Quaternionic projective invariance of the k-Cauchy-Fueter complex and applications I

The k-Cauchy-Fueter complex in quaternionic analysis is the counterpart of the Dolbeault complex in complex analysis. In this paper, we find the explicit transformation formula of these complexes under $\mathrm{SL}(n+1,H)$, which acts on $H^n$ as quaternionic fractional linear transformations. These transformation formulae have several interesting applications to k-regular functions, the quaternionic counterpart of holomorphic functions, and geometry of domains. They allow us to construct the k-Cauchy-Fueter complex over locally quaternionic projective flat manifolds explicitly and introduce various notions of pluripotential theory on this kind of manifolds. We also construct a quaternionic projectively invariant operator from the quaternionic Monge-Ampère operator, which can be used to find projectively invariant defining density of a domain, generalizing Fefferman's construction in complex analysis.