For a simple real Lie group \(G\) with Heisenberg parabolic subgroup \(P\), the author studies the corresponding degenerate principal series representations. For a certain induction parameter the kernel of the conformally invariant system of second order differential operators constructed by Barchini, Kable and Zierau is a subrepresentation which turns out to be the minimal representation. To study this subrepresentation, the author takes the Heisenberg group Fourier transform in the non-compact picture and shows that it yields a new realization of the minimal representation on a space of \(L^2\)-functions. The Lie algebra action is given by differential operators of order \(\leq3\) and the author finds explicit formulas for the functions constituting the lowest \(K\)-type.

These \(L^2\)-models were previously known for the groups \(\mathrm{SO}(n,n)\), \(E_{6(6)}, \), \(E_{7(7)}\) and \(E_{8(8)}\) by Kazhdan and Savin, for the group \(G_{2(2)}\) by Gelfand, and for the group \(\widetilde{\mathrm{SL}}(3,\mathbb{R}))\) by Torasso, using different methods. This new approach provides a uniform and systematic treatment of these cases and also constructs new \(L_{2}\)-models for \(E_{6(2)}, E_{7(-5)}\) and \(E_{8(-24)}\) for which the minimal representation is a continuation of the quaternionic discrete series, and for the groups \( \widetilde{\mathrm{SO}}(p,q)\) with either \(p\geq q=3\) or \(p,q\geq4\) and \(p+q\) even.

As a byproduct of our construction, the author finds an explicit formula for the group action of a non-trivial Weyl group element that, together with the simple action of a parabolic subgroup, generates \(G\).