Berezin quantization on generalized flag manifolds - Cahen, Benjamin (MATHEMATICA SCANDINAVICA, 2009)
Let $M=G/H$ be a generalized flag manifold where $G$ is a compact, connected, simply-connected Lie group with Lie algebra $\mathfrak{g}$ and $H$ is the centralizer of a torus. Let $π$ be a unitary irreducible representation of $G$ which is holomorphically induced from a character of $H$. Using a complex parametrization of a dense open subset of $M$, we realize $π$ on a Hilbert space of holomorphic functions. We give explicit expressions for the differential $dπ$ of $π$ and for the Berezin symbols of $π (g)$ ($g\in G$) and $dπ (X)$ ($X\in \mathfrak{g}$). In particular, we recover some results of S. Berceanu and we partially generalize a result of K. H. Neeb.