The tangential Cauchy–Fueter complex on the quaternionic Heisenberg group

The Cauchy–Fueter operator on the quaternionic space HnHn 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(2N,C), we construct an exact sequence on the quaternionic Heisenberg group, the tangential kk-Cauchy–Fueter complex, resolving the tangential kk-Cauchy–Fueter operator Q0(k). Q0(1) is the tangential Cauchy–Fueter operator. The complex gives the compatible conditions under which the non-homogeneous tangential kk-Cauchy–Fueter equations Q0(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.