All Questions
Tagged with geometric-representation-theory reference-request
21 questions
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 ...
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
13
votes
0
answers
797
views
Borel-Weil-Bott, Langlands and Hitchin
Let $G$ be a compact semi-simple Lie group and $G_\mathbb{C}$ be its complexification. We denote by $B$ a Borel subgroup of $G_\mathbb{C}$.
Given a dominant weight $\lambda$, one can construct a line ...
- 2,273
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
10
votes
1
answer
644
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
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
Reference request: Affine Grassmannian and G-bundles
Let $G$ be an affine algebraic group over an algebraically closed field $k$ of zero characteristic. The set of cosets $X_G=G(k((t))/G(k[[t]])$ is called the Affine Grassmannian of $G$ and can be given ...
- 2,027
8
votes
1
answer
1k
views
Quiver varieties and the affine Grassmannian
Recently I was watching a talk: http://media.cit.utexas.edu/math-grasp/Ben_Webster.html and at the end the lecturer gave a correspondence (I was having trouble with subscripts so changed the notation ...
- 741
7
votes
1
answer
1k
views
Modern proofs of the Verlinde formula?
Let $G$ be a semisimple algebraic group and $\Sigma$ a smooth proper curve. Then $\text{Bun}_G(\Sigma)$ comes equipped with a line bundle $\mathcal{L}$ which generates the torsion free part of $\text{...
- 5,701
6
votes
0
answers
173
views
Equivalence between $\mathcal{D}_\lambda$ modules and $\mathcal{D}_{0}$ modules
Fix $G$ a finite dimensional reductive group and $\lambda$ a weight. Apparently the category of $\mathcal{D}_\lambda$ modules on $G/B$ is equivalent to the category of $\mathcal{D}_0$ modules on $G'/B'...
- 5,701
5
votes
1
answer
525
views
what is the injective hull of indecomposable module of preprojective algebra
Let $Q$ be a ADE type quiver and $s_i$ ($i$ runs through the vertices of $Q$) be the simple $\Lambda$-module with 1-dimensional vector space at vertex $i$ and zero-dim at other vertices. Here $\Lambda$...
- 849
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
194
views
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 ...
- 173
4
votes
0
answers
99
views
Confusion about twisted Vermas in Feigin-Frenkel
Let $G$ be a finite dimensional semisimple algebraic group, and for $s\in W$ write $i_s: F_s=B^+sB^-/B^-\to F$ for the $s$th Bruhat cell in the flag variety $F$.
Then (in Affine Kac-Moody Algebras and ...
- 5,701
3
votes
0
answers
119
views
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
2
votes
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
2
votes
0
answers
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 ...
- 653
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
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 $T^*(G/B)$ ,
$\mu:T^*(G/B)\to \mathcal{N}$ ...
- 849
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 ...
- 893
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 ...
- 721