Principal series of quaternionic and real split exceptional Lie groups induced from Heisenberg parabolic subgroups

Let $G\mathrm{/}K$ be an irreducible quaternionic symmetric space of rank $4$. We study the principal series representation ${\pi }_{\nu }={\text{Ind}}_P^G(1\otimes e^{\nu }\otimes 1)$ of $G$ induced from the Heisenberg parabolic subgroup $P=MAN$ realized on $L^2(K\mathrm{/}L)$, $L=K\cap M$. We find the $K$-types in the induced representation via a double cover $K\mathrm{/}{L}_0\to K\mathrm{/}L$ and a circle bundle $K\mathrm{/}{L}_0\to K\mathrm{/}{L}_1$ over a compact Hermitian symmetric space $K\mathrm{/}{L}_1$. We compute the Lie algebra $\mathfrak{g}$-action of $G$ on the representation space. We find the complementary series, reducible points, and unitary subrepresentations in this family of representations.