Minimal coadjoint orbits and symplectic induction

Let $(X,\omega )$ be an integral symplectic manifold and let $(L,\mathrm{\nabla })$ be a quantum line bundle, with connection, over $X$ having $\omega$ as curvature. With this data one can define an induced symplectic manifold $(\widetilde {X},\omega_{\widetilde {X}})$where$dim \widetilde {X} = 2 + dim X$. It is then shown that prequantization on $X$ becomes classical Poisson bracket on $\tilde{X}$. We consider the possibility that if $X$ is the coadjoint orbit of a Lie group $K$ then $\tilde{X}$ is the coadjoint orbit of some larger Lie group $G$. We show that this is the case if $G$ is a non-compact simple Lie group with a finite center and $K$ is the maximal compact subgroup of $G$. The coadjoint orbit $X$ arises (Borel-Weil) from the action of $K$ on $\text{p}$ where $\text{g}=\text{k}+\text{p}$ is a Cartan decomposition. Using the Kostant-Sekiguchi correspondence and a diffeomorphism result of M. Vergne we establish a symplectic isomorphism $(\widetilde {X},\omega_{\widetilde {X}})\cong (Z,\omega_Z)$where$Z$isanon-zerominimal\mathrm{"}nilpotent\mathrm{"}coadjointorbitof$G$. This is applied to show that the split forms of the 5 exceptional Lie groups arise symplectically from the symplectic induction of coadjoint orbits of certain classical groups.