# Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations

## TL;DR

This paper develops a framework connecting algebraic and analytic indices for elliptic operators linked to discrete groups of quantized canonical transformations, extending index theory to new classes of operators.

## Contribution

It introduces a localized algebraic index for elliptic operators and proves its equality with the localized analytic index using semiclassical calculus and trace asymptotics.

## Key findings

- Localized algebraic and analytic indices coincide.
- Expressed Fredholm index in terms of algebraic index for certain groups.
- Extended index theory to discrete groups of quantized canonical transformations.

## 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.