Canonical Representations and Overgroups for Hyperboloids

V. F. Molchanov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 4, pp. 48-61.

Résumé

For the hyperboloid $\mathcal{X}=G/H$, where $G=\operatorname{SO}_0(p,q)$ and $H=\operatorname{SO}_0(p,q-1)$, we define canonical representations $R_{\lambda,\nu}$, $\lambda\in\mathbb{C}$, $\nu=0,1$, as the restrictions to $G$ of representations $\widetilde{R}_{\lambda,\nu}$, associated with a cone, of the group $\widetilde{G}=\operatorname{SO}_0(p+1,q)$. They act on functions on the direct product $\Omega$ of two spheres of dimensions $p-1$ and $q-1$. The manifold $\Omega$ contains two copies of $\mathcal{X}$ as open $G$-orbits. We explicitly describe the interaction of the Lie operators of the group $\widetilde{G}$ in $\widetilde{R}_{\lambda,\nu}$ with the Poisson and Fourier transforms associated with the canonical representations. These transforms are operators intertwining the representations $R_{\lambda,\nu}$ with representations of $G$ associated with a cone.

Détail

Mots-clés : Lie group, pseudo-orthogonal group, Poisson and Fourier transforms.
Keywords: Lie algebra, symmetric space, hyperboloid, canonical representation

Bibliographie

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

[2] Vershik A. M., Gelfand I. M., Graev M. I., “Predstavleniya gruppy $\mathrm{SL}(2,R)$, gde $R$ — koltso funktsii”, UMN, 28:5 (1973), 83–128 | MR

[3] Molchanov V. F., “Predstavleniya psevdoortogonalnoi gruppy, svyazannye s konusom”, Matem. sb., 81:3 (1970), 358–375 | Zbl

[4] Molchanov V. F., “Sfericheskie funktsii na giperboloidakh”, Matem. sb., 99:2 (1976), 139–161 | MR | Zbl

[5] Molchanov V. F., “Formula Plansherelya dlya giperboloidov”, Trudy MIAN, 147, 1980, 65–85 | MR

[6] Molchanov V. F., “Garmonicheskii analiz na odnorodnykh prostranstvakh”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 59, VINITI, M., 1990, 5–144 | MR

[7] Molchanov V. F., “Quantization on para-Hermitian symmetric spaces”, Contemporary Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, 175, 1996, 81–95 | MR | Zbl

[8] Molchanov V. F., Volotova N. B., “Finite dimensional analysis and polynomial quantization on a hyperboloid of one sheet”, Proc. Tambov Summer School-Seminar “Harmonic analysis on homogeneous spaces” (Aug. 26–31, 1996), Vestnik Tambovskogo univ., 3,, no. 1, 1998, 65–78

[9] Molchanov V. F., Canonical representations and overgroups, Preprint Math. Inst. Univ. Leiden, No. MI 2002-05, 2002 | MR

[10] Molchanov V. F., “Canonical representations and overgroups”, Lie Groups and Symmetric Spaces, Amer. Math. Soc. Transl. Ser. 2, 210, 2003, 213–224 | MR | Zbl

[11] Molchanov V. F., “Canonical representations and overgroups for hyperboloids of one sheet and Lobachevsky spaces”, Acta Appl. Math., 86:1 (2005), 115–129 | DOI | MR | Zbl

[12] Neretin Yu. A., “Deistvie nadalgebry v plansherelevskom razlozhenii i operatory sdviga v mnimom napravlenii”, Izv. RAN, ser. matem., 66:5 (2002), 171–182 | DOI | MR | Zbl