*-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
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