Abstract We consider elliptic operators associated with discrete groups of quantized canonical transformations. In order to be able to apply results from algebraic index theory, we define the localized algebraic index of the complete symbol of an elliptic operator. With the help of a calculus of semiclassical quantized canonical transformations, a version of Egorov's theorem and a theorem on trace asymptotics for semiclassical Fourier integral operators we show that the localized analytic index and the localized algebraic index coincide. As a corollary, we express the Fredholm index in terms of the algebraic index for a wide class of groups, in particular, for finite extensions of Abelian groups.

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与离散量化正则变换群相关的椭圆算子的解析和代数指数

摘要 我们考虑与离散量化规范变换组相关的椭圆算子。为了能够应用代数指数理论的结果，我们定义了椭圆算子的完整符号的局部代数指数。借助半经典量化正则变换演算、叶戈罗夫定理的一个版本和半经典傅立叶积分算子的迹渐近定理，我们证明了局部解析指数和局部代数指数重合。作为推论，我们用代数指数来表达弗雷德霍姆指数，用于广泛的群，特别是阿贝尔群的有限扩展。