Questions tagged [springer-fibres]

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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{...
Owen Colman's user avatar
8 votes
0 answers
257 views

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 ...
Cheng-Chiang Tsai's user avatar
6 votes
0 answers
278 views

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_{*}\...
prochet's user avatar
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5 votes
0 answers
184 views

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 ...
Lucas Seco's user avatar
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4 votes
0 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....
Yellow Pig's user avatar
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4 votes
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118 views

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 ...
Filip's user avatar
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4 votes
0 answers
253 views

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 ...
Cheng-Chiang Tsai's user avatar
4 votes
0 answers
213 views

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 ...
Cheng-Chiang Tsai's user avatar
3 votes
0 answers
291 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 $\...
zygomatic's user avatar
3 votes
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267 views

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 ...
Cheng-Chiang Tsai's user avatar
2 votes
0 answers
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Multiplicities of components of a Springer fibre

Given a Springer fibre of type A, are multiplicities of its irreducible components known in general, or at least in the special cases of two-row/hook types? By multiplicities I mean considering a ...
Filip's user avatar
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2 votes
0 answers
162 views

Are parabolic Springer fibers equal dimensional?

Let $G$ be a simple algrbraic group ( of type BCDEFG ) over the complex number $\mathbb{C}$, let $P$ be a parabolic subgroup of $G$, suppose we have a resolution of singularities $\mu: T^*(G/P)\to \...
fool rabbit's user avatar
2 votes
0 answers
180 views

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) ...
Tommaso Scognamiglio's user avatar
2 votes
0 answers
165 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)...
Tommaso Scognamiglio's user avatar
1 vote
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59 views

Affine Springer fibers for symmetric spaces

Springer fibers are defined to be the varieties of "isotropic" full flags which are fixed by a certain element in the symmetric space. In a similar manner, affine Springer fibers can be ...
211's user avatar
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0 answers
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Extension of a type A Springer fibre

Given a decomposition $p=(p_1,\dots,p_n)$ of $n$, one can associate its corresponding partial flag variety $$\mathcal{B}_p=\{F=(0=F_0\subset F_1\subset \dots \subset F_n=\mathbb{C}^n) \mid \dim F_i/F_{...
Filip's user avatar
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