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Background:

There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in Symplectic Reflection Algebras, and Ginzburg & Etingof's list given at a workshop in 2007 here.

Gordon titled these "Problems for next year". Now, more than years later, some of them are certainly solved. For example

2) in 1 or 0.1 in 2: Aspherical locus, conjugation-invariant functions $c$ for which $U_{1,c}$ and $H_{1,c}$ are Morita equivalent. There were characterized by a theorem of Bezrukavikov and Etingof in this paper as precisely those for which $U_{1,c}$ has finite homological dimension. Those aspherical values were then characterized in this paper by Dunkl and Griffeth.

6) in 1: Derived equivalences of the category $\mathcal{O}_c$ and $\mathcal{O}_{c'}$ where $c-c'$ is an integer conjugation-invariant function on the set of relections. This conjecture by Rouquier was settled in a recent paper by Losev.

My questions are:

A) Which of the questions posted in 1 or 2 still remain open?

(Or even questions from the original paper, Chapter 17).

I am particularly interested in the following ones:

1) in 1: Has there been progress in studying symplectic reflection algebras for symplectic reflection groups that are neither complex nor wreath products?

3) in 1 or 0.4 in 2: Is the Bernstein inequality for finitely generated $H_{1,c}$-modules that the Gelfand-Kirillov dimension is at least $1/2$ that of the reflection representation still open?

8) in 1: Is $\mathcal{O}_c$ Koszul?

Disclaimer: Due to my unfamiliarity with this vast field, I may be picking out the wrong ones or being imprecise in my review on things that are now proved. Apologies for that. Thank you!

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  • 2
    $\begingroup$ It's worth asking Iain Gordon himself to comment on this list of questions, since he probably has up-to-date information. (Or other specialists like Ivan Losev.) $\endgroup$ – Jim Humphreys Jul 27 '14 at 15:02
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My paper with Charles classifies only aspherical values for the monomial groups $G(r,p,n)$; for most exceptional reflection groups the calculation of the aspherical locus is still open (and our technique, based on explicit norm calculations with orthogonal functions, doesn't easily generalize).

Looking only at the list of problems in section 9 of Iain's paper, I believe that problem 12 has been partially solved (Bezrukavnikov and Finkelberg used the Cherednik algebra to generalize Macdonald positivity to wreath Macdonald polynomials), and problem 6 has been solved by Losev.

Essentially none of the (other) problems dealing with Cherednik algebras for arbitrary groups have been solved, though I am not sure about the status of the problems dealing with GK dimension or a hyper-kaehler structure on the symplectic resolution for the $G_4$ singularity.

A particularly embarrassing unsolved problem that does not appear on Iain's list: we do not have a classification of the finite dimensional irreducibles (except for the symmetric groups, and arguably, the monomial groups $G(r,p,n)$, though even there the combinatorics involved is of the Kazhdan-Lusztig type, rather than something simple like dominant weights in classical Lie theory).

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  • $\begingroup$ At least for G(r,1,n), one doesn't need Kazhdan-Lusztig combinatorics to see which simples are finite dimensional, though one does need them to calculate the dimension. The description of finite dimensional should follow from Bezrukavnikov-Losev. $\endgroup$ – Ben Webster Aug 14 '14 at 16:02
  • $\begingroup$ Well, ok, when is the trivial lowest weight irreducible finite dimensional? It's my impression from talking to Ivan that even this question is not so easy to answer. It would be interesting for me, at least, to know if the set of parameters for which it is can have components of codimension more than two in the parameter space! $\endgroup$ – Stephen Aug 14 '14 at 17:19
  • $\begingroup$ @BenWebster Perhaps I should have written "crystal combinatorics" instead of "KL combinatorics"? I would be interested to know if it's possible to recover Pavel Etingof's classification of when $L_c(\mathrm{triv})$ is finite dimensional for $G(2,1,n)$ from what Roman and Ivan did (or from what Ivan has done since). $\endgroup$ – Stephen Aug 15 '14 at 16:17
  • $\begingroup$ I'd certainly accept "crystal combinatorics." I'll think about your question about $L_c(\mathrm{triv})$; I'll admit that working through the combinatorics is some work, but I would say it's an order of magnitude less than affine KL polynomials. $\endgroup$ – Ben Webster Aug 16 '14 at 13:19
  • $\begingroup$ OK, thanks! I should edit the answer a bit when I have some time, and among other thigns I'll fix that. $\endgroup$ – Stephen Aug 16 '14 at 15:10
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For G(r,1,n), category $\mathcal{O}$ is Koszul. This is a consequence of a comparison theorem with a version of parabolic category $\mathcal{O}$ of affine type, recently proven by Rouquier-Shan-Varagnolo-Vasserot and by Losev when combined with slightly older work of Shan-Varagnolo-Vasserot. This technique doesn't have much hope of generalizing to other types (except maybe G(r,p,n)).

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  • $\begingroup$ Your link to Shan-Varagnolo-Vasserot points to the wrong place. $\endgroup$ – Peter McNamara Apr 10 '15 at 11:02
  • $\begingroup$ @PeterMcNamara Good catch. Fixed now. $\endgroup$ – Ben Webster Apr 10 '15 at 12:51

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