We illustrate the equivariant asymptotic index described in [6, 8] in the case of spheres , equipped with a unitary action of a compact group, for which this theory is more explicit. The article is mostly a review article, except for the last section (§5) in which we describe conjecturally some very natural generators of the relevant K-theory for a torus action on a sphere, generalizing in our Toeplitz operator context the generators proposed by M. F. Atiyah  for the transversally elliptic pseudodierential theory.
Cite this article
Louis Boutet de Monvel, Asymptotic Equivariant Index of Toeplitz Operators on the Sphere. Publ. Res. Inst. Math. Sci. 47 (2011), no. 1, pp. 141–151