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Saturated commutative spaces of Heisenberg type. (English) Zbl 1047.22021
Let \(X=G/K\) be a connected Riemannian space with isometry group \(G\) and denote by \({\mathcal D}(X)^G\) the algebra of \(G\)-invariant differential operators. If \({\mathcal D}(X)^G\) is commutative then \(X\) is also called commutative and the pair \((G,K)\) is called a Gelfand pair. Several other characterizations of commutative spaces, for instance in terms of the Poisson bracket operation on the universal enveloping algebra, are discussed.
\(X\) is said to be of Heisenberg type if in the Levi decomposition \(G=N\rtimes L\) one has \(L=K\). It is called saturated if \(N_F(K)^0=K\). A list of all saturated commutative spaces of Heisenberg type is given, for which \(\mathfrak{n}/\left[\mathfrak{n},\mathfrak{n}\right]\) is reducible.

22F30 Homogeneous spaces
14L30 Group actions on varieties or schemes (quotients)
53C30 Differential geometry of homogeneous manifolds
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