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Questions tagged [geometric-representation-theory]

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114 views

Sources of derived schemes in geometric representation theory

What are some derived schemes naturally arising in geometric representation theory? Some examples include: Steinberg scheme Hilbert scheme Moduli stack of local systems. Now, this looks like a ...
2
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1answer
86 views

Core components of quiver varieties as fiber bundles of flag varieties

Is there an example of Nakajima quiver variety of type A which has all core components smooth, such that at least one of them is NOT an iterated fibre bundle of flag manifolds (i.e. a space obtained ...
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81 views

Could we characterize injective objects in the category of $G$-equivariant sheaves?

Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$. Let $G$ be a topological group which act on $X$ continuously from the left....
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92 views

Explicit computation for perverse cohomology

To construct the convolution product for two ($G(O)$-equivariant) perverse sheaves $\mathcal{F}, \mathcal{G}$ on affine grassmanian, the first thing we need to compute is $^PH^0(\mathcal{F} \boxtimes^...
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3answers
442 views

How should I think about the Grothendieck-Springer alteration?

Given a simple complex Lie algebra $\mathfrak{g}$, recall the Springer resolution of its nilpotent cone $\widetilde{\mathcal{N}}\to \mathcal{N}$. Several times I have seen someone explaining Springer ...
3
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1answer
133 views

Equivariant quantum cohomology of conical symplectic resolutions

There is a couple of papers on this [Braverman, Maulik, Oblomkov, Okounkov, Pandharipande] where authors calculate quantum cohomology for various conical symplectic resolutions. The language in these ...
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1answer
70 views

Lagrangian cores of quiver variety in different GIT chambers

Let $\pi: \mathfrak{M}_{(\theta,0)}(Q,\text{v},\text{w}) \rightarrow \mathfrak{M}_{(0,0)}(Q,\text{v},\text{w})$ be the projective morphism between Nakajima quiver varieties when complex moment ...
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0answers
100 views

A question about affine Grassmannian

I am reading Sorger's Lectures on the Moduli of $G$-bundles, and I am confused about a detail in the proof of proposition 5.3.2., where he proves that the $G$-bundle description of affine Grassmannian ...
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129 views

Tannakian theory for Lie algebras

Let $G$ be a reductive (just in case) linear algebraic group over $\mathbb{C}$ and let $\mathfrak{g}$ be the Lie algebra of $G$. Consider the category $\operatorname{Rep}(G)$ of finite dimensional ...
2
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1answer
86 views

Frobenius coordinate expansion of character

Let $\lambda$ be the partition of integer $d$. The Frobenius coordinate of $\lambda$ is given $$ (a_1 ,\ldots,a_{d(\lambda)}|b_1,\ldots,b_{d(\lambda)}),$$ where $d(\lambda)$ denote the diagonal of $\...
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71 views

Is there Thom isomorphism for equivariant K groups in algebraic geometry, not necessarily complex number field?

In Chriss and Ginzburg's fantastic book 'representation theory and complex geometry', they use the following Thom Isomorphism: $\pi:E\rightarrow X$, is a G-equivariant affine linear bundle, then $\pi^...
2
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1answer
91 views

Descending chain property for compact Lie goups

I'm searching for a good reference that prove the descending chain propriety for compact Lie groups (i.e. every sequence $K_1\supset K_2\supset...$ of closed subgroups $K_i$ of $G$ is eventually ...
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0answers
150 views

Plucker coordinates of flag varieties

I am interested in understanding Lemma A.2 in the paper "Moduli spaces of principal F-bundles" by varshavsky which you can find here. It uses so called "Plücker" coordinates of the flag variety for ...
3
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188 views

Moduli stacks of flat bundles and of local systems are not algebraically equivalent?

Let $G$ be a complex reductive algebraic group, $X$ be a smooth compex projective curve. In the remark 1.1 here, it's claimed that the derived moduli stack of principal $G$-bundles equipped with a ...
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1answer
360 views

Decategorification of Gaitsgory's strange functional equation?

Let $G$ a complex reductive algebraic group, $X$ be a smooth compact complex curve. The moduli stack $Bun_G$ of principal $G$-bundles on $X$ is generally speaking not quasi-compact (so we do not have ...
6
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1answer
255 views

Examples of function fields Langlands for small genus (<= 2)

See Edward Frenkel's article "Lectures on the Langlands program and conformal field theory" for an exposition of the function fields Langlands correspondence (now a theorem of Drinfel'd, L.Lafforgue &...
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0answers
85 views

Prehomogeneous vector spaces for reductive groups

Recall that a prehomogeneous vector space, is a representation $V$ of a linear algebraic group $G$ having an open $G$-orbit. Let $Z$ be the neutral connected component of the stabilizer of a point of ...
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179 views

Geometric interpretation of a formula for the induced character (fix point localization?)

