All Questions
110 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 ...
24
votes
1
answer
2k
views
How many ways are there to globalize Harish Chandra modules?
Suppose $G$ a reductive Lie group with finitely many connected components, and suppose in addition that the connected component $G^0$ of the identity can be expressed as a finite cover of a linear Lie ...
- 4,909
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 ...
20
votes
0
answers
598
views
Your favourite alternative proof of Borel–Weil–Bott
There is a really nice proof of Borel–Weil–Bott, essentially using parabolic induction (see Proof of Borel-Weil-Bott Theorem, Lurie - A proof of the Borel–Weil–Bott theorem or Demazure - A very simple ...
18
votes
4
answers
5k
views
Why are the holomorphic line bundle sections finite dimensional?
I'm trying to understand the Borel--Weil theorem at the moment (not the whole Bott--Borel--Weil theorem as has been asked elsewhere). However, I am having a little difficulty finding a direct proof, ...
- 1,462
15
votes
1
answer
549
views
Branching rule of $S_n$ and Springer theory
Let $u\in\mathrm{GL}_n$ be a unipotent element, let $\mathcal{B}_u$ be the variety of Borel subgroups containing $u$, and let $d=\dim \mathcal{B}_u$. Then Springer theory tells us that $H^{2d}(\...
- 1,666
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
14
votes
3
answers
2k
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 ...
- 2,516
14
votes
1
answer
4k
views
How to understand character sheaves
There's a well-known series of articles by Lusztig about Character Sheaves. They have important connections to many things in (geometric) representation theory, e.g. 0904.1247
How to understand these ...
- 15.1k
14
votes
1
answer
1k
views
Morphisms between Verma modules
Let $\mathcal{O}_0$ be the principal block of the BGG category $\mathcal{O}$ for a finite dimensional simple Lie algebra over $\mathbb{C}$. For an element $w$ in the Weyl group $W$, let $\Delta_w$ ...
- 1,595
12
votes
2
answers
887
views
Representation viewpoint on Chern–Weil (cohomology computations done with rep theory?)
$\DeclareMathOperator\Sym{Sym}$Let $G$ be a compact lie group. Chern–Weil theory tells us that there's a homomorphism:
$$H^{*}(BG;\mathbb{R}) \to (\Sym^{\bullet} \mathfrak{g^*})^G$$
which in our case ...
- 7,789
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
717
views
Sign error in Chriss-Ginzburg?
On page 118, Theorem 2.7.26 (iii) in Chriss-Ginzburg "Representation Theory and Complex Geometry" there is a formula for the convolution of the classes of conormal bundles of $Y_{12}\subset ...
- 3,772
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
10
votes
1
answer
643
views
Is the affine closure of the basic affine space of a reductive algebraic group Cohen–Macaulay?
Let $G$ be a reductive algebraic group with choices of Borel subgroup and maximal torus $B \supseteq T$ and unipotent radical $U$ over an algebraically closed field $k$ of characteristic zero. Is the ...
- 422
9
votes
1
answer
2k
views
Explanation for Satake correspondence
Some time ago I was told there's an interesting classical Satake correspondence which I will write as
$$[\mathop{\mathrm{disk}} \Rightarrow G] \,\backslash\, [\mathop{\mathrm{disk}^\times} \...
- 15.1k
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
2
answers
743
views
Confusion about Subcategories of Category $\mathcal{O}$
So, in learning about category $\mathcal{O}$ representations of a semisimple Lie algebra $\mathfrak{g}$, I've come across two natural kinds of subcategories, and I think I'm confused about their ...
- 183
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
1
answer
694
views
Kazhdan-Lusztig Polynomials and Intersection Cohomology
I hope this question has not been asked before.
I would like to know which Ideas led (Deligne), Kazhdan and Lusztig believe, that Kazhdan–Lusztig polynomials can be expressed via intersection ...
- 2,554
8
votes
1
answer
1k
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 ...
- 3,051
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
2
answers
826
views
Weil's theorem about maps from a discrete group to a Lie group.
