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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.
##### MSC:
 2.2e+31 Analysis on real and complex Lie groups
##### Keywords:
Gelfand pairs; commutative homogeneous space
Zbl 0996.53034
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