Let $H < G$ be a subgroup of a finite group $G$. Let $X:=G/H$ and $\mathcal{F} \in Sh_G(X)$ be an equivariant sheaf on $X$ (w.r.t. left multiplication) associated to a finite dimensional ...
5
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1answer
222 views

Quiver variety analogue of Grothendieck-Springer resolution

A standard example of Nakajima quiver varieties are type A Springer resolutions $\widetilde{\mathcal{N}} \to \mathcal{N}$. In the theory of Springer resolutions it is often beneficial to consider the ...
5
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1answer
200 views

BGG resolution for characters of reductive groups

Take $G$ a split reductive group (over a field of char 0) with Borel $B$, opposite Borel $\overline{B}$ and maximal split torus $T\subset B$. We write $X = G/\overline{B}$. Let $O \subset X$ be the ...
4
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1answer
185 views

Computation of multiplicity of irreducible representation in some representation via geometric Satake correspondence

Let $G$ be a reductive algebraic group over $\mathbb{C}$. Let $\operatorname{Gr}_{G}$ be the corresponding affine Grassmannian ($\operatorname{Gr}_{G}(\mathbb{C})=G(\mathbb{C}((z)))/G(\mathbb{C}[[z]])$...
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1answer
419 views

Global Langlands function fields

Has V. Lafforgue proved the automorphic-to-Galois direction in the Global Langlands conjectures for general reductive groups over function fields? What is the current status, more generally? Related ...
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0answers
105 views

Computation of nearby cycles, monodromy action and action of $sl_{2}$ on $\operatorname{gr}(Ψ)$ for Picard-Lefshetz family

Let $f: \mathbb{A}^{2} \rightarrow \mathbb{A}^{1}$ be a map that sends $(x,y)$ to $xy$. Let $U \hookrightarrow \mathbb{A}^{2}$ be the preimage $f^{-1}(\mathbb{A}^{2} \setminus \{0\})$ and $X:=f^{−1}(0)...
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184 views

Dimension of affine springer fiber

I have a few questions with respect to Bezrukavnikov's proof of the dimension formula for affine springer fibers in Fixed point set on affine flag manifolds. The setting is as follows: Let G be a ...
18
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1answer
829 views

Proofs of Beilinson-Bernstein

The Beilinson-Bernstein localization theorem states roughly that the category of $D$-modules on the flag variety $G/B$ is equivalent to the category of modules over the universal enveloping algebra $U\...
2
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1answer
87 views

Orbit decomposition of the restriction of an equivariant sheaf?

All sets and groups in the question are finite. In order to understand equivariant sheaves better I'm trying to prove some basic facts from Mackey theory using equivariant sheaves. The main obstacle ...
4
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1answer
163 views

Functors between categories of equivariant sheaves are equivariant sheaves on the product?

This is a follow up question to this question which remained unanswered (satisfactorily) even after a large bounty. I have made a litlle progress and I have no a more specific question which might be ...
7
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1answer
795 views

A combinatorial expression of Hall-Littlewood polynomials

This is related to the question Hall-Littlewood functions and functions on the nilpotent cone, and arises in the construction of Coulomb branches of gauge theories. The motivation is explained at the ...
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1answer
203 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 ...
8
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1answer
376 views

Interactions (functors) between equivariant sheaves for different groups?

Let $G$ be a finite group and $k$ a field (alg. closed char 0 for simplicity). To every $G$ set $X$ we can assign the category of $G$-equivariant sheaves of $k$-vector spaces $Sh_G(X)$. It is ...
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Identifying and reconstructing the derived category from its auto-equivalences

Background: Given a smooth irreducible algebraic variety $Y$ with $\omega_Y$ or $\omega_Y^{-1}$ ample. Then Bondal-Orlov theorem states that if there exists any other smooth algebraic variety $Y'$ ...
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124 views

Projective and Quasiprojective quotients

Let $G$ be a finite group acting on a projective variety $X$. Then $G$ also acts on $X-X^G$, where $X^G$ is the fixed locus. The GIT quotient varieties $X/G$ and $(X-X^G)/G$ are projective and quasi-...
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1answer
679 views

