  For a simple real Lie group $G$ with Heisenberg parabolic subgroup $P$, we
study 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, we take the Heisenberg group Fourier transform in
the non-compact picture and show 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 we find explicit formulas
for the functions constituting the lowest $K$-type.
  These $L^2$-models were previously known for the groups
$\operatorname{SO}(n,n)$, $E_{6(6)}$, $E_{7(7)}$ and $E_{8(8)}$ by Kazhdan and
Savin, for the group $G_{2(2)}$ by Gelfand, and for the group
$\widetilde{\operatorname{SL}}(3,\mathbb{R})$ by Torasso, using different
methods. Our 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
$\widetilde{\operatorname{SO}}(p,q)$ with either $p\geq q=3$ or $p,q\geq4$ and
$p+q$ even.
  As a byproduct of our construction, we find 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 $G$.
