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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

22E60 Lie algebras of Lie groups
17B35 Universal enveloping (super)algebras
Zbl 0516.22012
Full Text: DOI Numdam EuDML
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