Solvability of second-order left-invariant differential operators on the Heisenberg group

We present some recent results, obtained jointly with Detlef Muller, on solvability of operators of the form 2n ^ a^kVjVk^iaU j.k=i where the Vj are left-invariant vector fields on the Heisenberg group, such that [Vj^ Vj-^-n] = U (1 < j< n) are the only nontrivial relations, and A = (ajk) is a complex symmetric matrix with semi-definite real part. The presentation also contains references on the work done in the past few years in this area.