Derived equivalences for rational Cherednik algebras.

*(English)*Zbl 1414.16006Let \(\mathfrak{h}\) 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 \(KZ_c, KZ_{c^\prime}\) .

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 |