扭Cherednik代数的层作为泛滤形式变形
Sheaves of Twisted Cherednik Algebras as Universal Filtered Formal Deformations
摘要：
根据Pavel Etingof的陈述，在有限群$G$具有忠实作用的仿射簇$X$的特殊情况下，具有形式参数$c, ψ$的（扭曲）Cherednik代数$\mathcal{H}_{1, c, ψ, X, G}$的层是$\mathcal{D}_X\rtimes G$的通用形式变形，其中$\mathcal{D}_X$是$X$上的微分算子层。本文将Etingof的结果推广到非仿射情形。证明了对于一般光滑解析簇或代数簇$X$，具有形式$c$和$ψ$的层$\mathcal{H}_{1, c, ψ, X, G}$是$\mathcal{D}_X\rtimes G$的泛滤形式变形.为此，我们首先构造了$\mathcal{D}_X\rtimes G$的Hochschild（co）链复形与$X$中$g$-不动点子流形的余切丛上的全纯微分形式层的$G$中所有元素$g$上的直和的$G$ -不变部分之间的拟同构.最后，我们结合这些准同构的过滤结合代数层的代数扩展理论的结果，建立一个双射之间的对应空间的过滤无穷小变形的$\mathcal{D}_X\rtimes G$和参数空间的$\mathcal{H}_{1, c, ψ, X, G}$的同构类。
英文摘要：
According to a statement by Pavel Etingof, in the special case of an affine variety $X$ with a faithful action by a finite group $G$, the sheaf of (twisted) Cherednik algebras $\mathcal{H}_{1, c, ψ, X, G}$ with formal parameters $c, ψ$ is a universal formal deformation of $\mathcal{D}_X\rtimes G$ where $\mathcal{D}_X$ is the sheaf of differential operators on $X$. In the current note, we generalize Etingof's result to the non-affine case. We prove that for a generic smooth analytic or algebraic variety $X$, the sheaf $\mathcal{H}_{1, c, ψ, X, G}$ with formal $c$ and $ψ$ is a universal filtered formal deformation of $\mathcal{D}_X\rtimes G$. To that aim, we first construct quasi-isomorphisms between the Hochschild (co)chain complex of $\mathcal{D}_X\rtimes G$ and the $G$-invariant part of the direct sum over all elements $g$ in $G$ of sheaves of holomorphic differential forms on the cotangent bundles of the $g$-fixed point submanifolds in $X$. Finally, we combine these quasi-isomorphisms with results from the theory of algebraic extensions for sheaves of filtered associative algebras to establish a bijective correspondence between the space of isomorphism classes of filtered infinitesimal deformations of $\mathcal{D}_X\rtimes G$ and the parameter space of $\mathcal{H}_{1, c, ψ, X, G}$.
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