# Questions tagged [geometric-representation-theory]

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### 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 ...
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### 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 ...
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### 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 ...
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### Geometric Arthur-Selberg

What should be the analogue of the Arthur-Selberg trace formula in the geometric Langlands theory?
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### 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 ...
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### 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 ...
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### Hypertoric varieties in dimension 4?

Are the only smooth hypertoric varieties in real dimension 4 obtained as minimal resolutions of type A simple singularities $\mathbb{C}^2/\mathbb{Z}_{/n}$?
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### 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 $e$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
### 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 ...
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 ...