Geodesic
Invariant subspaces in some function spaces on symmetric spaces.~III
Izvestiya. Mathematics , Tome 66 (2002) no. 1, pp. 165-200.

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We describe the structure of closed linear subspaces in some topological vector spaces that consist of functions on the exceptional non-compact symmetric space $M=F_4/{\operatorname{Spin}(9)}$ (the Cayley space) and are invariant under the natural quasi-regular representation of the group $F_4$. The class of function spaces under consideration contains the spaces $C^d(M)$ of $d$-times continuously differentiable functions ($d=0,1,\dots,\infty$) and the spaces of functions of exponential growth on $M$. 
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S. S. Platonov. Invariant subspaces in some function spaces on symmetric spaces.~III. Izvestiya. Mathematics , Tome 66 (2002) no. 1, pp. 165-200. http://geodesic.mathdoc.fr/item/IM2_2002_66_1_a7/

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