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.
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Louis Boutet de Monvel, Asymptotic Equivariant Index of Toeplitz Operators on the Sphere. Publ. Res. Inst. Math. Sci. 47 (2011), no. 1, pp. 141–151DOI 10.2977/PRIMS/33