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 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 ≤ 3 and we find explicit formulas for the functions constituting the lowest K-type. These L 2-models were previously known for the groups 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 ˜SL(3, 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 ˜SO(p, q) with either p ≥ q = 3 or p, q ≥ 4 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.