Nonlocal elliptic operators for compact lie groups

The class of nonlocal operators corresponding to shifts along orbits under the action of a compact Lie group on a smooth manifold is considered. A Lie group invariant metric on the manifold and a normalized Haar measure corresponding to a bi-invariant metric on the group is assumed. It is found that the nonlocal operator is elliptic if there exists an element is invertible in the algebra. If a nonlocal operator is elliptic, then it is found to determine a Fredholm operator. For an orthonormal basic in the Lie algebra, the corresponding vector fields on the manifold are defined. An element is found to induce an orthogonal endomorphism of the bundle and the exterior form bundle. The restriction of operator to the subspace of sections invariant with respect to representation is isomorphic to the given nonlocal operator.