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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

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
Full Text: DOI
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