In this paper we consider (polynomial) solution spaces for the symplectic Dirac operator (with a focus on $1$-homogeneous solutions). This space forms an infinite-dimensional representation space for the symplectic Lie algebra $\mathfrak{sp}(2m)$. Because $\mathfrak{so}(m)\subset \mathfrak{sp}(2m)$, this leads to a branching problem which generalises the classical Fischer decomposition in harmonic analysis. Due to the infinite nature of the solution spaces for the symplectic Dirac operators, this is a non-trivial question: both the summands appearing in the decomposition and their explicit embedding factors will be determined in terms of a suitable Mickelsson-Zhelobenko algebra.

Plane-based Geometric Algebra (PGA) has revealed points in a $d$-dimensional pseudo-Euclidean space $\mathbb{R}_{p,q,1}$ to be represented by $d$-blades rather than vectors. This discovery allows points to be factored into $d$ orthogonal hyperplanes, establishing points as pseudoscalars of a local geometric algebra $\mathbb{R}_{pq}$. Astonishingly, the non-uniqueness of this factorization reveals the existence of a local $\text{Spin}(p,q)$ geometric gauge group at each point. Moreover, a point can alternatively be factored into a product of the elements of the Cartan subalgebra of $\mathfrak{spin}(p,q)$, which are traditionally used to label spinor representations. Therefore, points reveal previously hidden geometric foundations for some of quantum field theory's mysteries. This work outlines the impact of PGA on the study of spinor representations in any number of dimensions, and is the first in a research programme exploring the consequences of this insight.

In this paper, we will prove that the reproducing kernels Z_k(x,u) for the spaces ℋ_k(ℝ^m,ℂ) of k-homogeneous harmonics can be seen as elements of an infinite-dimensional ladder operator representation for a cubic polynomial angular momentum algebra which is known as the Higgs algebra. This algebra will be shown to be one of two direct summands in a transvector algebra which is related to the harmonic Fischer decomposition in two vector variables.

In this paper the Cayley–Laplace operator Δ _xu is considered, a rotationally invariant differential operator which can be seen as a generalisation of the classical Laplace operator for functions depending on wedge variables X_ab (the minors of a matrix variable). We will show that the Bessel–Clifford function appears naturally in the framework of two-wedge variables, and explain how this function somehow plays the role of the exponential function in the framework of Grassmannians. This will be used to obtain a generalisation of the series expansion for the Newtonian potential, and to investigate a new kind of binomial polynomials related to Nayarana numbers.

In this paper, we develop the Hermitian refinement of symplectic Clifford analysis, by introducing a complex structure on the canonical symplectic manifold . This gives rise to two symplectic Dirac operators and (in the sense of Habermann), leading to a ‐invariant system of equations on . We discuss the solution space for this system, culminating in a Fischer decomposition for the space of (harmonic) polynomials on with values in the symplectic spinors. To make this decomposition explicit, we will construct the associated embedding factors using a transvector algebra.

In this paper we study the $$\mathfrak {sp}(2m)$$ -invariant Dirac operator $$D_s$$ which acts on symplectic spinors, from an orthogonal point of view. By this we mean that we will focus on the subalgebra $$\mathfrak {so}(m) \subset \mathfrak {sp}(2m)$$ , as this will allow us to derive branching rules for the space of 1-homogeneous polynomial solutions for the operator $$D_s$$ (hence generalising the classical Fischer decomposition in harmonic analysis for a vector variable in $${\mathbb {R}}^m$$ ). To arrive at this result we use techniques from representation theory, including the transvector algebra $${\mathcal {Z}}(\mathfrak {sp}(4),\mathfrak {so}(4))$$ and tensor products of Verma modules.

This paper is devoted to the role played by the Higgs algebra H_3 in the generalisation of classical harmonic analysis from the sphere S^m-1 to the (oriented) Grassmann manifold Gr_o(m,2) of 2-planes. This algebra is identified as the dual partner (in the sense of Howe duality) of the orthogonal group SO(m) acting on functions on the Grassmannian. This is then used to obtain a Pizzetti formula for integration over this manifold. The resulting formulas are finally compared to formulas obtained earlier for the Pizzetti integration over Stiefel manifolds, using an argument involving symmetry reduction.

In this paper we present a symplectic analogue of the Fueter theorem. This allows the construction of special (polynomial) solutions for the symplectic Dirac operator $$D_s$$ , which is defined as the first-order $$\mathfrak {sp}(2n)$$ -invariant differential operator acting on functions on $$\mathbb {R}^{2n}$$ taking values in the metaplectic spinor representation.

