A New Proof of Branson's Classification of Elliptic Generalized Gradients

TL;DR

This paper provides a new, representation-theoretic proof of Branson's classification of elliptic generalized gradients, extending the approach to broader G-structures and offering a potentially more versatile method.

Contribution

It introduces a novel proof based on representation theory and Kato constants, broadening the classification to include generalized gradients for various G-structures.

Findings

Extended Branson's classification using representation theory.

Recovered classification results up to one special case.

Proposed a method applicable to other G-structures.

Abstract

We give a representation theoretical proof of Branson's classification of minimal elliptic sums of generalized gradients. The original proof uses tools of harmonic analysis, which as powerful as they are, seem to be specific for the structure groups SO(n) and Spin(n). The different approach we propose is based on the relationship between ellipticity and optimal Kato constants and on the representation theory of so(n). Optimal Kato constants for elliptic operators were computed by Calderbank, Gauduchon and Herzlich. We extend their method to all generalized gradients (not necessarily elliptic) and recover Branson's result, up to one special case. The interest of this method is that it is better suited to be applied for classifying elliptic sums of generalized gradients of G-structures, for other subgroups G of the special orthogonal group.