Appears in collection : Symmetry in Geometry and Analysis

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 $\leq 3$ and we find explicit formulas for the lowest $K$-type. This realization can be viewed as an analogue of the Schrödinger model for the minimal representation of $O(p, q)$ as constructed by Kobayashi and Ørsted.