**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](http://arxiv.org/abs/0712.1568), and Ginzburg & Etingof's list given at a workshop in 2007 [here](http://www.icms.org.uk/downloads/cherpdf/openpro.pdf).


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


**2) in [1](http://arxiv.org/abs/0712.1568) or 0.1 in [2](http://www.icms.org.uk/downloads/cherpdf/openpro.pdf):** 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](http://arxiv.org/pdf/1011.4584v4.pdf) as precisely those for which $U\_{1,c}$ has finite homological dimension. Those aspherical values were then characterized in [this paper](http://arxiv.org/pdf/1002.4607.pdf) by Dunkl and Griffeth.


**6) in [1](http://arxiv.org/abs/0712.1568):** 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](http://arxiv.org/pdf/1406.7502.pdf).


My **questions** are:


***A) Which of the questions posted in [1](http://arxiv.org/abs/0712.1568) or [2](http://www.icms.org.uk/downloads/cherpdf/openpro.pdf) still remain open?***


(Or even questions from the original [paper](http://arxiv.org/abs/math/0011114), Chapter 17).


I am particularly interested in the following ones:


**1) in [1](http://arxiv.org/abs/0712.1568):** Has there been progress in studying symplectic reflection algebras for symplectic reflection groups that are neither complex nor wreath products?


**3) in [1](http://arxiv.org/abs/0712.1568) or 0.4 in [2](http://www.icms.org.uk/downloads/cherpdf/openpro.pdf):** 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](http://arxiv.org/abs/0712.1568):** 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!



Update on list of open problems for Cherednik/Symplectic Reflection Algebras
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).


