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On the classification of Gel’fand pairs. (English. Russian original) Zbl 1053.22006
Russ. Math. Surv. 58, No. 3, 619-621 (2003); translation from Usp. Mat. Nauk 58, No. 3, 195-196 (2003).
Let \(G\) be a real Lie group, let \(K\) be a compact subgroup of \(G\) and let the Riemannian homogeneous space \(X=G/K\) be connected.
Suppose that \((G,K)\) is a Gelfand pair. Then one has a Levi decomposition \(G=N.L\) where \(N\) is a nilpotent group and \(L\) a reductive subgroup containing \(K\) [E. B. Vinberg, Russ. Math. Surv. 56, 1–60 (2001; Zbl 0996.53034)]. The commutative spaces \(X\) such that \(L=K\) have been classified by E. B. Vinberg [loc. cit.] and [E. B. Vinberg, Moscow Math. Soc. 64 (2002)] under some additional conditions.
Here, the author considers the opposite case \(L\not=K\). The main result announced in this note is a classification of the principal commutative simply connected spaces \(X\) satisfying: \(L\not=K\) and the action of \(L\) on the Lie algebra of \(N\) is locally effective.
22E30 Analysis on real and complex Lie groups
Zbl 0996.53034
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