A note on Atiyah's $\Gamma$-index theorem in Heisenberg calculus

TL;DR

This paper extends Atiyah's $\Gamma$-index theorem to Heisenberg elliptic operators on Galois coverings, providing a new index theorem for non-elliptic operators and an example with non-trivial $\Gamma$-index.

Contribution

It introduces an index theorem for Heisenberg elliptic operators on Galois coverings, generalizing Atiyah's theorem to a non-elliptic setting.

Findings

Proved an index theorem for Heisenberg elliptic operators on Galois coverings.

Provided an example of Heisenberg operators with non-trivial $\Gamma$-index.

Extended the applicability of $\Gamma$-index theory to non-elliptic operators.

Abstract

In this note, we prove an index theorem on Galois covering for Heisenberg elliptic differential operators, which is not elliptic, analogous to Atiyah's $\Gamma$-index theorem. This note also contains an example of Heisenberg differential operators with non-trivial $\Gamma$-index.