Questions tagged [geometric-representation-theory]

Filter by
Sorted by
Tagged with
1
vote
0answers
61 views

Coefficient ring of Satake isomorphism

Let $G$ be a split reductive group over local field $F$, $G^L$ be the (complex) Langlands dual group of $G$. Denote $H$ to be the $\mathbb{Z}$-Hecke algebra of $G$, that is the ring of $G(\mathcal{O}...
3
votes
0answers
62 views

Freeness of completed homology over universal deformation ring

In Theorem 7.4 of the paper "patching and the $p$-adic Langlands program for $\mathrm{GL}2(\mathbf{Q}_p)$" (arXiv link), it is proved that in the minimally ramified case, the completed homology of ...
3
votes
2answers
185 views

Nontrivial Poisson relations for affine Poisson algebras

Let $A$ be a polynomial algebra over a field of characteristic $0$ in the variables $x_1,\dots,x_n$. Consider polynomials $f_1,\dots,f_m\in A$ and let $I$ be the ideal they generate in $A$. Moreover, ...
4
votes
0answers
69 views

Relations between double coinvariants and affine Springer fibers

Diagonal coinvariants have an interpretation from https://arxiv.org/abs/math/0201148 in terms of the Hilbert scheme. There are two recent papers https://arxiv.org/pdf/1801.09033.pdf and https://arxiv....
6
votes
0answers
94 views

Presentation of $H^2(\overline{M}_{0,n},\mathbb{Z})$ as an $S_n$-module?

Let $\overline{M}_{0,n}$ be the moduli space of genus zero curves with $n$ marked points. Let $I=\{\{S,S^c\}|S\subset\{1,\dots,n\},|S|\geq2, |S^c|\geq2\}$ be the set of partitions of $\{1,\dots n\}$ ...
1
vote
0answers
404 views

Areas of algebraic geometry useful for geometric representation theory

What topics/areas of algebraic geometry (aside from perverse sheaves/D-modules, etale cohomology, and possibly derived algebraic geometry) is it useful to learn/master if one is interested in doing ...
11
votes
0answers
279 views

Applications of character sheaves

There are many important recent works (for example, by Lusztig, Bezrukavnikov-Finkelberg-Ostrik, Ben-Zvi-Nadler, Boyarchenko-Drinfeld, Lusztig-Yun, Vilonen-Xue) on character sheaves (which are certain ...
3
votes
0answers
135 views

Hochschild cohomology of (generalizations) of Khovanov's arc algebra

Backgroud: In his seminal paper A functor-valued invariant of tangles, Khovanov (among many other things) introduced the arc algebra $H^{n}$ and several functors between $H^{n}$ and $H^{m}$ related to ...
2
votes
0answers
129 views

Projection of conormal bundle of Schubert variety under Springer resolution

Let $G=\mathrm{GL}_n(\mathbb{C})$ and $X_{\omega}=\overline{B_-wB/B}\subset G/B$ be a Schubert variety. Denote by $C(X_\omega)$ the conormal variety inside $T^*(G/B)$ , $\mu:T^*(G/B)\to \mathcal{N}$ ...
1
vote
0answers
70 views

Factoriality of schubert cells in affine flag variety

Take for simplicity $G=SL_n$ and consider the affine flag variety $Fl=G(\mathbb{C}((t)))/I$ for $I$ the Iwahori corresponding to the Borel of upper triangular matrices of determinant one. For each $...
1
vote
0answers
120 views

Spherical perverse sheaves on the affine Grassmannian and critically twisted $D$-modules

Let $G$ be a reductive algebraic group and let $Gr_G=G((z))/G[[z]]$ be its affine Grassmannian. Define $\mathcal{D}(Gr_G)_{crit}-mod$ to be the category of right $D$-modules on $Gr_G$ twisted by the ...
8
votes
1answer
322 views

Convolution in K-Theory via an Example (From StackExchange)

I've spent lots of time in Chriss and Ginzburg's "Complex Geometry and Representation Theory" and despite convolution (in Borel-Moore homology or K-theory) being very central, I feel like I'm still ...
2
votes
0answers
173 views

Non-Archimedean Soergel

If you are studying finite groups, and you have classified all finite simple groups, your job is not done yet since potentially there might be some interesting phenomena in how two finite simple ...
5
votes
0answers
91 views

Geometric Arthur-Selberg

What should be the analogue of the Arthur-Selberg trace formula in the geometric Langlands theory?
4
votes
0answers
77 views

Intersection of components in Springer fibre of type A

From the standard results on Springer fibers of type A, we know that given a Springer fiber, say $\mathcal{B}_\lambda,$ its irreducible components are all equidimensional and parametrized by standard ...
3
votes
1answer
118 views

Computing Deligne-Lusztig Characters in General

The goal for this question is to try to find a relatively explicit way of computing the Deligne-Lusztig characters. I understand that the $R_{T,\theta}$ can be computed if we know the values of the ...
3
votes
0answers
58 views

Hypertoric varieties in dimension 4?

Are the only smooth hypertoric varieties in real dimension 4 obtained as minimal resolutions of type A simple singularities $\mathbb{C}^2/\mathbb{Z}_{/n}$?
2
votes
1answer
114 views

Characterisation of even nilpotent elements in $\mathfrak{sl}_n$

Is there a ''nice'' classification of even nilpotent elements in $\mathfrak{sl}_n,$ using the correspondence between nilpotent elements and partitions of n? By an even element, I mean an element $e$, ...
6
votes
0answers
170 views

Two seemingly different definitions of a left cell

This is a question about two seemingly different notions of a left cell in a finite Weyl group and why they are the same. My question arose from reading a paper of W. McGovern titled "Left cells and ...
4
votes
1answer
78 views

Image of a quiver variety under natural morphism

We know that the natural morphism $\pi:\mathfrak{M}_{\theta}(Q,\mathbf{v},\mathbf{w})\rightarrow \mathfrak{M}_0(Q,\mathbf{v},\mathbf{w})$ between a smooth and affine quiver variety is not necessarily ...
3
votes
1answer
210 views

Core of the Jordan quiver variety

It is known that, given the Jordan quiver, dimension vectors $\textbf{v}=n,\textbf{w}=1$ and a stability condition $\theta<0,$ the corresponding quiver variety $\mathcal{M}_{\theta}(n,1)\cong \...
1
vote
0answers
169 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
votes
1answer
156 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 ...
4
votes
0answers
103 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....
7
votes
0answers
130 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^...
8
votes
3answers
683 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
votes
1answer
187 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 ...
4
votes
1answer
127 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 ...
2
votes
0answers
125 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 ...
5
votes
0answers
154 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
votes
1answer
144 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 $\...
3
votes
0answers
86 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
votes
1answer
101 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 ...
4
votes
0answers
214 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 ...
11
votes
1answer
429 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
votes
1answer
320 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 &...
4
votes
0answers
100 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 ...
6
votes
0answers
220 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
votes
1answer
308 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
votes
1answer
275 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
votes
1answer
210 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 ($\operatorname{Gr}_{G}(\mathbb{C})=G(\mathbb{C}((z)))/G(\mathbb{C}[[z]])$...
3
votes
1answer
588 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 ...
3
votes
0answers
118 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)...
4
votes
0answers
221 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 ...
21
votes
1answer
1k 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
votes
1answer
103 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
votes
1answer
199 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
votes
1answer
873 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 ...
-1
votes
1answer
224 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
votes
1answer
429 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 ...