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On the commutativity of weakly commutative Riemannian homogeneous spaces. (English. Russian original) Zbl 1050.22028
Funct. Anal. Appl. 37, No. 2, 114-122 (2003); translation from Funkts. Anal. Prilozh. 37, No. 2, 41-51 (2003).
Let $$G$$ be a real Lie group with Lie algebra $${\mathfrak g}$$, let $$H \subset G$$ be a compact subgroup with Lie algebra $${\mathfrak h}$$. The homogeneous space $$X=G/H$$ is said to be commutative if the algebra $$D(X)^G$$ of $$G$$-invariant differential operators on $$X$$ is commutative and weakly commutative if the algebra $$P(X)^G$$ of $$G$$-invariant symbols $$P(X)\in \text{gr} \,D(X)$$ is commutative with respect to the Poisson bracket. The commutativity clearly implies the weak commutativity and the author proves the inverse implication by subtle analysis of the structure of the algebra $$U({\mathfrak g})^H$$ of $$H$$-invariant elements in the universal enveloping algebra. The final result is closely related to the conjecture of M. Duflo [Conf. on analysis on homogeneous spaces, August 25–30, Kataka, Japan, 186, 1–5] ensuring the isomorphism between the center of the associative algebra $$(U( {\mathfrak g})/U({\mathfrak g}) {\mathfrak h})^H$$ isomorphic to $$D(X)^G$$ and the center of the Poisson algebra $$(S({\mathfrak g})/S({\mathfrak g}) {\mathfrak h})^H$$ isomorphic to $$P(X)^G$$. However, this conjecture remains open for a general group $$G$$.

##### MSC:
 22F30 Homogeneous spaces 43A85 Harmonic analysis on homogeneous spaces 17B35 Universal enveloping (super)algebras
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