## Invariant Differential Operators on the Schr\"{o}dinger Model for the Minimal Representation of the Conformal Group

J. Math. Sci. Univ. Tokyo
Vol. 18 (2011), No. 3, Page 355--395.

Kowata, Atsutaka; Moriwaki, Masayasu
Invariant Differential Operators on the Schr\"{o}dinger Model for the Minimal Representation of the Conformal Group
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Abstract:
We consider the Schr\"{o}dinger model of the minimal representation for the conformal group $O(2n,2) \, (n>1)$ which was constructed by Kobayashi-\O rsted [Adv. Math. 2003], and enriched by a series of papers by Kobayashi-Mano [Memoirs of AMS 2011, etc]. We get the joint spectra of the differential operators on the model for generators of the center of the Lie algebra of $U(k) \times U(n-k) \times U(1)$ for $k=1, \dots, n-1$. Further, we obtain the generators of the algebra consisting of all invariant differential operators for two compact subgroups $I_{1} \times SO(2n-1) \times SO(2)$ and $SO(2n) \times SO(2)$ of $O(2n,2)$.

Mathematics Subject Classification (2010): Primary 22E45; Secondary 47F05.