Questions tagged [geometric-representation-theory]
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208 questions
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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 ...
- 2,964
3
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119
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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 ...
- 403
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110
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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 ...
- 1,677
3
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58
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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 $\...
user108998
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413
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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
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126
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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
$$P(X)=\sum_{k\in \mathbb{N}}dim ...
- 71
3
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117
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Fundamental representation bases and generalized minors
Let $G$ be a simple simpy connected complex algebraic group.
I was wondering if there is a clear relationship between the generalized minors (defined by Berenstein, Fomin and Zelevinsky) and bases of ...
- 201
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197
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Are there six functors for twisted D modules?
Is there a notion of holonomic D module which admits the six functor formalism in the world of twisted D modules?
Recall that twisted D modules on $X$ are well-defined for any $T$ torsor $\tilde{X}\...
- 5,701
3
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126
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Nakajima reflection functors and the flavour/framing group action
Nakajima has constructed so-called reflection functors that are isomorphisms between different quiver varieties that have the same framing $\mathbf{w}:$
$$\Phi_{\sigma}:\mathfrak{M}_\zeta(Q,\mathbf{v}...
- 1,677
3
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86
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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 ...
- 497
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229
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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 ...
- 3,019
3
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0
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83
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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 $\mathbb{C}^2/\mathbb{Z}_{/n}$?
- 1,677
3
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0
answers
144
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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)...
- 103
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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 $H^{3}(\mathfrak{g}((t))((s)),\mathbb{C})$?
- 3,472
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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 ...
- 9,019
2
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1
answer
227
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Component group of stabilizer group of a nilpotent element
Let $G$ be a semisimple complex algebraic group, and $\mathfrak{g}$ be its Lie algebra. $G$ acts on $\mathfrak{g}$ by adjoint action. Let $x$ be an nilpotent element in $\mathfrak{g}$, and $G(x)\...
- 653
2
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2
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959
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Around the socle filtration of a Verma module
Work in the context of a finite dimensional simple Lie algebra over $\mathbb{C}$.
Write $W$ for the Weyl group and $\leq$ for the Bruhat order. For $w\in W$ let $\Delta_w$ denote the Verma module of ...
- 1,595
2
votes
1
answer
415
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 ...
- 1,061
2
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1
answer
144
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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 $e$, ...
- 1,677
2
votes
1
answer
200
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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
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1
answer
152
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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 ...
- 73
2
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1
answer
184
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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 ...
- 7,789
2
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1
answer
137
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Representation variety in $\mathrm{SU}(p,q)$
$\DeclareMathOperator\SU{SU}$Let $\Gamma$ be a cocompact oriented Fuchsian group, and consider the representation variety $\textrm{Hom}(\Gamma, \SU(p,q))$. Consider a point $\rho : \Gamma \to \SU(p,q)$...
- 601
2
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1
answer
202
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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 ...
- 1,677
2
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1
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687
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Derived Push-Forward of Morphism of Perverse Sheaves and Translation Functors
I hope this question is not too vague.
Let $G$ be a complex reductive group, $B$ a Borel subgroup of $G$, and $P$ a parabolic containing $B$.
Denote by $\pi:G/B\to G/P$ the canonical map. Consider ...
- 2,554
2
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0
answers
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 ...
- 653
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129
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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 ...
- 653
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101
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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 ...
- 1,677
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123
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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 ...
- 834
2
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180
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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 \...
- 653
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124
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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 ...
- 57
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169
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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)...
- 1,061
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209
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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 ...
- 959
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640
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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 ...
- 2,964
2
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150
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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}$ ...
- 849
2
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220
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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 ...
user141500
2
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0
answers
154
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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 ...
- 335
2
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0
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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-...
- 213
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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 ...
- 893
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1
answer
324
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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 ...
- 163
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244
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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
1
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1
answer
90
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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
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0
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79
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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_{...
- 1,677
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0
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162
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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", ...
- 291
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0
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133
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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 $...
- 5,701
1
vote
0
answers
216
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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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0
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101
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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 ...
- 721
1
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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, &...
- 159
1
vote
1
answer
238
views
Two different formulations of the Bott–Samelson resolution
There seem to be two formulations of the Bott–Samelson resolution flowing around. For concreteness, let $ G = \mathrm{GL}_{n} ( \mathbb{C} ) $ with the Borel subgroup $ B \subset G $ of upper ...
- 295
1
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0
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121
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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}...
- 677