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Questions tagged [geometric-representation-theory]

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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^...
Bin Wang's user avatar
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486 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 ...
cccp's user avatar
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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 ...
Roman Fedorov's user avatar
4 votes
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381 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 ...
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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....
Rauan Akylzhanov's user avatar
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597 views

Euler characteristic, character of group representation and Riemann Roch theorem

I am considering the following set up:Let $G$ be a finite group,let $Rep(G)$ denote the category of finite dimensional representations over $\mathbb{C}$. Let $V,W$ be representations of $G$ in $Rep(G)$...
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Canonical basis in equivariant K-theory of the Springer resolution

In Definition 15.0.2 of the notes from a course by Bezrukavnikov there is a characterization of canonical basis in K-theory of a Springer fiber which is due to Lusztig. This characterization is in ...
Yellow Pig's user avatar
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Different definitions of the thick affine flag variety

I have seen several different definitions of the so called "thick" affine flag variety associated to an affine Lie algebra, and I am having trouble seeing why they are the same. Some ...
Qixian Zhao's user avatar
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110 views

additive vs multiplicative quiver/hypertoric varieties - properties

It is a standard fact that a smooth Nakajima quiver variety / hypertoric variety $X$ has the following properties: It is holomorphic symplectic $(X,\omega)$, in fact 1'. hyperkahler It has a ...
Filip's user avatar
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3 votes
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Locally finite positive energy modules generated by singular vectors at positive levels?

This is question is about whether or not certain modules for an affine Lie algebra are generated by their singular vectors. I begin with some background. Backround on affine Lie algebras. Let $\...
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413 views

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 $\...
zygomatic's user avatar
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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 ...
Yingjin Bi's user avatar
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117 views

Fundamental representation bases and generalized minors

Let $G$ be a simple simpy connected complex algebraic group. I was wondering if there is a clear relationship between the generalized minors (defined by Berenstein, Fomin and Zelevinsky) and bases of ...
F.H.A's user avatar
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197 views

Are there six functors for twisted D modules?

Is there a notion of holonomic D module which admits the six functor formalism in the world of twisted D modules? Recall that twisted D modules on $X$ are well-defined for any $T$ torsor $\tilde{X}\...
Pulcinella's user avatar
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Nakajima reflection functors and the flavour/framing group action

Nakajima has constructed so-called reflection functors that are isomorphisms between different quiver varieties that have the same framing $\mathbf{w}:$ $$\Phi_{\sigma}:\mathfrak{M}_\zeta(Q,\mathbf{v}...
Filip's user avatar
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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 ...
little dog's user avatar
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229 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 ...
Exit path's user avatar
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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 $\mathbb{C}^2/\mathbb{Z}_{/n}$?
Filip's user avatar
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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)...
Din's user avatar
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Double loop groups and cohomology

Let $G$ be a connected reductive group over $\mathbb{C}$ of Lie algebra $\mathfrak{g}$. What is the value of $H^{3}(\mathfrak{g}((t))((s)),\mathbb{C})$?
prochet's user avatar
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268 views

What's the relation of the Hecke algebra of a pair and the flag variety?

Let $G$ be a real semisimple Lie group and $K$ a maximal compact subgroup. Let $\mathfrak{g}$ and $\mathfrak{k}$ be the complexified Lie algebra of $G$ and $K$, respectively. Then the Hecke algebra ...
Zhaoting Wei's user avatar
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2 votes
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97 views

An injective map in equivariant algebraic K-theory

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ and $\mathfrak{g}$ be its Lie algebra. Let $\mathcal{N}$ be the nilpotent cone of $\mathfrak{g}$ consists of all nilpotent ...
fool rabbit's user avatar
2 votes
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129 views

Flag variety type Beilinson resolution

The Beilinson resolution is a locally free sheaves resolution for sheaf $\Delta_*\mathcal{O}_{\mathbb{P}}$,where $\Delta: \mathbb{P}\to \mathbb{P}\times\mathbb{P}$ is the diagonal embedding of ...
fool rabbit's user avatar
2 votes
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101 views

