Field extensions and isotropic subspaces in symplectic geometry
- Cite this article as:
- Kim, D.S. & Rabau, P. Geom Dedicata (1990) 34: 281. doi:10.1007/BF00181690
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Let L/k be a finite field extension and let (V, B) be a finite dimensional symplectic space over L. We examine the action of the symplectic group SpL(V) on the set of B′-isotropic k-subspaces of V, where B′=ψ°B is the k-symplectic form induced by a ‘trace’ map ψ:L→k. The orbits are completely classified in the case of a quadratic extension and for maximal B′-isotropic subspaces in the case of a cubic extension; the number of orbits of maximal B′-isotropic subspaces is shown to be infinite if the degree of the extension is at least 4.