Conformally invariant differential operators on Heisenberg groups and minimal representations.
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 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 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, for the group $\widetilde{\operatorname{SL}}(3,\mathbb{R})$ by Torasso, and for $\widetilde{\operatorname{SO}}(4,3)$ by Sabourin. Our new approach provides a uniform and systematic treatment of these cases and also gives 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.