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For a simple real Lie group $G$ with Heisenberg parabolic subgroup $P$, westudy the corresponding degenerate principal series representations. For acertain induction parameter the kernel of the conformally invariant system ofsecond order differential operators constructed by Barchini, Kable and Zierauis a subrepresentation which turns out to be the minimal representation. Tostudy this subrepresentation, we take the Heisenberg Fourier transform in thenon-compact picture and show that it yields a new realization of the minimalrepresentation on a space of $L^2$-functions. The Lie algebra action is givenby differential operators of order $\leq3$ and we find explicit formulas forthe 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 andSavin, for the group $G_{2(2)}$ by Gelfand, for the group$\widetilde{\operatorname{SL}}(3,\mathbb{R})$ by Torasso, and for$\widetilde{\operatorname{SO}}(4,3)$ by Sabourin. Our new approach provides auniform and systematic treatment of these cases and also gives new $L^2$-modelsfor $E_{6(2)}$, $E_{7(-5)}$ and $E_{8(-24)}$ for which the minimalrepresentation is a continuation of the quaternionic discrete series, and forthe groups $\widetilde{\operatorname{SO}}(p,q)$ with either $p\geq q=3$ or$p,q\geq4$ and $p+q$ even.

Conformally invariant differential operators on Heisenberg groups and minimal representations