David Eelbode

The Higher Spin Laplace Operator

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 polynomial solutions. Finally, we will also construct the fundamental solution using the theory of Riesz potentials.

Fischer decomposition for osp(4|2)‐monogenics in quaternionic Clifford analysis

Spaces of spinor-valued homogeneous polynomials, and in particular spaces of spinor-valued spherical harmonics, are decomposed in terms of irreducible representations of the symplectic group Sp(p). These Fischer decompositions involve spaces of homogeneous, so-called osp(4|2)-monogenic polynomials, the Lie super algebra osp(4|2) being the Howe dual partner to the symplectic group Sp(p). In order to obtain Sp(p)-irreducibility, this new concept of osp(4|2)-monogenicity has to be introduced as a refinement of quaternionic monogenicity; it is defined by means of the four quaternionic Dirac operators, a scalar Euler operator E underlying the notion of symplectic harmonicity and a multiplicative Clifford algebra operator P underlying the decomposition of spinor space into symplectic cells. These operators E and P, and their Hermitian conjugates, arise naturally when constructing the Howe dual pair osp(4|2)×Sp(p), the action of which will make the Fischer decomposition multiplicity free. Copyright

Fundaments of quaternionic Clifford analysis II: splitting of equations

Quaternionic Clifford analysis is a recent new branch of Clifford analysis, a higher dimensional function theory which refines harmonic analysis and generalizes to higher dimension the theory of holomorphic functions in the complex plane. So-called quaternionic monogenic functions satisfy a system of first-order linear differential equations expressed in terms of four interrelated Dirac operators. The conceptual significance of quaternionic Clifford analysis is unraveled by showing that quaternionic monogenicity can be characterized by means of generalized gradients in the sense of Stein and Weiss. At the same time, connections between quaternionic monogenic functions and other branches of Clifford analysis, viz Hermitian monogenic and standard or Euclidean monogenic functions are established as well.

Decomposition of the polynomial kernel of arbitrary higher spin Dirac operators

In a series of recent papers, we have introduced higher spin Dirac operators, which are generalisations of the classical Dirac operator. Whereas the latter acts on spinor-valued functions, the former acts on functions taking values in arbitrary irreducible half-integer highest weight representations for the spin group. In this paper, we describe how the polynomial kernel spaces of such operators decompose in irreducible representations of the spin group. We will hereby make use of results from representation theory.

Polynomial Solutions For Arbitrary Higher Spin Dirac Operators

In a series of recent papers, we have introduced higher spin Dirac operators, which are far-reaching generalisations of the classical Dirac operator. Whereas the latter acts on spinor-valued functions, the former acts on functions taking values in arbitrary irreducible half-integer highest weight representations for the spin group. In this paper, we describe a general procedure to decompose the polynomial kernel spaces for these operators in irreducible summands for the regular action of the spin group. We will do this in an inductive way, making use of twisted higher spin operators.

The higher spin Laplace operator in several vector variables

In this paper, an explicit expression is obtained for the conformally invariant higher spin Laplace operator

Generalised Maxwell Equations in Higher Dimensions

mathcal{D}_{lambda}

The harmonic transvector algebra in two vector variables

, which acts on functions taking values in an arbitrary (finite-dimensional) irreducible representation for the orthogonal group with integer valued highest weight. Once an explicit expression is obtained, a special kind of (polynomial) solutions of this operator is determined.