Geodesic
Invariant finite-dimensional exponential families on homogeneous spaces
Matematičeskie zametki, Tome 7 (1970) no. 6, pp. 707-715.

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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. 
@article{MZM_1970_7_6_a6,
     author = {P. N. Sapozhnikov},
     title = {Invariant finite-dimensional exponential families on homogeneous spaces},
     journal = {Matemati\v{c}eskie zametki},
     pages = {707--715},
     publisher = {mathdoc},
     volume = {7},
     number = {6},
     year = {1970},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_7_6_a6/}
}
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P. N. Sapozhnikov. Invariant finite-dimensional exponential families on homogeneous spaces. Matematičeskie zametki, Tome 7 (1970) no. 6, pp. 707-715. http://geodesic.mathdoc.fr/item/MZM_1970_7_6_a6/