Asymptotic equivariant index of Toeplitz operators on the sphere (Q533370)

Summary: We illustrate the equivariant asymptotic index described in [\textit{L. Boutet de Monvel}, RIMS Kôkyûroku Bessatsu B10, 33--45 (2008; Zbl 1179.58013) and \textit{L. Boutet de Monvel, E. Leichtman, X. Tang}] and [\textit{A. Weinstein}, \url{arXiv:0808.136501}] in the case of spheres \(\mathbb {S}^{2N-1}\subset\mathbb {C}^N\), 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 ({\S}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 \textit{M. F. Atiyah} [Elliptic operators and compact groups. Lecture Notes in Mathematics. 401. Berlin-Heidelberg-New York: Springer-Verlag (1974; Zbl 0297.58009)] for the transversally elliptic pseudodifferential theory.