure of the basic affine space of a reductive algebraic group Cohen–Macaulay?

Let $G$ be a reductive algebraic group with choices of Borel subgroup and maximal torus $B \supseteq T$ and unipotent radical $U$ over an algebraically closed field $k$ of characteristic zero. Is the ...

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votes

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### Making Virasoro uniformization explicit for elliptic curves

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Lie{Lie}\DeclareMathOperator\Der{Der}\newcommand\el{\mathrm{ell}}$Let $\mathcal{M}_{g,1^{\infty}}$ denote the space ...

4
votes

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### On the order of the head of product of two simple modules over Quiver Hecke Algebras

My question is:
We assume the underlying quiver is a Dynkin quiver. Let $L(\lambda)$ and $L(\mu)$ be two simple modules over Quiver Hecke algebra $R$ where $\lambda$ and $\mu$ are two Konstant ...

3
votes

1
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251
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### Iterating specialization of sheaves?

This is a question about the operation of taking the specialization of sheaves along a subspace. I'll recall the settings in which I've encountered a notion of specialization of sheaves:
The real, ...

3
votes

1
answer

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### Geometric explanation for the $\widehat{\mathfrak{sl}}_2$ free field realisation (alias $L_1(\mathfrak{sl}_2)\to V(\sqrt{2}\mathbf{Z})$)

Let $\sqrt{n}\mathbf{Z}$ be the one dimensional lattice, whose generator has length $2$. Associated to this is a lattice vertex algebra
$$V(\sqrt{2}\mathbf{Z}).$$
We also have the simple quotient of ...

2
votes

1
answer

281
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### Hyperkahler and symplectic complex geometry: reference?

I would need some references regarding symplectic and hyperkahler (complex) geometry. My background is mostly from algebraic geometry and I know a little bit the basics on Kahler manifolds.
I would be ...

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0
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### Coincidence of notation in the classification of representations of affine Hecke algebras

This is spurred by a short discussion I had in the comments of this MO question.
In Ginzburg's 1998 paper, https://arxiv.org/abs/math/9802004v3, or equivalently in the book by Chriss and Ginzburg, &...