All Questions
41 questions
40
votes
1
answer
4k
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 ...
- 7,789
28
votes
2
answers
3k
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\...
- 5,958
25
votes
3
answers
6k
views
Introductory References for Geometric Representation Theory
Would anyone be able to recommend text books that give an introduction to Geometric Representation Theory and survey papers that give an outline of the work that has been done in the field? I'm ...
22
votes
4
answers
5k
views
motivating geometric representation theory
I am wondering if there is a good motivation for geometric representation theory from within the questions of classical representation theory.
In other words, I'd be curious to see something using ...
15
votes
0
answers
542
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 ...
- 2,964
12
votes
1
answer
842
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 $...
- 7,789
12
votes
0
answers
388
views
Perverse sheaves and representation theory
At my university we are having a working group on perverse sheaves, with the aim of applying them to representation theory (Lusztig canonical bases for quivers/quantum groups etc). We are still ...
- 1,061
11
votes
2
answers
978
views
Reference for combinatorics with view towards representation theory/algebraic geometry
I'm making this post to ask for a reference about combinatorics: I'm a PhD student in representation theory/algebraic geometry. My background is mostly in algebra and geometry (and also mostly ...
- 1,061
11
votes
1
answer
1k
views
Characteristic Classes in Geometric Representation Theory
Geometric respectively topological methods are widely applied in representation theory. As far as I know mainly cohomological methods are used.
I wonder if there are concrete applications of the ...
- 2,554
9
votes
1
answer
1k
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 ...
- 7,789
9
votes
3
answers
1k
views
Is there a good account of D-affinity and localization theorem for partial flag varieties?
Recall that a topological space is called $A$-affine for a sheaf of algebras $A$ if taking global sections of coherent sheaves of $A$-modules is an equivalence of categories to finitely generated $\...
- 44.7k
8
votes
3
answers
2k
views
Smoothness properties of the Springer fiber
The Springer fiber, recall, is defined (briefly) with reference to a chosen unipotent matrix $U \in \mathrm{GL}_n$, and consists of all flags $0 = F_0 \subset F_1 \subset \dots \subset F_n = \mathbb{C}...
- 7,273
8
votes
3
answers
529
views
Intuitive reason that the regular representation is a uniform function
Corollary 12.14 of Digne-Michel's book Representations of finite groups of Lie type gives various decompositions of the regular representation $\operatorname{reg}_G$ in terms of the Deligne-Lusztig ...
- 721
8
votes
2
answers
2k
views
Global Affine Flag Variety and Affine Flag Variety
There is a construction of a global affine flag variety over $\mathbb{A}^1$ (or another curve) $Fl_{\mathbb{A}_1}$ such that each fiber above $\epsilon \neq 0$ is isomorphic to a direct product of the ...
- 1,719
8
votes
1
answer
530
views
Restriction to Levi Subgroups and the Affine Grassmannian
Let $G$ be a complex reductive group, $L\subset G$ a Levi subgroup and $Rep(G)$ the category of rational representations of $G$.
My Question:
What is the geometric analogue of the restriction ...
- 2,554
8
votes
1
answer
599
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 ...
- 3,772
7
votes
1
answer
555
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 &...
- 2,216
6
votes
0
answers
193
views
Bundles equivariant with respect to a transitive Lie algebra action
Suppose an algebraic group $G$ transitively acts on a variety $X$. Then it is well known that $G$-equivariant vector bundles on $X$ are in correspondence with representations of the stabilizer of a ...
- 151
6
votes
0
answers
304
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 ...
- 7,789
5
votes
1
answer
304
views
Beilinson-Bernstein for nonintegral levels
If one wants to understand representations of $\mathfrak{g}$ (a finite dimensional semisimple Lie algebra) of weight $\lambda$, the happiest you could be is if $\lambda+\rho$ is (integral) regular ...
- 5,701
5
votes
1
answer
675
views
Is it possible to describe the action of the Weyl group on the cohomology of the fibers of the Grothendieck-Springer resolution?
I am confused about the following: can one describe the action of the Weyl group on the cohomology of each fiber of the Grothendieck-Springer resolution? I only need the case of ${\mathfrak sl}_n$. ...
- 2,964
5
votes
1
answer
266
views
Two identities involving Ext functors in the context of D-modules
I have several questions regarding proposition 2.3 in "Cherednik and Hecke algebras of varieties with a finite group action", by Pavel Etingof. Let $X$ be a complex affine algebraic variety ...
- 541
5
votes
1
answer
509
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 $ADE$...
- 849
5
votes
0
answers
223
views
Making Virasoro uniformization explicit for elliptic curves
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Lie{Lie}\DeclareMathOperator\Der{Der}\newcommand\el{\mathrm{ell}}$Let $\mathcal{M}_{g,1^{\infty}}$ denote the space ...
user108998
5
votes
0
answers
244
views
Borel–Weil–Bott theorem and tensor product
Let $G=\operatorname{SL}(2)$ and $V_n$ be the n+1 dimensional irreducible representation of $G$. This gives a representation for $G(O)=\operatorname{Map}(\operatorname{Spec} k[[t]], G)$ and hence an ($...
- 189
5
votes
0
answers
324
views
A question about equivariant derived categories and [BBD]
Let $G$ be an algebraic group (over $\mathbb{C}$) acting algebraically on a variety $X$. Bernstein and Lunts then define in [BL94] the equivariant derived category $D^b_G(X,\mathbb{C})$ (of $\mathbb{C}...
- 2,554
4
votes
1
answer
296
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 ($\operatorname{Gr}_{G}(\mathbb{C})=G(\mathbb{C}((z)))/G(\mathbb{C}[[z]])$...
- 103
4
votes
0
answers
115
views
Tensors of minimal rank in Schur modules $S_{\lambda}V \subset V^{\otimes |\lambda|}$
It is well known that for a vector space $V$ with $\dim(V)=n+1$ the $GL(V)-$module $V^{\otimes d}$ splits as a sum of irreducible representations (with suitable multiplicities) $S_{\lambda}V$, where $\...
- 1,343
4
votes
0
answers
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 ...
- 41
4
votes
0
answers
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 $Rep(G)$...
- 1,982
3
votes
1
answer
304
views
A question on algebraic loop groops
Setup:
Let $\mathcal{K}=\mathbb{C}((t))$, $\mathcal{O}:= \mathbb{C}[[t]]$ and $G$ be a reductive algebraic group (over $\mathbb{C}$). Let further $\mathcal{K}_n$ denote the $\mathcal{O}$-ideal in $\...
- 2,554
3
votes
0
answers
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 $\...
- 31
3
votes
0
answers
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 ...
- 3,019
3
votes
0
answers
144
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)...
- 103
2
votes
1
answer
202
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 ...
- 1,677
2
votes
1
answer
687
views
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
votes
0
answers
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)...
- 1,061
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 ...
- 2,964
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 $...
- 3,472
1
vote
2
answers
360
views
When representation of two different coadjoint orbits are equivalent?
Let $G$ be a compact connected Lie group and $\mu:T\to S^1$ be a representation of a maximal torus $T \subset G$ and $\lambda=d\mu$ be a weight for some $\lambda\in\mathfrak{t}^*$ (where $\mathfrak{t}$...
user21574
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 ...
