The subject of this thesis lies at the crossroads of several mathematical domains. Whereas the initial motivation stems from physics, the techniques used to answer the problems we have focused on are more algebraic and geometric in nature. The starting point is the observation that the description of physical phenomena (be it related to particles or fields) is often governed by equations which are described in terms of a differential operator. A well-known example, which lies at the core of this thesis, is the Laplace operator. This is a second-order differential operator which appears in a variety of problems in physics and engineering such as celestial mechanics, the heat transfer problem, fluid dynamics, Maxwell’s equations describing electromagnetism and even quantum mechanics. From a purely mathematical point of view, this operator has given rise to a branch of analysis which is called harmonic analysis.

The Laplace operator has certain invariance properties, which can best be explained in terms of ’symmetry’. Mathematically speaking, this is usually done in terms of groups and algebras. In the first part of this thesis, we have focussed on the rotational invariance of the Laplace operator. We have shown that this operator is part of an algebraic structure which is crucial in the theory of Howe duality. In particular, we have unravelled a new symmetry, hereby using a so-called transvector algebra rather than a Lie algebra. This has certain advantages, as explained in the thesis. What matters for the second part is the observation that the Laplace operator is not only rotationally invariant, but it is an example of a whole range of differential operators which are conformally invariant. The existence of these operators follows from purely mathematical considerations, but they appear in modern theoretical physics when describing the behaviour of so-called higher spin particles. In this thesis, we have focused on a particular family of such operators: the higher spin Laplace operators, for which the easiest example (the lowest spin number) is precisely the classical Laplace operator. Most of our results are describing a particular case (a toy model, so to speak, for which the explicit calculations are still manageable), but also for the most general case we have obtained some partial results.