This course is devoted to topics lying on the crossroad of multivariable analysis, representation theory for Lie algebras and theoretical physics. Starting from 2 differential equations playing a crucial role in physics (the Klein-Gordon and Dirac equation for massless particles), we will introduce notions which are useful in both abstract algebra and theoretical physics. For each of these equations, we will study the symmetries, the vector spaces of homogeneous polynomial solutions and their role as representation spaces for the orthogonal Lie algebra, the dual Howe symmetry behind these equations and a few formulas generalising what one already knows from complex analysis (e.g. Cauchy's formula).
In order to do this, we will have to define the Clifford algebra's and their representations: this will allow the introduction of spinors (important when studying the Dirac equation), and simple models for the root system of the orthogonal Lie algebra.