Representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities
Let \(\Gamma \) be a finite subgroup of \(SL(2,\mathbb{C}).\) The wreath product \(\Gamma _n=S_n\ltimes \Gamma ^n\) acts on \((\mathbb{C}^2)^n\) preserving the product symplectic form. The symplectic reflection algebra \({\mathbb{H}_c}\) is the \(\mathbb{C}\)-algebra, defined as the quotient of the skew group ring \(T(\mathbb{C}^2)^n\star \Gamma _n\) by the relations \(x\otimes y-y\otimes x=\sum _{\gamma } c_{\gamma }\omega _{\gamma }(x,y)\gamma \) for all \(x,y \in (\mathbb{C}^2)^n\) and the symplectic reflections \(\gamma \) and their corresponding forms \(\omega _{\gamma }.\) The authors prove that there is a crepant resolution (i.e. preserving the symplectic forms) \(\pi _{\mathbf c}:Y_{\mathbf c}\to X_{\mathbf c}\) of the spectrum of the centre of the algebra \({\mathbb{H}_c}\) such that there is an equivalence between the bounded derived category of finitely generated \({\mathbb{H}_c}\)-modules and the bounded derived category of coherent sheaves on \(Y_{\mathbf c}.\) As the consequence they get that for all \(x\in X_{\mathbf c}\), there is an isomorphism of Grothendieck groups \(K({\mathbb{H}_c}/m_x{\mathbb{H}_c})\cong K(\pi _c^{-1}(x)).\)
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