Let K be a group (with discrete topology), G be a Lie group. Let $\operatorname{Hom}(K,G)$ be the space of all group homomorphisms from K to G. This is a closed subvariety of the space of all the maps ...
- 3,203
8
votes
1
answer
698
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 ...
- 7,789
8
votes
1
answer
372
views
Two versions of parabolic category O
Let $ \mathfrak{g} $ be a semisimple Lie algebra with corresponding complex semisimple group $ G$. Let $ P \subset G $ be a parabolic subgroup. Let $ W^P $ be the set of shortest coset ...
- 4,612
8
votes
1
answer
530
views
Are there cases in which the Weyl group _does_ act on the flag variety/springer fiber?
In nearly every reference on the classical springer correspondence (for example Chriss/Ginzburg's book on Complex Geometry) it is stated that the action of the Weyl Group on the homology of the ...
- 241
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
8
votes
1
answer
204
views
Relative de Rham cohomology of flag varieties
Let $B \subset P \subset G$ be a parabolic, Borel, and reductive (split) group over the complex numbers. Consider the projection $\pi: G/B \to G/P$, I am interested in computing $R\pi_{*} \Omega^{\...
- 461
8
votes
0
answers
388
views
Representation theory of Chevalley groups as a categorical trace
Dennis Gaitsgory's 2016 preprint, From Geometric to Function-Theoretic Langlands (or How to Invent Shtukas) includes in the third section a very compressed but suggestive discussion of the ...
- 858
7
votes
3
answers
3k
views
Beilinson-Bernstein and Koszul duality
For geometric representation theorists down here.
Consider the Beilinson-Bernstein theorem:
Functor of global sections establishes
the correspondence between twisted
D-modules with fixed ...
- 15.1k
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
7
votes
0
answers
223
views
Twisted D-module structure on pushfoward of structure sheaf of Bruhat cell
Apologies for the basic question, but my experience on math.stackexchange tells me that this will go unanswered there.
Background: In the principal block, the dual Verma modules (with highest weight $...
- 311
7
votes
0
answers
160
views
comparison of polynomial loop group and smooth loop group
I have a question about Section 8.6 of Pressley-Segal's Loop groups book. Let $G$ be a compact, connected Lie group. Proposition 8.6.6 concerns the comparison of homotopy type between its polynomial ...
- 959
7
votes
0
answers
167
views
How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in the framework of D-modules?
Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be ...
- 9,019
6
votes
1
answer
559
views
Irreducible representations of product of profinite groups
It is a standard fact in the representation theory of finite groups that for $G,H$ finite groups, all of the irreducible representations of $G \times H$ are the external tensor product of irreps of $G$...
- 721
6
votes
0
answers
116
views
Properties of a functor from Soergel bimodules to Soergel modules
I am looking for an extension of a result of Riche-Soergel about a functor which maps Soergel bimodules to Soergel modules.
Fix a given Coxeter system $(W,S)$, together with a (reflection faithful) ...
- 211
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
442
views
Conceptual proof of braid group actions on quantum groups
Roughly 1990, Lusztig wrote a series of papers on quantum groups. Perhaps the result that the braid groups acts on $U_q(\mathfrak{g})$ is the proof which is least conceptual.
The original paper ...
- 719
6
votes
0
answers
360
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 ...
- 915
6
votes
0
answers
214
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 ...
- 1,073
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
2
answers
491
views
Kostant's $G$-invariant part in the sym power ring of adjoint representation?
Let $g$ be a Lie algebra, say $sl_n(\mathbb C)$. It is considered as the adjoint representation of $G=SL_n(\mathbb C)$.
A famous theorem of Kostant from "Lie Group Representations on Polynomial ...
- 3,804
5
votes
1
answer
187
views
Interesting properties of "coadjoint" orbits inside $V\in \operatorname{Rep}G$
Let $G$ be a reductive group over $\mathbf{C}$. It acts on the dual of its Lie algebra $\mathfrak{g}^*$ by conjugation.
One can describe the orbits of $\mathfrak{g}^*$ explicitly (e.g. using Jordan ...
- 5,701