  This paper deals with a certain class of second-order conformally invariant
operators acting on functions taking values in particular (finite-dimensional)
irreducible representations of the orthogonal group. These operators can be
seen as a generalisation of the Laplace operator to higher spin as well as a
second order analogue of the Rarita-Schwinger operator. To construct these
operators, we will use the framework of Clifford analysis, a multivariate
function theory in which arbitrary irreducible representations for the
orthogonal group can be realised in terms of polynomials satisfying a system of
differential equations. As a consequence, the functions on which this
particular class of operators act are functions taking values in the space of
harmonics homogeneous of degree k. We prove the ellipticity of these operators
and use this to investigate their kernel, focusing on both polynomial solutions
and the fundamental solution.
