# Questions tagged [springer-fibres]

The springer-fibres tag has no usage guidance.

26
questions

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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 $\...

2
votes

0
answers

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### Springer sheaf and Deligne-Lusztig induction

Let $G=Gl_n$ be the general linear group over the algebraic closure of a finite field $\overline{\mathbb{F}}_q $ and let $F:G \to G$ be the standard Frobenius. On $G$ there is the Springer (perverse) ...

5
votes

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### Invariants of cohomology of Springer sheaf

Let $G=Gl_n(\mathbb{C})$ and $\mathcal{N}$ be the nilpotent cone associated to it i.e nilpotent matrices inside $\mathfrak{g}=\mathfrak{gl}_n(\mathbb{C})$.
We have the variety $\tilde{\mathcal{N}}$ ...

2
votes

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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)...

11
votes

1
answer

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### Reference for character sheaves over $\mathrm{GL}_n(q)$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$I know a little bit about complex representation theory of finite reductive groups as $\GL_n(q),\SO_n(q)$ etc via Deligne-Lusztig induction and ...

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### A Lie-theoretic question regarding $B\ltimes \mathfrak{g}/\mathfrak{b}$

I am stuck on a seeming elementary Lie-theoretic question arising from a study of components of affine Springer fibers. Will be very grateful if somebody would like to share some insight, or ...

2
votes

1
answer

153
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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?

11
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### Two-sided cells, special nilpotent orbits and special representations

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. This question concerns three classical objects of representation theory: the two-sided Kazhdan-Lusztig cells of the Weyl group $W$ of $\mathfrak{...

7
votes

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answer

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### 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 ...

4
votes

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answers

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### Relations between double coinvariants and affine Springer fibers

Diagonal coinvariants have an interpretation from https://arxiv.org/abs/math/0201148 in terms of the Hilbert scheme.
There are two recent papers https://arxiv.org/pdf/1801.09033.pdf and https://arxiv....

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### Intersection of components in Springer fibre of type A

From the standard results on Springer fibers of type A, we know that given a Springer fiber, say $\mathcal{B}_\lambda,$ its irreducible components are all equidimensional and parametrized by standard ...

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### Homeomorphisms of Springer fibers

Let $V$ be a complex $n$-dimensional vector space and denote by ${\cal F}$ its space of complete flags. Let $g \in Gl(V)$ be unipotent and consider the Springer fiber ${\cal F}_g$ of its fixed points ...

3
votes

1
answer

460
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### Computing affine Springer fibers

$\DeclareMathOperator\diag{diag}\DeclareMathOperator\Gr{Gr}\DeclareMathOperator\SL{SL}$I'm having some trouble computing affine Springer fibers, even in simple cases. For example, consider the group $...

11
votes

3
answers

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### 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 ...

4
votes

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answers

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### Families of Hessenberg varieties for $GL_n$

In short, the question is
What do we know about the sheaf $\pi_*\underline{\bar{\mathbb{Q}}_{\ell}}$ given by the family of (very original, see below) Hessenberg varieties for $GL_n$? As a sum of ...

12
votes

2
answers

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### Hitchin fibration and Springer resolution

Let C be a curve and let us assume $G=GL_N$ and $\mathfrak{g}=\mathfrak{gl}_N$ for simplicity. The moduli space $\mathcal{M}_H(C,G)$ of $G$-Higgs bundles admits the Hitchin fibration $\pi: \mathcal{M}...

6
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### Springer fibers and Weyl group

Let $\pi:\tilde{\mathfrak{g}}\rightarrow\mathfrak{g}$ the Grothendieck-Springer resolution of a semisimple Lie algebra $\mathfrak{g}$, over $\mathbb{C}$.
We know it's a small map, and that $\pi_{*}\...

4
votes

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### Centralizer action on components of Springer fibers

Let $G$ be a complex adjoint group. Let $u\in G$ be unipotent. The group $A(u):=\pi_0(Z_G(u))$ acts on the set of components of the Springer fiber $\mathcal{B}_u$, the variety of Borel subgroups that ...

5
votes

1
answer

211
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### $G$-orbits in Springer resolution (or, stabilizer actions on Springer fibers)

This may be an elementary question, but I'm having trouble coming up with an answer: Let $\tilde{N} = T^*(G/B)$ be the Springer resolution of the nilpotent cone. Does it have finitely many $G$-orbits?...

8
votes

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409
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### Can we count the number of simple modules for a reduced enveloping algebra?

Let $G$ be a reductive algebraic group over a field of positive characteristic $p$, which I'll assume to be very good for $G$. Then the Lie algebra $\mathfrak{g}$ is restricted and each simple $\...

3
votes

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257
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### Computing Springer action on the homology of affine Springer fibers

Lusztig defined (in Sec. 5, also Sage) a Springer action of the affine Weyl group on the homology of affine Springer fibers (Iwahori one, i.e. in an affine flag variety). In the regular semisimple ...

3
votes

1
answer

555
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### 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 ${\mathfrak sl}_n$. ...

3
votes

2
answers

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### counting points on nilpotent Springer fiber

Computing $p$-adic orbital integral I come to the following question. My ground field $k$ is the residue field of a non-arch local field, i.e. a finite field. I am happy to put any assumption on $\...

2
votes

1
answer

281
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### Computing tangent spaces of resolutions to Slodowy slices

This question is about (a special case of) the varieties discussed here: Does the preimage of the Slodowy slice in $T^*G/P$ have a name?.
Let $G = SL_n(\mathbb{C}), \mathfrak{g} = \mathfrak{sl}_n(\...

8
votes

1
answer

350
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### What's a good example/reference for cohomology classes on Springer fibers that aren't restricted from the flag variety

As usual, by Springer fiber, I mean the fixed points $X^u$ of a unipotent element $u$ of the group $G$ on the flag variety $X=G/B$. It's a lovely theorem that when $G=SL_n$, the induced map on ...

2
votes

1
answer

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### Springer Action on Centre of Parabolic Category O (after Brundan)

I recently learned of a result of Brundan describing the centre of the regular block of parabolic category $\mathcal{O}$ for $\mathfrak{gl}_{n}$ as the cohomology of a corresponding Springer fibre (...