A new proof of Branson’s classification of elliptic generalized gradients

Abstract

We give a representation theoretical proof of Branson's classification, [4], 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 a local one, 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, [8]. 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.