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.


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)


Zbl 0820.58024
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