Spherical functions on the group of diffeomorphisms preserving the volume (Q806120)

Let X be a compact n-dimensional \(C^{\infty}\)-manifold with a smooth positive measure v, D the connected component of unity in the group of diffeomorphisms preserving v. Let \(Y\subset X\) such that \(Y\simeq R^ n\), \(v(X\setminus Y)=0\) and let \(D_ Y\) be the connected component of unity in the subgroup of D consisting of diffeomorphisms which are the identity outside of a compact set contained in Y. In this note, the author presents a description of a pair \((S_ 0,\phi_ 0)\) consisting of a topological space \(S_ 0\) and a map \(\phi_ 0: D\to S_ 0\) with dense image. This pair is useful in the study of the space L of the continuous functions f: \(D\to C\), bilaterally invariant with respect to \(D_ Y\).