Multiplicities of components of a Springer fibre

Given a Springer fibre of type A, are multiplicities of its irreducible components known in general, or at least in the special cases of two-row/hook types? By multiplicities I mean considering a ...
Filip's user avatar
  • 1,677
2 votes
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123 views

Lie Algebra representations outside of generalized central characters

For a simple Lie algebra $\mathfrak{g}$, we can view its category of representations as fibered over $\operatorname{Spec}Z(\mathfrak{g})$ (a representation will lie over a point if the center's action ...
E. KOW's user avatar
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0 answers
180 views

Are parabolic Springer fibers equal dimensional?

Let $G$ be a simple algrbraic group ( of type BCDEFG ) over the complex number $\mathbb{C}$, let $P$ be a parabolic subgroup of $G$, suppose we have a resolution of singularities $\mu: T^*(G/P)\to \...
fool rabbit's user avatar
2 votes
0 answers
124 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 ...
user492133's user avatar
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169 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)...
Tommaso Scognamiglio's user avatar
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209 views

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 ...
onefishtwofish's user avatar
2 votes
0 answers
640 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 ...
Yellow Pig's user avatar
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2 votes
0 answers
150 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 $T^*(G/B)$ , $\mu:T^*(G/B)\to \mathcal{N}$ ...
Ben's user avatar
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2 votes
0 answers
220 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 ...
user avatar
2 votes
0 answers
154 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 ...
C.Niculescu's user avatar
2 votes
0 answers
172 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-...
Mark Shiffor's user avatar
2 votes
0 answers
228 views

References for crystal bases and Demazure modules in representation theory

I was wondering what are some standard general references/books/survey articles about: (1) crystal bases, and string parameterizations and (2) Demazure modules, and Schubert varieties (containing ...
Kiu's user avatar
  • 893
1 vote
0 answers
79 views

Extension of a type A Springer fibre

Given a decomposition $p=(p_1,\dots,p_n)$ of $n$, one can associate its corresponding partial flag variety $$\mathcal{B}_p=\{F=(0=F_0\subset F_1\subset \dots \subset F_n=\mathbb{C}^n) \mid \dim F_i/F_{...
Filip's user avatar
  • 1,677
1 vote
0 answers
162 views

Definition of nearby cycle over an affine line

In some famous papers like Gaitsgory's "Construction of central elements in the affine Hecke algebra via nearby cycles" and Beilinson-Bernstein's "A proof of Jantzen conjectures", ...
Allen Lee's user avatar
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133 views

What is Pic of the torus global affine Grassmannian?

Let $T$ be a torus and $X$ a proper smooth curve over characteristic $0$ algebraically closed field $k$. What is $\text{Pic}(\text{Gr}_{T,X^n})$? Here $\text{Gr}_{T,X^n}$ is the BD Grassmannian over $...
Pulcinella's user avatar
  • 5,701
1 vote
0 answers
216 views

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{\...
Gaussler's user avatar
  • 295
1 vote
0 answers
101 views

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 ...
Martin Skilleter's user avatar
1 vote
0 answers
66 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, &...
mi.f.zh's user avatar
  • 159
1 vote
0 answers
121 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}...
userabc's user avatar
  • 677
1 vote
0 answers
146 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 $...
prochet's user avatar
  • 3,472
1 vote
0 answers
249 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 ...
paul's user avatar
  • 375
0 votes
0 answers
82 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 ...
zhichengzhang's user avatar
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0 answers
231 views

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 ...
Abel 's user avatar
  • 61
-5 votes
0 answers
124 views

Is a quiver variety a moduli stack of quiver representations?

As the title, I was just wondering is a quiver variety a moduli stack of quiver representations? I am familiar with quiver varieties but not that familiar with moduli stacks, so I was just wondering ...
user236626's user avatar

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