Abstract.We use Berezin's dequantization procedure to define a formal *-product on a dense subalgebra of the algebra of smooth functions on a compact homogeneous Kahler manifold M. We prove that this formal »-product is convergent when M is a hermitian symmetric space.
IntroductionIn part I of this paper [7] we showed how to quantize certain compact Kahler manifolds (M, co, J). This means the following: Let (L, V, h) be a quantization bundle over M (i.e., a holomorphic line bundle L with connection V admitting an invariant hermitian structure h, such that the curvature is curv(V) = -2inoe).Let %? be the Hilbert space of holomorphic sections of L. To any linear operator A on %? is associated a symbol  which is a real analytic function on M. Denote by Ê(L) the space of these symbols. For any positive integer k , (Lk = ®fc L, V(A:), A(/c)) is a quantization bundle for (M, kco, J). If Xfc is the Hilbert space of holomorphic sections of Lk , we denote by Ê(Lk) the space of symbols of linear operators on %?k . If, for every k, a certain characteristic function e(/c) (which depends on L and k and which is real analytic on M) is constant, the space Ê(L') is contained in the space Ê(Lk) for any k > I. Furthermore U/^i Ê(Ll) (denoted by fé¿) is a dense subspace of the space of continuous functions on M. Any function / in ^l belongs to a particular Ê{Ll) and is thus the symbol of an operator AÍ ' acting on ^k for k > I. One has thus constructed, for a given /, a family of quantum operators parametrized by an integer k .From the point of view of deformation theory [1], where quantization is realised at the level of the algebra of functions, one can say that one has constructed a family of associative products on Ê(L'), with values in 8¿, parametrized by an integer k ; indeed f*kg = AfÀf, f,g£Ê(Ll),k>l.
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