# Differential operators on $G/K$

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.