Traces and triangles in symmetric symplectic spaces. (English) Zbl 0820.58024

Weinstein, Alan

Maeda, Yoshiaki (ed.) et al., Symplectic geometry and quantization. Papers presented at the 31st Taniguchi international symposium on symplectic geometry and quantization problems held at Sanda, Japan, July 12-17, 1993 and a satellite symposium held at Keio University, Yokohama, Japan, from July 21-24, 1993. Providence, RI: American Mathematical Society. Contemp. Math. 179, 261-270 (1994).

The author’s abstract: “One may describe a noncommutative deformation of the multiplication on a symplectic manifold \(M\) in terms of its kernel, a function of triples \((x,y,z)\) of points of \(M\). The method of symplectic groupoids, which suggests a way to construct such a kernel in the WKB approximation, is applied in this paper to the case in which \(M\) admits a nice family of symplectic symmetries. The phase function of the kernel is shown to be the symplectic area of a “triangle” in \(M\) for which \(x\), \(y\) and \(z\) are the midpoints of the sides. The demonstration is purely- geometric, but the argument is motivated by reasoning using an Atiyah- Bott-Lefschetz type trace formula applied to a geometric quantization line bundle”.
For the entire collection see [Zbl 0810.00022].

Reviewer: P.R.Rodrigues (Rio de Janeiro)

MSC:

37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
22E67	Loop groups and related constructions, group-theoretic treatment
58J20	Index theory and related fixed-point theorems on manifolds
53D50	Geometric quantization
58H05	Pseudogroups and differentiable groupoids

Keywords:

symplectic geometry; symmetric spaces