Minimal coadjoint orbits and symplectic induction

Abstract

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 $(\tilde X,\omega _{\tilde X} )$, where dim $\tilde X = 2 + \dim X$. 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 noncompact 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 $\mathfrak{p}$, where $\mathfrak{g} = \mathfrak{k} + \mathfrak{p}$ is a Cartan decomposition. Using the Kostant-Sekiguchi correspondence and a diffeomorphism result of M. Vergne we establish a symplectic isomorphism $(\tilde X,\omega _{\tilde X} ) \cong (Z,\omega _Z )$, where Z is a nonzero minimal “nilpotent” coadjoint orbit of G. This is applied to show that the split forms of the five exceptional Lie groups arise symplectically from the symplectic induction of coadjoint orbits of certain classical groups.

This research was supported in part by NSF contract DMS-0209473 and the KG&G Foundation.

To Alan Weinstein, with admiration and respect.