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 $\varepsilon_{(α,g(μ;ν))}$ of $\big(\big(\prod_{j=1}^kΩ_j\big)^{\mathbb{B}^{d_0}}(μ),g(μ;ν)\big)$, where $g(μ;ν)$ is a Kähler metric associated with the Kähler potential $-\sum_{j=1}^kν_j\ln N_{Ω_j}(z_j,\overline{z_j})^{μ_j}-\ln(\prod_{j=1}^kN_{Ω_j}(z_j,\overline{z_j})^{μ_j}-\|w\|^2)$ on the generalized Cartan-Hartogs domain $\big(\prod_{j=1}^kΩ_j\big)^{\mathbb{B}^{d_0}}(μ)$ and obtain necessary and sufficient conditions for $\varepsilon_{(α,g(μ;ν))}$ to become a polynomial in $1-\|\widetilde{w}\|^2$. Secondly, we study the Berezin quantization on $\big(\prod_{j=1}^kΩ_j\big)^{\mathbb{B}^{d_0}}(μ)$ with the metric $ g(μ;ν)$.