In this paper, we generalize Fueter's theorem to the higher spin setting. To do so, we consider an alternative proof for the celebrated theorem that uses the Fischer decomposition. This decomposition is then extended to spaces of polynomials that depend on wedge variables, after which we can finish the proof of our higher spin Fueter theorem.

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

The Cauchy–Kovalevskaya extension for polynomials in several variables is a crucial result in Clifford analysis which describes theoretically how to construct simplicial monogenics in m dimensions starting from certain polynomial spaces in \(m-1\) dimensions, hereby using the so-called branching rules. In [13] it was shown that the Cauchy–Kovalevskaya extension map is an isomorphism of the kernel of the wedge operator onto the space of spherical monogenics in two variables. The aim of this paper is to show that this extension maps simplicial polynomials of the kernel of the wedge operator just onto the simplicial monogenics.

The decomposition of polynomials of one vector variable into irreducible modules for the orthogonal group is a crucial result in harmonic analysis which makes use of the Howe duality theorem and leads to the study of spherical harmonics. The aim of the present paper is to describe a decomposition of polynomials in two vector variables and to obtain projection operators on each of the irreducible components. To do so, a particular transvector algebra will be used as a new dual partner for the orthogonal group leading to a generalisation of the classical Howe duality. The results are subsequently used to obtain explicit projection operators and formulas for integration of polynomials over the associated Stiefel manifold.

In this paper we generalise the harmonic Gegenbauer polynomials to the higher spin setting. To do so we will consider the space of simplicial harmonics and look for polynomials that are invariant with respect to a particular subalgebra of the orthogonal Lie algebra. Analogue to the classic case we will construct a ladder operator which generates our special functions and use them to construct Appell sequences.

In recent work a deformation of the classical Dirac operator in \(\mathbb {R}^m\) was introduced. The key idea behind this deformation is a family of new realizations of the Lie superalgebra \(\mathfrak {osp}(1|2)\), by means of a so-called radially deformed Dirac operator \(\mathbf D \) depending on a deformation parameter c, such that for \(c=0\) the classical Dirac operator is reobtained. In this paper, we investigate various properties of this deformation. We first determine the conformal structure of \(\mathbf D \) and obtain a version of Stokes’ theorem. Subsequently we derive an explicit form for the kernel of the associated Fourier transform in terms of trigonometric functions.

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.

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.

We prove the Fischer decomposition for the space of spinor-valued polynomials, defined on Euclidean space of four-fold dimension, in terms of irreducible modules for the symplectic group, consisting of so-called osp(4|2)-monogenics.

This paper deals with the generalisation of the classical Maxwell equations to arbitrary dimension \(m\) and their connections with the Rarita–Schwinger equation. This is done using the framework of Clifford analysis, a multivariate function theory in which arbitrary irreducible representations for the spin group can be realised in terms of polynomials satisfying a system of differential equations. This allows the construction of generalised wave equations in terms of the unique conformally invariant second-order operator acting on harmonic-valued functions. We prove the ellipticity of this operator and use this to investigate the kernel, focusing on both polynomial solutions and the fundamental solution.

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)xSp(p), the action of which will make the Fischer decomposition multiplicity free. Copyright (c) 2016 John Wiley & Sons, Ltd.

This chapter focuses on the use of Clifford analysis techniques as an encompassing and unifying tool to study higher spin generalizations of the classical Dirac operator. These operators belong to a complete family of conformally invariant first-order differential operators, acting on functions taking their values in an irreducible representation for the spin group (the double cover for the orthogonal group). Their existence follows from a standard classification result due to Fegan (Q. J. Math. 27:513–538, 1976), and a canonical way to construct them is to use the technique of Stein–Weiss gradients. This then gives rise to two kinds of differential operators defined on irreducible tensor fields, the standard language used in, e.g., theoretical physics, where higher spin operators appear in the equations of motion for elementary particles having arbitrary half-integer spin: on the one hand, there are the (elliptic) generalizations of the Dirac operator, acting as endomorphisms on the space of smooth functions with values in a fixed module (i.e., preserving the values), and on the other hand there are the invariant operators acting between functions taking values in different modules for the spin group (the so-called twistor operators and their duals). In this chapter, both types of higher spin operators will be defined on spinor-valued functions of a matrix variable (i.e., in several vector variables): this has the advantage that the resulting equations become more transparent, and it allows using techniques for Clifford analysis in several variables. In particular, it provides an elegant framework to develop a function theory for the aforementioned operators, such as a full description of the (polynomial) null solutions and analogues of the classical Cauchy integral formula.