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*-products on some Kähler manifolds. (English) Zbl 0618.53049
Let $$(M,2n,ds^ 2)$$ be a connected Kähler manifold of complex dimension n, $$(z,\bar z)$$ complex coordinates, D a bounded domain in $${\mathbb{C}}^ n$$, $$A(D)$$ the group of analytic diffeomorphisms g of D, and $$H_{\omega}(D)$$ the Hilbert space of analytic functions f on D with the scalar product defined by F. A. Berezin [Math. USSR, Izv. 9(1975), 341-379 (1976); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 39, 363- 402 (1975; Zbl 0312.53050), and ibid. 8(1974), 1109-1165 (1975), translation from Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1116-1175 (1974; Zbl 0312.53049)]. Let $$\{\phi_ K: K=0,1,2,...\}$$ be an orthonormal basis in $$H_{\omega}(D)$$ and let vectors $$\phi_{\bar z}\in H_{\omega}(D)$$ correspond to the point z.
The author shows that if D is irreducible and symmetric, these vectors are coherent states of a unitary irreducible representation of the connected Lie group $$A_ 0(D)$$, characterized by the value $$\omega$$. Let $$\hat A$$ and $$\hat B$$ be two bounded operators on $$H_{\omega}(D)$$, A and B the covariant symbols of $$\hat A$$ and $$\hat B,$$ defined by Berezin, and (A*B) the covariant symbol of the operator $$(\hat A.\hat B)$$. For the asymptotic expansion of A*B the author also uses the notation (*). His result is: the *-product is invariant by $$A_ 0(D)$$ if D is irreducible and symmetric. A recursion formula to calculate any 2-cochain in the *- product for spaces of type I and IV is obtained.
Reviewer: P.Stavre

MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53D50 Geometric quantization 32K15 Differentiable functions on analytic spaces, differentiable spaces 22E10 General properties and structure of complex Lie groups
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References:
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