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Derived equivalences for rational Cherednik algebras. (English) Zbl 1414.16006
Let $$\mathfrak{h}$$ denote the reflection representation of a complex reflection group $$W$$. A rational Cherednik algebra $$H_c$$ is a flat deformation of the skew-group ring $$S(\mathfrak{h}\oplus \mathfrak{h}^\ast)\# W$$ depending on a parameter $$c\in \mathfrak{p}: = \mathbb{C}^{S/W}$$. This algebra admits a triangular decomposition $$H_c = S(\mathfrak{h}^\ast)\otimes \mathbb{C}W\otimes S(\mathfrak{h})$$, where $$S(\mathfrak{h}^\ast), \mathbb{C}W, S(\mathfrak{h})$$ are embedded as subalgebras. $$\mathcal{O}_c$$ is a full subcategory in $$H_c$$-mod consisting of all modules that are finitely generated over $$S(\mathfrak{h}^\ast)$$ and have locally nilpotent action of $$\mathfrak{h}$$.
The category $$\mathcal{O}_c$$ has a so called highest weight structure that axiomatizes certain upper triangularity properties similar to those of the BGG categories $$\mathcal{O}$$. One consequence of being highest weight is that $$\mathcal{O}_c$$ has finite homological dimension. Moreover, there is a quotient functor $$KZ_c: \mathcal{O}_c \twoheadrightarrow \underline{\mathcal{H}}_q$$-mod introduced in [V. Ginzburg et al., Invent. Math. 154, No. 3, 617–651 (2003; Zbl 1071.20005)] that is fully faithful on the projective objects. We therefore can view $$\mathcal{O}_c$$ as a “resolution of singularities” for $$\underline{\mathcal{H}}_q$$-mod. Here, $$q$$ is recovered from $$c$$ by some kind of exponentiation: there is a $$\mathbb{Z}$$-lattice $$\mathfrak{p}\mathbb{Z}\subset \mathfrak{p}$$ such that the set of Hecke parameters is identified with $$\mathfrak{p}/\mathfrak{p}_{\mathbb{Z}}$$ and $$q=c+\mathfrak{p}_{\mathbb{Z}}$$.
Now let $$c, c^\prime$$ be two Cherednik parameters with $$c-c^\prime\in \mathfrak{p}_{\mathbb{Z}}$$ $$\mathcal{O}_c$$ and $$\mathcal{O}_{c^\prime}$$ are two resolutions of singularities for $$\mathcal{H}$$-mod. A natural question to ask $$q$$ is whether these two resolutions are derived equivalent. R. Rouquier conjectured that this is so in [Mosc. Math. J. 8, No. 1, 119–158 (2008; Zbl 1213.20007)]. The main goal of this paper is to prove this conjecture. It turns out that there is a derived equivalence between $$D^b(\mathcal{O}_c)$$ and $$D^b(\mathcal{O}_{c^\prime})$$ intertwining the functors $$KZ_c, KZ_{c^\prime}$$ .
Reviewer: Wei Feng (Beijing)

##### MSC:
 16E35 Derived categories and associative algebras 16G99 Representation theory of associative rings and algebras 20C08 Hecke algebras and their representations
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