The tangential Cauchy–Fueter complex on the quaternionic Heisenberg group

Abstract

The Cauchy–Fueter operator on the quaternionic space H n induces the tangential Cauchy–Fueter operator on the boundary of a domain. The quaternionic Heisenberg group is a standard model of the boundaries. By using the Penrose transformation associated to a double fibration of homogeneous spaces of Sp ( 2 N , C ) , we construct an exact sequence on the quaternionic Heisenberg group, the tangential k -Cauchy–Fueter complex, resolving the tangential k -Cauchy–Fueter operator Q 0 ( k ) . Q 0 ( 1 ) is the tangential Cauchy–Fueter operator. The complex gives the compatible conditions under which the non-homogeneous tangential k -Cauchy–Fueter equations Q 0 ( k ) u = f are solvable. The operators in this complex are left invariant differential operators on the quaternionic Heisenberg group. This is a quaternionic version of ∂ ¯ b -complex on the Heisenberg group in the theory of several complex variables.