Let $G$ be a connected Lie group and $K$ a compact subgroug of $G$. The question is about the algebra of the differential operators $Diff(G/K)$ on $G/K.$ Let $U(\mathfrak g)$ denote the universal envelopeing algebra for the Lie algebra $\mathfrak g $ for $G.$ Three natural ways to produce differential operators are: $L : U(\mathfrak g) \rightarrow Diff(G/K) $ is the map that send $D$ to left differentiation by $D$. $R: U(\mathfrak g)^K \rightarrow Diff(G/K)$ is the map that sends $D$ to a left invariant differential operator. And finally $M : C^\infty (G/K) \rightarrow Diff(G/K)$ the map that sends $f$ in multiplication by $f$. The the union of the respective image of $ L,R,M$ generates a subalgebra of $Diff(G/K)$. Is this subalgebra the whole algebra?. Certainly, locally is a true.



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