The Heisenberg ultrahyperbolic equation: K-finite and polynomial solutions

Abstract

A family of partial differential operators on the Heisenberg group is introduced and studied. These operators may be regarded as analogues of the ultrahyperbolic operator on Euclidean space. Each of them is conformally invariant under the special linear group. The main focus is on the space of smooth solutions that extend to smooth sections of a suitable line bundle over a generalized flag manifold that contains the Heisenberg group as a dense open subset. The space of polynomial solutions is also considered from the point of view of conformal invariance.