A Clifford Algebraic Framework for $\mathfrak{sp}(m)$-Invariant Differential Operators

Abstract.

We introduce a framework for studying differential operators which are invariant with respect to the real (complex) symplectic Lie algebra $\mathfrak{sp}(m)$ ($\mathfrak{sp}_{2m}({\mathbb{C}})$), associated to a quaternionic structure on a vector space ${\mathbb{R}}^{4m}$. To do so, these algebras are realized within the orthogonal Lie algebra $\mathfrak{so}(4m)$. This leads in a natural way to a refinement of the recently introduced notion of complex Hermitean Clifford analysis, in which four variations of the classical Dirac operator play a dominant role.