Does there exists a Fukaya category with no objects

... and really without even the possibility of having objects, so it's not a matter of just finding the "correct" flavour of Fukaya category to use. Question: Does there exist interesting symplectic ...
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366 views

Open conjectures on the Fukaya category coming from physics

This is a slightly vague question (for which I apologize in advance): can somebody give examples to open conjectures on the behavior of the $Fuk(M,\omega)$ that come from string theory and can be ...
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0answers
252 views

Automorphism that the Fukaya category is “blind” to

Given a symplectic manifold $(M,\omega)$, there is a natural map $$ Symp(M,\omega) \to Auteq(D^\pi Fuk(M,\omega))$$ which sends a symplectic automorphism to the $A_\infty$-functor it induces on the ...
8
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1answer
378 views

Affine vs Yokonuma

Let $G=GL_n$. Let us start with the Hecke algebra $H_n$. It acts on K(constructible sheaves on $G/B$) by Hecke correpondences and on K(coherent sheaves on $G/B$) by Lusztig's construction [1]. Now we ...
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0answers
114 views

Reference request: Prequantization of canonical transformations and Lie group action

Hello to MathoverFlow community I have some seemingly technical questions on applications of geometric quantisation to Lie group representation theory. We shall start by giving background definitions....
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1answer
569 views

What kind of algebraic object is $\mathcal{D}_X$? (algebra of diifferential operators). What's special about modules over it?

Let $R$ be a regular ring over a field of char 0. Let $X=Spec R$ and $D=\mathcal{D}_X$ the algebra of differential operators over it. The overall vague question is what kind of algebraic object is $...
5
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2answers
208 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 ...
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580 views

Borel-Weil-Bott, Langlands and Hitchin

Let $G$ be a compact semi-simple Lie group and $G_\mathbb{C}$ be its complexification. We denote by $B$ a Borel subgroup of $G_\mathbb{C}$. Given a dominant weight $\lambda$, one can construct a line ...
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1answer
127 views

$G$-orbits in Springer resolution (or, stabilizer actions on Springer fibers)

This may be an elementary question, but I'm having trouble coming up with an answer: Let $\tilde{N} = T^*(G/B)$ be the Springer resolution of the nilpotent cone. Does it have finitely many $G$-orbits?...
6
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1answer
453 views

Geometric construcion of Proj as a quotient by a $\mathbb{G}_m$ action

I'm trying to translate the Proj construction as a kind of quotient by a $\mathbb{G}_m$ action. Here's what I have so far: Let $X=Spec A$ be an affine scheme (after this case is setteled I imagine it ...
11
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2answers
616 views

Representation viewpoint on Chern Weil (cohomology computations done with rep theory?)

Let $G$ be a compact lie group. Chern-Weil theory tells us that there's a homomorphism: $$H^{*}(BG;\mathbb{R}) \to (Sym^{\bullet} \mathfrak{g^*})^G$$ Which in our case is an isomorphism since $G$ ...
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1answer
999 views

What do Hecke eigensheaves actually look like?

Let $\mathbb F_q$ be a finite field, $C$ a curve over $\mathbb F_q$ of genus $g\geq 2$, $\rho: \pi_1(C) \to GL_2(\overline{\mathbb Q}_\ell)$ an irreducible local system. The geometric Langlands ...
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1answer
2k 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 ...
4
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1answer
373 views

what is the injective hull of indecomposable module of preprojective algebra

Let $Q$ be a ADE type quiver and $s_i$ ($i$ runs through the vertices of $Q$) be the simple $\Lambda$-module with 1-dimensional vector space at vertex $i$ and zero-dim at other vertices. Here $\Lambda$...
5
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1answer
418 views

analog of Lusztig nilpotent scheme

Fix a quiver $Q$ without loop. Denote the set of vectices of $Q$ by $I$. Let $\Lambda_V$ be the Lusztig nilpotent scheme with associated vector space $V$ over $I$. Briefly speaking, when $Q$ is a $...
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1answer
496 views

Why should intersection cohomology and quantum cohomology be related for a symplectic resolution?

In http://arxiv.org/pdf/1410.6240.pdf M. McBreen and N. Proudfoot conjectured a precise relationship between the quantum cohomology of a symplectic resolution and the intersection cohomology of the ...
4
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1answer
243 views

Finite-dimensional representations of DAHA

It is shown by Berest-Etingof-Ginzburg that there exist finite-dimensional irreducible representations of rational Cherednik algebra $H_c(S_n)$ of $A_{n-1}$ type if and only if the deformation ...
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2answers
490 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 ...