Invariant finite-dimensional exponential families on homogeneous spaces

A general form is obtained for the finite-dimensional exponential family invariant with respect to a locally compact group $G$ when it is defined on the measurable quotient spade $(G/\Gamma, \mathcal{A}, \mu)$ of this group with respect to a subgroup $\Gamma$. Conditions for the existence of such families are derived. Examples are given of exponential families on a compact homogeneous space, and the general form of families in $R_n$ invariant with respect to $GL(n)$ is obtained.