  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 $L^2({\mathcal X})$.
