Factorization of Laplace Operators on Higher Spin Representations

Abstract

This paper deals with the problem of factorizing integer powers of the Laplace operator acting on functions taking values in higher spin representations. This is a far-reaching generalization of the well-known fact that the square of the Dirac operator is equal to the Laplace operator.

Using algebraic properties of projections of Stein-Weiss gradients, i.e. generalized Rarita-Schwinger and twistor operators, we give a sharp upper bound on the order of polyharmonicity for functions with values in a given representation with half-integral highest weight.

Keywords

• Laplace operators
• Dirac operators
• Spin group
• Clifford algebras