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Zbl 0551.22007
Benabdallah, Abdel-Ilah
Générateurs de l'algèbre ${\cal U}(G)\sp K$ avec $G=SO(m)$ ou $SO\sb 0(1,m-1)$ et $K=SO(m-1)$. (French)
[J] Bull. Soc. Math. Fr. 111, 303-326 (1983). ISSN 0037-9484

Let $G=SO(m)$ or $SO\sb 0(l,m-1)$, $K=SO(m-1)$ and let ${\cal U}(G)\sp K$ be the real algebra of differential operators on G, left invariant by G and right invariant by K. The author shows that ${\cal U}(G)\sp K$ is generated by the centers of ${\cal U}(G)$ and ${\cal U}(K)$ where ${\cal U}(G)$ (resp. ${\cal U}(K))$ is the real algebra of differential operators on G (resp. on K) invariant by left translations of G (resp. of K). In particular ${\cal U}(G)\sp K$ is commutative. The proof makes heavy use of the isomorphism between ${\cal U}(G)\sp K$ and the algebra of polynomial functions on the Lie algebra of G, invariant under the adjoint representation of K. $\{$ A short version of this paper has been published in Publ. Dép. Math., Nouv. Sér., Univ. Claude Bernard, Lyon 41B, Exp. No.3, 2 p. (1982; Zbl 0516.22012)$\}$.
[W.Miller jr]

MSC 2000:: *22E60 Lie algebras of Lie groups
17B35 Universal enveloping algebras (Lie algebras)

Keywords: invariant differential operators; generators; homogeneous fibre space; algebra of polynomial functions; adjoint representation

Citations: Zbl 0516.22012

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