仿射亚辛群的非交换轨道折叠和等变Zeta函数的局部指标公式
Local Index Formulae on Noncommutative Orbifolds and Equivariant Zeta Functions for the Affine Metaplectic Group
摘要：
我们考虑由等距仿射正则变换的量子化生成的$L^2(\mathbb{R}^n)$上的有界算子的代数$A$。代数$A$包括所有非交换环面和环面orbifold作为子代数。我们定义了谱三元组$(A, H, D)$，其中包含$H=L^2(\mathbb R^n, Λ(\mathbb R^n))$和Euler算子$D$，一阶微分算子的指数为$1$ .证明了该谱三元组具有单维谱：对于由Shubin型伪微分算子和$A$的元素生成的代数$Ψ(A,H,D)$中的每一个算子$B$，zeta函数$ζ_B(z) = {\rm Tr} (B|D|^{-2z})$具有至多单极点的$\mathbb C$的亚纯扩张.我们的主要结果是一个明确的代数表达式的Connes-Moscovici循环上循环。作为推论，我们得到了非交换环面和环面orbifolds的局部指数公式。
英文摘要：
We consider the algebra $A$ of bounded operators on $L^2(\mathbb{R}^n)$ generated by quantizations of isometric affine canonical transformations. The algebra $A$ includes as subalgebras all noncommutative tori and toric orbifolds. We define the spectral triple $(A, H, D)$ with $H=L^2(\mathbb R^n, Λ(\mathbb R^n))$ and the Euler operator $D$, a first order differential operator of index $1$. We show that this spectral triple has simple dimension spectrum: For every operator $B$ in the algebra $Ψ(A,H,D)$ generated by the Shubin type pseudodifferential operators and the elements of $A$, the zeta function $ζ_B(z) = {\rm Tr} (B|D|^{-2z})$ has a meromorphic extension to $\mathbb C$ with at most simple poles. Our main result then is an explicit algebraic expression for the Connes-Moscovici cyclic cocycle. As a corollary we obtain local index formulae for noncommutative tori and toric orbifolds.
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