Geodesic
Canonical representations on two-sheeted hyperboloids
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIV, Tome 331 (2006), pp. 91-124 
V. F. Molchanov. Canonical representations on two-sheeted hyperboloids. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIV, Tome 331 (2006), pp. 91-124. http://geodesic.mathdoc.fr/item/ZNSL_2006_331_a7/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The two-sheeted hyperboloid $\mathcal L$ in $\mathbb R^n$ can be identified with the unit sphere $\Omega$ in $\mathbb R^n$ without the equator. Canonical representations of the group $G=\mathrm{SO}_0(n-1,1)$ on $\mathcal L$ are defined as the restrictions to $G$ of the representations of the overgroup $\widetilde G=\mathrm{SO}_0(n,1)$ associated with a cone. They act on functions and distributions on the sphere $\Omega$. We decompose these canonical representations into irreducible constituents and decompose the Berezin form.

[1] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii. Gipergeometricheskaya funktsiya, funktsii Lezhandra, Nauka, M., 1965

[2] G. Beitmen, A. Erdeii, Tablitsy integralnykh preobrazovanii, t. II, Nauka, M., 1970

[3] F. A. Berezin, “Kvantovanie v kompleksnykh simmetricheskikh prostranstvakh”, Izv. Akad. Nauk SSSR, ser. matem., 39:2 (1975), 363–402 | MR | Zbl

[4] A. M. Vershik, I. M. Gelfand, M. I. Graev, “Predstavleniya gruppy $SL(2,R)$, gde $R$ – koltso funktsii”, Uspekhi matem. nauk, 28:5 (1973), 83–128 | MR

[5] N. Ya. Vilenkin, Spetsialnye funktsii i teoriya predstavlenii grupp, Nauka, M., 1965 | MR

[6] L. I. Grosheva, “Kanonicheskie i granichnye predstavleniya na prostranstve Lobachevskogo”, Vestnik Tambovskogo un-ta, 9:3 (2004), 306–311

[7] V. F. Molchanov, “Garmonicheskii analiz na odnorodnykh prostranstvakh”, Itogi nauki i tekhn. Sovr. probl. matem. Fundam. napr., 59, VINITI, 1990, 5–144 | MR

[8] G. van Dijk, S. Hille, “Canonical representations related to hyperbolic spaces”, J. Funct. Anal., 147 (1997), 109–139 | DOI | MR | Zbl

[9] V. F. Molchanov, “Quantization on para-Hermitian symmetric spaces”, Amer. Math. Soc. Transl., Ser. 2, 175(31), AMS, Providence, RI, 1996, 81–95 | MR

[10] V. F. Molchanov, “Canonical and boundary representations on a hyperboloid of one sheet”, Acta Appl. Math., 81:1–3 (2004), 191–204 | DOI | MR

[11] V. F. Molchanov, “Canonical representations on the two-sheeted hyperboloid”, Indag. Math., 16:3–4 (2005), 609–630 | DOI | MR | Zbl

[12] V. F. Molchanov, L. I. Grosheva, “Canonical and boundary representations on the Lobachevsky plane”, Acta Appl. Math., 79:1–2 (2003), 59–77 | MR