A relationship between symplectic geometry and information geometry is studied. The square of a dually flat space admits a natural symplectic structure that is the pullback of the canonical symplectic structure on the cotangent bundle of the dually flat space via the canonical divergence. With respect to the symplectic structure, there exists a moment map whose image is the dually flat space. As an example, we obtain a duality relation between the Fubini-Study metric on a projective space and the Fisher metric on a statistical model on a finite set. Conversely, a dually flat space admitting a symplectic structure is locally symplectically isomorphic to the cotangent bundle with the canonical symplectic structure of some dually flat space. We also discuss nonparametric cases.2010 Mathematics subject classification: primary 53D20; secondary 60D05.
In this paper we consider a homogeneous holomorphic line bundle over an elliptic adjoint orbit of a real semisimple Lie group, and set a continuous representation of the Lie group on a certain complex vector subspace of the complex vector space of holomorphic cross-sections of the line bundle. Then, we demonstrate that the representation is irreducible unitary.
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