In KazhdanLusztig theory BeilinsonBernstein localization plays an important role. There are results about localization for quantizations of sympectic resolutions by Losev. There are also results about localization for the rational Cherednik algebra due to GordonStafford and KashiwaraRouquier. Does this mean that there should be a generalization of KazhdanLusztig theory to quantizations of symplectic resolutions, in particular the rational Cherednik algebra? For example, do certain sheaves on the Hilbert scheme tell us things about algebraic representation theory of the rational Cherednik algebra?
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Phew, here goes. This is related to stuff I've thought about for a long time, so there's a much longer version of this answer, but let me try to do the short version:
 Category $\mathcal{O}$ for a Cherednik algebra of the algebraic group $G(\ell,1,r)$ does have a KazhdanLusztig theory in the sense that you can compute multiplicities of simples in Vermas by a barinvariant GramSchmidt algorithm. The cleanest way to say this is that there is an isomorphism of the Grothendieck group summed over all $r$ to a graded lift to a higher level quantum Fock space (in the sense of https://arxiv.org/abs/math/9901032); in the notation of that paper $\ell$ is as I used it here, and $n$ is the order of $q=e^{2\pi i h}$ as a root of unity. This was conjectured by Rouquier, and proven roughly simultaneously by RouquierShanVaragnoloVasserot, Losev and myself.
 On the other hand, if you think that "KazhdanLusztig theory" means specifically identifying these multiplicities with local intersection homology, then no, we have no idea how to do that. There is a very interesting theory about connecting symplectic resolutions to modules over algebras (in addition to the papers of Losev, my coauthors and I wrote some papers about that), but geometrically characterizing simples as IC sheaves do is one of the pieces that has proven stubbornly difficult to find.

$\begingroup$ Thank you very much! Additionally, is there an analog of the Hecke algebra for symplectic resolutions/Cherednik algebra and computing the KazhdanLusztig polynomials using the Hecke algebra? (I mean, Cherednik algebra is related to the double affine Hecke algebra, maybe it has to do with that?) $\endgroup$ May 31 at 6:49

1$\begingroup$ I think DAHA is a red herring here (though I can't promise). There is a generalization of the Hecke algebra, is the convolution of the algebra of the resolution (of course, that's just the Weyl algebra for the flag variety, but a graded lift should qdeform it). In the general case, though, the Grothendieck group of category O will be a module over this algebra, but not a free one. Instead, it should have a commuting action of the same algebra for the dual singularity. $\endgroup$– Ben Webster ♦May 31 at 14:43

1$\begingroup$ For the Cherednik algebra of $G(\ell,1,r)$, this is the commuting actions of the $U_q(\mathfrak{\widehat{sl}}_{\ell})$ and $U_q(\mathfrak{\widehat{sl}}_{n})$ on the level $\ell$ Fock space. $\endgroup$– Ben Webster ♦May 31 at 14:44