Questions tagged [geometric-representation-theory]
The geometric-representation-theory tag has no usage guidance.
183
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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 ...
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votes
1
answer
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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 ...
- 5,112
3
votes
0
answers
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 $\...
- 31
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 $\mathbf{P}^1$...
- 5,112
8
votes
1
answer
239
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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 ...
- 5,112
7
votes
0
answers
161
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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 $...
- 181
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 ...
- 567
11
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0
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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 ...
- 329
2
votes
0
answers
67
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 ...
- 65
4
votes
0
answers
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 ...
- 5,112
4
votes
0
answers
91
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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<...
- 179
1
vote
1
answer
63
views
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 ...
- 11
4
votes
1
answer
233
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 $\...
- 121
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votes
0
answers
73
views
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 ...
1
vote
0
answers
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{\...
- 295
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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 ...
- 51
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 ...
- 41
2
votes
0
answers
153
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 :
$$1)...
4
votes
1
answer
149
views
Equivalence of categories of modules over $U(\mathfrak{g})$ on which $Z(\mathfrak{g})$ acts by central characters
$\newcommand{\g}{\mathfrak{g}}$Setting: $\mathfrak{g}$ is a semisimple complex Lie algebra. Here $\chi_\lambda$ denotes the central character corresponding to the action of $Z(\g)$ on a highest weight ...
- 181
5
votes
0
answers
136
views
Concrete computation of pullback of the $D$-module $\mathbb{C}[t,t^{-1}]$
I have some problems in calculating some example explicitly. Consider
$$ \{0\} \overset{i}{\rightarrow} \mathbb{C} \overset{j}{\leftarrow} \mathbb{C}^*.$$
Then $Rj_+\mathbb{C}[t,t^{-1}] = \mathbb{C}[t,...
- 1,034
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 $\...
- 657
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 ...
- 5,112
16
votes
0
answers
399
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
235
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 ...
7
votes
0
answers
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 ...
- 253
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{...
- 71
2
votes
1
answer
141
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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
0
answers
185
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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 ...
- 253
12
votes
1
answer
381
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}(\...
- 1,504
10
votes
0
answers
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 ...
- 3,532
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0
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308
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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
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 ...
- 143
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votes
2
answers
743
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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
votes
0
answers
100
views
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 ...
- 71
4
votes
0
answers
210
views
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....
- 435
6
votes
0
answers
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 (...
- 273
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 ...
- 5,112
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 ...
- 59
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 ...
- 13
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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 ...
- 692
8
votes
3
answers
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 ...
- 692
6
votes
1
answer
362
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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 $G$...
- 692
3
votes
2
answers
259
views
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 ...
- 692
10
votes
1
answer
432
views
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 ...
- 422
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votes
0
answers
178
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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 ...
- 1,765
4
votes
0
answers
59
views
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 ...
- 71
3
votes
1
answer
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, ...
- 273
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votes
1
answer
175
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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 ...
- 5,112
2
votes
1
answer
281
views
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 ...
1
vote
0
answers
51
views
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, &...
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