Stein–Weiss Operators and Ellipticity

Abstract

Stein and Weiss (1968,Am. J. Math.90, 163–196) introduced the notion ofgeneralized gradients: equivariant first order differential operatorsGbetween irreducible vector bundles with structure group SO(n) or Spin(n). Among other things, they proved ellipticity for certain systems, analogous to the Cauchy–Riemann equations, and to the (Riemannian signature) Maxwell and Dirac equations, built from these generalized gradients. In this paper, we classify all systems of this type which are elliptic; the answer is valid for all Riemannian or (if spin structure enters) Riemannian spin manifolds. In particular, we find that ellipticity may be attained by assembling surprisingly few generalized gradients. The method employed yields a side benefit: the spectral resolution ofG*Gon the standard sphereSn, for each generalized gradientG. This spectral resolution was previously understood only for operators on “small bundles”—for example, the differential form operatorsδd, anddδ, and the square of the Dirac operator.