The equivariant multiplicative McKay conjecture

Let VV be a symplectic vector space, let GG act symplectically on VV, and let XV/GX\to V/G be a symplectic resolution. The natural dilation action of C\mathbb C^* on VV lifts to XX, and uu denotes the degree-two generator of H(BC)=C[u]H^*(B\mathbb C^*)=\mathbb C[u]. If Fi(ZG)F_i(\mathsf ZG) is the filtration on the center of the group algebra and

Rees(ZG)=iFi(ZG)uiZG[u],\operatorname{Rees}_\bullet(\mathsf ZG)=\sum_i F_i(\mathsf ZG)u^i\subset \mathsf ZG[u],

then the equivariant multiplicative McKay conjecture. There is a canonical graded C[u]\mathbb C[u]-algebra isomorphism

HC(X,C)Rees(ZG).H^\bullet_{\mathbb C^*}(X,\mathbb C)\cong \operatorname{Rees}_\bullet(\mathsf ZG).

This is an equivariant refinement of the multiplicative McKay correspondence; the source states that the corresponding nonequivariant algebra isomorphism is proved, but gives no resolution status for this equivariant formulation.

Sources & referencesView supporting material

Primary source

Victor Ginzburg and Dmitry Kaledin, “Poisson deformations of symplectic quotient singularities”, arXiv:math/0212279 (2010).

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