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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
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