Asymptotic equivariant index of Toeplitz operators on the sphere. (English) Zbl 1215.58011
Summary: We illustrate the equivariant asymptotic index described in [L. Boutet de Monvel, RIMS Kôkyûroku Bessatsu B10, 33–45 (2008; Zbl 1179.58013) and L. Boutet de Monvel, E. Leichtman, X. Tang] and [A. Weinstein, 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 (§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 [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.
MSC:
| 58J20 | Index theory and related fixed-point theorems on manifolds |
| 19L47 | Equivariant \(K\)-theory |
| 32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |
| 53D10 | Contact manifolds (general theory) |
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
References:
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