Asymptotic Equivariant Index of Toeplitz Operators on the Sphere

This article is dedicated to M. Sato, whose work has always been a great source of inspiration. We illustrate the equivariant asymptotic index described in [6, 8] in the case of spheres S2N−1 ⊂ C , 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 [2] for the transversally elliptic pseudodifferential theory.