The Complex Geometry and Representation Theory of Statistical Transformation Models

Given a measure space ${\mathcal X}$, we can construct a number of induced structures: eg. its $L^2$ space, the space ${\mathcal P}({\mathcal X})$ of probability distributions on ${\mathcal X}$. If, in addition, ${\mathcal X}$ admits a transitive measure-preserving Lie group action, natural actions are induced on those structures. We expect relationships between these induced structures and actions. We study, in particular, the relations between $L^2({\mathcal X})$ and exponential transformation models on ${\mathcal X}$, which are special "submanifolds" of ${\mathcal P}({\mathcal X})$ closed under the induced action, whose tangent bundles are K\"ahler manifolds (given by Molitor). Geometrically, we show the tangent bundle has, locally, the "same" K\"ahler metric with the Fubini-Study metric on the projectivization of $L^2({\mathcal X})$. Moreover we show the action on the tangent bundle is equivariant with that on $L^2({\mathcal X})$, which is a unitary representation. Finally, in some cases, when the symplectic action on the tangent bundle is Hamiltonian, we show that any coadjoint orbit in the image of its moment map induces, via Kirillov's correspondence from orbit method, irreducible unitary representations that are subrepresentations of the aforementioned representation in $L^2({\mathcal X})$.