论文标题
正式Cherednik代数的痕量密度和代数索引定理
Trace Densities and Algebraic Index Theorems for Sheaves of Formal Cherednik Algebras
论文作者
论文摘要
我们展示了Cherednik代数的新颖构造如何通过作者先前的作品中正式的几何形状在商Orbifold Y = x/g上进行，从而导致了Cherednik代数的结果，直到最近才被视为棘手。首先，对于$ x $中的每个轨道类型层，我们为Cherednik代数的hochschild链络合物定义了痕量密度图，该图概述了标准的Engeli-Felder的痕量密度构造，用于差异操作员的融合。其次，通过新获得的痕量密度图，我们证明了$ \ mathbb c_y [[[\ hbar]] $模块的派生类别中的同构类别，该模块计算了Hochschschild链链复杂$ \ MATHCAL {C} c} c} c} _______ {\ chered seef的hochschschild Chain Complex $ \ Mathcal {C}我们表明，这种过度酒精学对Orbifold $ x/g $的Chen-Ruan共同体是同构，其值与正式Power系列$ \ Mathbb C [[\ hbar] $中的值。我们推断，偏斜组代数$ \ MATHCAL {D} _X \ rtimes G $具有明确定义的Euler特征，与$ x/g $的拓扑欧拉特征成正比。最后，我们证明了代数索引定理。
We show how a novel construction of the sheaf of Cherednik algebras on a quotient orbifold Y=X/G by virtue of formal geometry in author's prior work leads to results for the sheaf of Cherednik algebra which until recently were viewed as intractable. First, for every orbit type stratum in $X$, we define a trace density map for the Hochschild chain complex of the sheaf of Cherednik algebras, which generalises the standard Engeli-Felder's trace density construction for the sheaf of differential operators. Second, by means of the newly obtained trace density maps, we prove an isomorphism in the derived category of complexes of $\mathbb C_Y[[\hbar]]$-modules which computes the hypercohomology of the Hochschild chain complex $\mathcal{C}_{\bullet}$ of the sheaf of formal Cherednik algebras. We show that this hypercohomology is isomorphic to the Chen-Ruan cohomology of the orbifold $X/G$ with values in the ring of formal power series $\mathbb C[[\hbar]]$. We infer that the Hochschild chain complex of the sheaf of skew group algebras $\mathcal{D}_X\rtimes G$ has a well-defined Euler characteristic which is proportional to the topological Euler characteristic of $X/G$. Finally, we prove an algebraic index theorem.
