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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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