All Questions
14 questions
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
3
votes
0
answers
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 ...
- 201
3
votes
3
answers
581
views
Reductive group with simply connected derived group has all root groups $\mathrm{SL}_2$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}$Motivation: I am trying to understand why the Deligne-Langlands conjectures are only stated for $p$-adic reductive groups with connected ...
- 143
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
2
votes
1
answer
137
views
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
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
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
5
votes
0
answers
253
views
Tannakian theory for Lie algebras
Let $G$ be a reductive (just in case) linear algebraic group over $\mathbb{C}$ and let $\mathfrak{g}$ be the Lie algebra of $G$. Consider the category $\operatorname{Rep}(G)$ of finite dimensional ...
4
votes
0
answers
120
views
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 ...
- 1,590
5
votes
1
answer
460
views
BGG resolution for characters of reductive groups
Take $G$ a split reductive group (over a field of char 0) with Borel $B$, opposite Borel $\overline{B}$ and maximal split torus $T\subset B$. We write $X = G/\overline{B}$. Let $O \subset X$ be the ...
- 123
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
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
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
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