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Invariant star products and representations of compact semisimple Lie groups. (English) Zbl 0681.53036
Starting from the work of F. A. Berezin [Izv. Akad. Nauk SSSR, Ser. Mat. 39, 363-402 (1975; Zbl 0312.53050)] an earlier paper by the author [Lett. Math. Phys. 11, 361-372 (1986; Zbl 0618.53049)] defined an invariant star product on every nonexceptional Kähler symmetric space. A recursion formula is obtained to calculate the corresponding invariant Hochschild 2-cochains for spaces of types II and III. An invariant star product is defined on every integral symplectic (Kähler) homogeneous space of simply-connected compact Lie groups (on every integral orbit of the coadjoint representation). The invariant 2-cochains are obtained from the Bochner-Calabi function of the space. The leading term of the \(\ell th\)-2-cochain is determined by the \(\ell\)-power of the Laplace operator.

MSC:
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C30 Differential geometry of homogeneous manifolds
22E46 Semisimple Lie groups and their representations
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