Minimal coadjoint orbits and symplectic induction

Original · EN

Let (X,ω) be an integral symplectic manifold and let (L,∇) be a quantum line bundle, with connection, over X having ω as curvature. With this data one can define an induced symplectic manifold (X,ω X) where dim X = 2 + dim X. It is then shown that prequantization on X becomes classical Poisson bracket on X. We consider the possibility that if X is the coadjoint orbit of a Lie group K then 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 where = + is a Cartan decomposition. Using the Kostant-Sekiguchi correspondence and a diffeomorphism result of M. Vergne we establish a symplectic isomorphism (X,ω X) (Z,ωZ) where Z is a non-zero minimal "nilpotent" coadjoint orbit of 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.