Complexes of invariant operators in several quaternionic variables§

Abstract

Much attention has recently been paid to a study of analogues of the Dolbeault complex for the Dirac equation in several vector variables. In this article, we study these questions in dimension 4, in several quaternionic variables. Instead of the Fueter equation and quaternionic (or spinor) valued functions, we consider invariant first-order equations for functions with values in higher spin representations. We present a classification of conformally invariant equations on with the property that the corresponding equation in n variables is invariant with respect to the symmetry group of the projective quaternionic space. We get two series of equations. For each of them, we construct complexes starting from these equations and we relate them to complexes constructed earlier on quaternionic manifolds by R. Baston and S. Salamon (see Baston, R.J., 1992, Quaternionic Complexes, J. Geom. Phys., 8(1–4), 29–52, and Salamon, S.M., 1986, Differential geometry of quaternionic manifolds, Ann. Scient. Éc. Norm. Sup., 4 o -serie, 19, pp. 31–55, respectively). §Dedicated to Richard Delanghe on the occasion of his 65th birthday.