Rawnsley's $\varepsilon$-function on some Hartogs type domains over bounded symmetric domains and its applications

Abstract

The purpose of this paper is twofold. Firstly, we will compute the explicit expression of the Rawnsley's $\varepsilon$-function $\varepsilon_{(\alpha,g(\mu;\nu))}$ of $\big(\big(\prod_{j=1}^k\Omega_j\big)^{{\mathbb{B}}^{d_0}}(\mu),g(\mu;\nu)\big)$, where $g(\mu;\nu)$ is a K\"ahler metric associated with the K\"ahler potential $-\sum_{j=1}^k\nu_j\ln N_{\Omega_j}(z_j,\overline{z_j})^{\mu_j}-\ln(\prod_{j=1}^kN_{\Omega_j}(z_j,\overline{z_j})^{\mu_j}-\|w\|^2)$ on the generalized Cartan-Hartogs domain $\big(\prod_{j=1}^k\Omega_j\big)^{{\mathbb{B}}^{d_0}}(\mu)$ and obtain necessary and sufficient conditions for $\varepsilon_{(\alpha,g(\mu;\nu))}$ to become a polynomial in $1-\|\widetilde{w}\|^2$. Secondly, we study the Berezin quantization on $\big(\prod_{j=1}^k\Omega_j\big)^{{\mathbb{B}}^{d_0}}(\mu)$ with the metric $ g(\mu;\nu)$.