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Générateurs de l’algèbre $${\mathcal U}(G)^ K$$ avec $$G=SO(m)$$ ou $$SO_ 0(1,m-1)$$ et $$K=SO(m-1)$$. (French) Zbl 0551.22007
Let $$G=SO(m)$$ or $$SO_ 0(l,m-1)$$, $$K=SO(m-1)$$ and let $${\mathcal U}(G)^ K$$ be the real algebra of differential operators on G, left invariant by G and right invariant by K. The author shows that $${\mathcal U}(G)^ K$$ is generated by the centers of $${\mathcal U}(G)$$ and $${\mathcal U}(K)$$ where $${\mathcal U}(G)$$ (resp. $${\mathcal U}(K))$$ is the real algebra of differential operators on G (resp. on K) invariant by left translations of G (resp. of K). In particular $${\mathcal U}(G)^ K$$ is commutative. The proof makes heavy use of the isomorphism between $${\mathcal U}(G)^ K$$ 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)$$\}$$.
Reviewer: W.Miller jr

##### MSC:
 22E60 Lie algebras of Lie groups 17B35 Universal enveloping (super)algebras
Zbl 0516.22012
Full Text:
##### References:
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