Search

In this paper we will study both the finite and infinite-dimensional representations of the symplectic Lie algebra $\mathfrak{sp}(2n)$ and develop a polynomial model for these representations. This means that we will associate a certain space of homogeneous polynomials in a matrix variable, intersected with the kernel of $\mathfrak{sp}(2n)$-invariant differential operators related to the symplectic Dirac operator with every irreducible representation of $\mathfrak{sp}(2n)$. We will show that the systems of symplectic Dirac operators can be seen as generators of parafermion algebras. As an application of these new models, we construct a symplectic analogue of the Rarita-Schwinger operator using the theory of transvector algebras.

In this paper we develop the Hermitian refinement of symplectic Clifford analysis, by introducing a complex structure $\mathbb{J}$ on the canonical symplectic manifold $(\mathbb {R}^{2n},\omega_0)$. This gives rise to two symplectic Dirac operators $D_s$ and $D_t$ (in the sense of Habermann), leading to a $\mathfrak{u}(n)$-invariant system of equations on $\mathbb{R}^{2n}$. We discuss the solution space for this system, culminating in a Fischer decomposition for the space of polynomials on $\mathbb {R}^{2n}$ 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 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.

In this paper we use the canonical complex structure $\mathbb{J}$ on $\mathbb{R}^{2n}$ to introduce a twist of the symplectic Dirac operator. As a matter of fact, these operators can be interpreted as the bosonic analogues of the Dirac operators on a Hermitian manifold. Moreover, we prove that the algebra of these symplectic Dirac operators is isomorphic to the Lie algebra $\mathfrak{su}(1,2)$ which leads to the Howe dual pair $(\mathsf{U}(n),\mathfrak{su}(1,2))$.

In this paper, we provide tools to study the dynamics of point vortex dynamics on $\mathbb{CP}^n$ and the flag manifold $\mathbb{F}_{1,2}(\mathbb{C}^3)$. These are the only K\"ahler twistor spaces arising from 4-manifolds. We give an explicit expression for Green's function on $\mathbb{CP}^n$ which enables us to determine the Hamiltonian $H$ and the equations of motions for the point vortex problem on $\mathbb{CP}^n$. Moreover, we determine the momentum map $\mu:\mathbb{F}_{1,2}(\mathbb{C}^3)\to \mathfrak{su}^*(3)$ on the flag manifold., Comment: 38 pages

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