Geodesic
Invariant subspaces in some function spaces on symmetric spaces. III
Izvestiya. Mathematics, Tome 66 (2002) no. 1, pp. 165-200 
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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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $M$.

[1] Platonov S. S., “Invariantnye podprostranstva v nekotorykh funktsionalnykh prostranstvakh na simmetricheskikh prostranstvakh, I”, Izv. RAN. Ser. matem., 59:5 (1995), 127–172 | MR | Zbl

[2] Platonov S. S., “Invariantnye podprostranstva v nekotorykh funktsionalnykh prostranstvakh na simmetricheskikh prostranstvakh, II”, Izv. RAN. Ser. matem., 62:2 (1998), 131–168 | MR | Zbl

[3] Platonov S. S., “Invariantnye podprostranstva v nekotorykh funktsionalnykh prostranstvakh na $n$-mernom prostranstve Lobachevskogo”, Matem. sb., 137:12 (1988), 435–461 | MR | Zbl

[4] Khelgason S., Differentsialnaya geometriya i simmetricheskie prostranstva, Mir, M., 1964 | Zbl

[5] Helgason S., Geometric Analysis on Symmetric Spaces, Amer. Math. Soc., Providence, R.I., 1997 | MR | Zbl

[6] Khelgason S., Gruppy i geometricheskii analiz, Mir, M., 1987 | MR

[7] Goto M., Groskhans F., Poluprostye algebry Li, Mir, M., 1981 | MR | Zbl

[8] Burbaki N., Gruppy i algebry Li, Gl. VII, VIII, Mir, M., 1978 | MR

[9] Helgason S., “A duality for symmetric spaces with applications to group representations, I”, Adv. Math., 5 (1970), 1–154 | DOI | MR | Zbl

[10] Berenstein C. A., “Spectral synthesis on symmetric spaces”, Contemp. Math., 63 (1987), 1–25 | MR | Zbl

[11] Wawrzynczyk A., “Spectral analysis and synthesis on symmetric spaces”, J. Math. Ann. and Appl., 127 (1987), 1–17 | DOI | MR | Zbl

[12] Platonov S. S., “Invariantnye podprostranstva v nekotorykh funktsionalnykh prostranstvakh na simmetricheskikh prostranstvakh ranga 1 (simplekticheskii i osobyi sluchai)”, DAN SSSR, 322:2 (1992), 243–247 | MR | Zbl

[13] Cartan E., Sur la structure des groupes de transformations finis et continus. Oeveres completes. Partie I, V. 1, Paris, 1952

[14] Kass N., “Explicit decompositions of some tensor products of modules for semisimple Lie algebras”, Commun. Algebra, 15 (1987), 2251–2261 | DOI | MR | Zbl