Let G be simple Lie group and P be a Heisenberg parabolical subgroup. We study the induced representations of G from P by a one-dimensional representation, their complementary series, composition series and subrepresentations. When G is a Hermitian Lie group they have been studied both in the compact and non-compact realizations by Frahm, Weiske and me. Here we shall consider the quaternionic and real split exceptional Lie groups G, and we combine our earlier method and another method developed by Branson-Olafsson-Orsted. Time permitting, I might also explain some connections with sub-Riemannian geometry.