Stein-Weiss operators and ellipticity (Q1378480)

Stein and Weiss introduced the notion of generalized gradients: equivariant first order differential operators \(G\) between irreducible vector bundles with structure group \(SO(n)\) or \(\text{Spin} (n)\). Among other things, they proved ellipticity for certain systems, analogous to the Cauchy-Riemann equations. In the paper under review, the author classifies all systems of this type which are elliptic. He obtains also the spectral resolution of \(G^*G\) on the standard sphere \(S^n\) for each generalized gradient, which was previously understood only for operators on ``small bundles'', e.g., for \(\delta d\), \(d\delta\) or the square of the Dirac operator.