On the index of maximally hypoelliptic differential operators

TL;DR

This paper derives a general index formula for maximally hypoelliptic differential operators on closed manifolds, extending classical results and providing explicit computations for Hörmander's sum of squares operators.

Contribution

It introduces a unified index formula for *-maximally hypoelliptic operators, generalizing Atiyah-Singer and van Erp, with new explicit calculations for Hörmander's operators.

Findings

Derived a comprehensive index formula for maximally hypoelliptic operators.

Extended classical index theorems to a broader class of hypoelliptic operators.

Provided explicit index calculations for Hörmander's sum of squares operators.

Abstract

We give an index formula for the class of all *-maximally hypoelliptic differential operators on any closed manifold with vector bundle coefficients, generalising previous index formulas by Atiyah-Singer and van Erp. Using this formula, we give new explicit index computations for Hormander's sum of squares operators of arbitrary rank.