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Strict quantization of solvable symmetric spaces. (English) Zbl 1032.53080

If \((M,\omega)\) is a symplectic manifold and \(\nabla\) is a torsion-free affine connection on \(M\) such that \(\nabla\omega=0\), then \((M,\omega,\nabla)\) is a symplectic symmetric space if at every point \(x\in M\) the local geodesic symmetry \(s_x\) extends globally to \(M\) as an affine symplectic transformation. If the group \(G(M)\) of transformations of \(M\) generated by the symmetries \(\{s_x\}\) is a solvable Lie group, then \((M,\omega,\nabla)\) is a solvable symplectic symmetric space. On a symplectic manifold \(M\) the multiplication \(*\) on functions on \(M\) is defined by \((u * v)(x)=\int_{M\times M}K(x,\cdot,\cdot)u\otimes v\), where \(x\in M\) and \(K\) is the three-point kernel defining \(*\).
A. Weinstein [in: Symplectic geometry and quantization, Contemp. Math. 179, 261-270 (1994; Zbl 0820.58024)] defined a WKB-quantization by the integral product formula, where kernels are of the form \(K=a_\hbar e^{\frac{i}{\hbar}S}\), where \(S\) is a real-valued smooth phase function on \(M\times M\times M\) and the amplitude \(a_\hbar\) is usually a power series in \(\hbar\). In this paper, the author determines explicitly the amplitude functions \(a_\hbar\) that define invariant WKB-quantizations of solvable symmetric spaces and oscillatory integral formulas for strongly invariant strict deformation quantizations. Also, the author defines deformed function algebras, that is, spaces of functions on a symmetric space that are stable under the oscillatory integral deformed product.

MSC:

53D50 Geometric quantization
53C35 Differential geometry of symmetric spaces
81S10 Geometry and quantization, symplectic methods
53D55 Deformation quantization, star products
53D05 Symplectic manifolds (general theory)

Citations:

Zbl 0820.58024
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