, Volume 34, Issue 3, pp 281-293

Field extensions and isotropic subspaces in symplectic geometry

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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 Sp L (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:Lk. 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.