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.