Research interests & Publications

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 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.

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 Kähler 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.

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

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 $\mathfrak{sp}(m)\subset \mathfrak{sp}(2m)$, 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.