12
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How should I think about the Grothendieck-Springer alteration?
For Question 1 I agree with dhy that Namikawa's work is the relevant place to start, and from there the booming field of symplectic representation theory, which from one perspective is all about ...
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9
votes
Accepted
Invariants of cohomology of Springer sheaf
You want to look at the partial Grothendieck-Springer resolution, i.e. the variety of pairs $ (g \in G/ P_\mu, v \in g \mathfrak p_\mu g^{-1})$.
The partial Grothendieck-Springer resolution is smooth, ...
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8
votes
How should I think about the Grothendieck-Springer alteration?
The answer to 1) is that this is a special case of a broader phenomenon for symplectic resolutions (though I think some features are specific to the Grothendieck-Springer case.) For instance you have ...
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7
votes
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Component group of stabilizer group of a nilpotent element
No. Example 16 of Sommers - A generalization of the Bala–Carter theorem for nilpotent orbits shows that, if $x$ is a subregular nilpotent element in $\mathsf G_2$, then $\operatorname C(x)$ is $\...
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7
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Hitchin fibration and Springer resolution
I will try to answer the first question only.
As in the remarks, the canonical reference is
Beauville, Narasimhan, Ramanan, Spectral curves and the generalised theta divisor. J. Reine Angew. Math. ...
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5
votes
Hitchin fibration and Springer resolution
I'll try to answer the second and third questions. My preferred way to organize this circle of ideas is to think of the following ladder of theories :
Representation theory of $\mathfrak{g}$ (or) ...
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4
votes
How should I think about the Grothendieck-Springer alteration?
For question 1, the precise statement is due to Namikawa, but perhaps best summarized in Proposition 2.7 of the paper by Braden-Proudfoot-Webster, https://arxiv.org/abs/1208.3863. To fit with the ...
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4
votes
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Reference for character sheaves over $\mathrm{GL}_n(q)$
I am not exactly sure what you have read already, but how about this set of notes, from a course given by Victor Ostrik in Luminy in 2010 (notes by Geordie Williamson)?
Character sheaves, tensor ...
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3
votes
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Nilpotent orbits of a parabolic subgroup
I thank Emile Okada for suggesting the following argument that there is a unique $P$-orbit in $q^{-1}(O) \cap p^{-1}(O_G)$ (all mistakes are due to me, however). The argument is inspired by Lemma 5.2....
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3
votes
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Geometric meaning of inducing a representation from a parabolic subgroup of a Weyl group
Thanks to colleagues, it turns out that the Springer correspondence is a functor which associates representations of the Weyl group to sheaves on the nilpotent cone, and this functor maps induction ...
3
votes
Are there cases in which the Weyl group _does_ act on the flag variety/springer fiber?
Here is a general fact: Given a group $G$ and a subgroup $H$, there is an isomorphism between
The space $\text{End}(G/H)$, consisting of $G$-equivariant maps $G/H\to G/H$.
The group $N/H$, where $N$ ...
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3
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Computing affine Springer fibers
I came across this question while studying affine Springer fibers myself, and I hope this answer can help future learners.
Let us fix a Borel subgroup $B\subset G$ and a maximal torus $T\subset G$. ...
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