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Berezin quantization on generalized flag manifolds. (English) Zbl 1183.22006
Let $$M=G/H$$ be a homogeneous manifold, where $$G$$ is a compact, connected, simply-connected Lie group with Lie algebra $$\mathfrak{g}$$ and $$H$$ is a centralizer of a torus. Then the generalized flag manifold $$M$$ is homogeneous Kählerian and algebraic [A. Borel, Proc. Natl. Acad. Sci. USA 40, 1147–1151 (1954; Zbl 0058.16002)]. Let $$\pi$$ be a unitary irreducible representation of $$G$$ induced from a character $$\chi$$ of $$H$$ on a Hilbert space $$\mathcal{H}$$, realized as a space of holomorphic functions defined on a dense open subset $$D$$ of $$M$$ (global sections of the $$G_{\mathbb{C}}$$-homogeneous holomorphic line bundle $$L_{\chi}$$ associated by means of the character $$\chi$$ to the principal $$H$$-bundle). The author calculates the derived representation $$d\pi(X)$$ ($$X\in\mathfrak{g}_{\mathbb{C}}$$) and the Berezin symbol of $$\pi(g)$$ ($$g\in G$$). The main tool is provided by a proposition which expresses the reproducing kernel $$K(z,w)$$ ($$z,w\in D\subset M$$) of the space $$\mathcal{H}$$ as a function of the character $$\chi$$ and two projection operators. Similar formulas are known in the larger context of the realization of highest weight representations on complex domains; see Chapter XII in the book [K.-H. Neeb, Holomorphy and convexity in Lie theory. de Gruyter Expositions in Mathematics 28, Berlin: de Gruyter (1999; Zbl 0936.22001)]. Proposition 5.1 in the paper under review gives an expression of $$d\pi(X)f(z)$$ ($$X\in\mathfrak{g}_{\mathbb{C}}, f\in\mathcal{H}$$) as a sum of two terms, the first one containing $$f(z)$$, multiplied by a function $$P$$ of $$\chi$$, the second one containing the differential $$df(z)$$ times a function $$Q$$. In the case of an abelian algebra $$\mathfrak{n}^+$$ which appears in the Gauss decomposition $$\mathfrak{g}_{\mathbb{C}}={\mathfrak{h}}_{\mathbb{C}}\oplus {\mathfrak{n}}^+\oplus{\mathfrak{n}}^-$$, B. Cahen proves that $$P,Q$$ are polynomials, as was also proved in Proposition XII.2.1 in the quoted book of Neeb for quasihermitian algebras (see definition VII.2.15 in the book of K.-H. Neeb). B. Cahen also mentions that he has recovered a result obtained in the paper [S. Berceanu, Realization of coherent state Lie algebras by differential operators, Boca, Florin-Petre (ed.) et al., Advances in operator algebras and mathematical physics, Proceedings of the 2nd conference on operator algebras and mathematical physics, Sinaia, Romania, June 26–July 4, 2003. Bucharest: Theta, Theta Series in Advanced Mathematics 5, 1–24 (2005; Zbl 1212.81012)] in the case of the coherent state Lie algebras (see the definitions in Chapter XV of the quoted book of K.-H. Neeb), where explicit formulas are presented for the polynomials $$P,Q$$ involving the Bernoulli numbers and the structure constants for semisimple Lie algebras.

##### MSC:
 22E46 Semisimple Lie groups and their representations 81S10 Geometry and quantization, symplectic methods 53B35 Local differential geometry of Hermitian and Kählerian structures 81R30 Coherent states 53D50 Geometric quantization
##### Citations:
Zbl 0058.16002; Zbl 0936.22001; Zbl 1212.81012
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