On the degenerate principal series of $G_{2(2)}$ induced from a Heisenberg parabolic subgroup

We study degenerate principal series representations of the split real group $G_{2(2)}$ induced from a character of a maximal parabolic subgroup whose unipotent radical is a Heisenberg group. Using the Lie algebra action on the space of $K$-finite vectors, we find the points of reducibility and the complementary series. The minimal representation and a limit of discrete series are identified as kernel of the corresponding Knapp-Stein intertwining operator. Moreover, we show that some quaternionic discrete series representations occur as the subrepresentation on which the family of intertwining operators vanishes of order two.