Symplectic Projective Orbits

Abstract

Let H be a complex Hilbert space and P(H) the corresponding projective space, i.e., the space of all one dimensional subspaces of H. If v is a non-zero vector in H we shall denote the corresponding point in P(H), i.e., the line throughv by [v]. Let G be a compact Lie group which is unitarily represented on H so that we may consider the corresponding action of G on P(H). In conjunction with the Hartree-Fock and other approximations, a number of physicists, [2], [4], and [5], have become interested in the following problem: For which smooth vectorsv is the orbit G [v] symplectic? Since G is compact, by projecting onto components we may reduce the problem to the case where the representation is irreducible and hence H is finite dimensional. We restate the question in this case: The unitary structure on H makesP(H) into a Kaehler manifold and thus, in particular, into a symplectic manifold. Each G orbit in P(H) is a submanifold. The question is: for which G orbits is the restriction of the symplectic form of P(H) nondegenerate so that the orbit becomes a symplectic manifold? We shall show that the only symplectic orbits are orbits through projectivized weight vectors. But not all such orbits are symplectic; there is a further restriction on the weight vector that we shall describe. The orbit through the projectivized maximal weight vector is not only symplectic but is also Kaehler, i.e., is a complex submanifold of P(H) and is the only orbit with this property.