Minimal coadjoint orbits and symplectic induction
| dc.creator | Kostant, Bertram | |
| dc.date | 2003-12-12 | |
| dc.date.accessioned | 2026-07-07T05:03:51Z | |
| dc.date.available | 2026-07-07T05:03:51Z | |
| dc.description | Let $(X,ω)$ be an integral symplectic manifold and let $(L,\nabla)$ be a quantum line bundle, with connection, over $X$ having $ω$ as curvature. With this data one can define an induced symplectic manifold $(\widetilde {X},ω_{\widetilde {X}})$ where $dim \widetilde {X} = 2 + dim X$. It is then shown that prequantization on $X$ becomes classical Poisson bracket on $\widetilde {X}$. We consider the possibility that if $X$ is the coadjoint orbit of a Lie group $K$ then $\widetilde {X}$ 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 $\p$ where $\g= \k +\p$ is a Cartan decomposition. Using the Kostant-Sekiguchi correspondence and a diffeomorphism result of M. Vergne we establish a symplectic isomorphism $(\widetilde {X},ω_{\widetilde {X}})\cong (Z,ω_Z)$ 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. | |
| dc.description | 38 pages, plain tex | |
| dc.identifier | https://arxiv.org/abs/math/0312252 | |
| dc.identifier | http://arxiv.org/abs/math/0312252 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/69581 | |
| dc.subject | Symplectic Geometry | |
| dc.subject | Group Theory | |
| dc.subject | 53Dxx, 53D50, 53D20, 81SXX, 57D25 | |
| dc.title | Minimal coadjoint orbits and symplectic induction | |
| dc.type | text |