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.


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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