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Quantization on symplectic symmetric spaces


Symplectic symmetric manifolds G/H with G simple are divided into four classes [17]: (a) Hermitian symmetric spaces; (b) semi-Kaehlerian irreducible symmetric spaces; (c) para-Hermitian symmetric spaces of the first category; (d) para-Hermitian symmetric spaces of the second category. The spaces of the three latter classes are not Riemannian, and each has a Riemannian form belonging to the class of Hermitian symmetric spaces. Berezin constructed quantization on spaces of class (a). We would like to outline a program for a quantization in the spirit of Berezin for other classes of symplectic homogeneous manifolds. In these lectures we restrict ourselves to class (c). The local classification of spaces of class (c) is given in paragraph 3. There is an inspiring analogy between (a) and (c), which starts at the coordinate level: z, z , see paragraph 3, and continues on the level of formulae and so on. On the other hand, it is well-known, that the passage from the Riemannian case to the non-Riemannian one drastically increases the difficulties. So, in this theory there are still many interesting open problems.

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