Off-diagonal estimates of the Bergman kernel associated to Siegel varieties

For $g\geq 2$, let $\Gamma\subset\mathrm{Sp}(2g,\mathbb{R})$ be a discrete subgroup, which is either a cocompact subgroup or an arithmetic subgroup without torsion elements, and let $\mathbb{H}_{g}$ denote the Siegel upper half space of genus $g$. Let $X_{\Gamma}:=\Gamma\backslash\mathbb{H}_{g}$ denote the quotient space, which is a complex manifold of dimension $g(g+1)/2$. Let $\Omega_{X_{\Gamma}}$ denote the cotangent bundle, and let $\ell:=\mathrm{det}(\Omega_{X_{\Gamma}})$ denote the determinant line bundle of $\Omega_{X_{\Gamma}}$. For any $Z,W\in X_{\Gamma}$, let $d_{\mathrm{S}}(Z,W)$ denote the geodesic distance between the points $Z$ and $W$ on $X_{\Gamma}$. \vspace{0.15cm}\noindent For any $k\geq 1$, let $H^{0}(X_{\Gamma},\ell^{\otimes k})$ denote the complex vector space of global sections of the line bundle $\ell^{\otimes k}$, and let $\|\cdot\|_{k}$ denote the point-wise norm on $\ell^{\otimes k}$. Let $\mathcal{B}_{X_{\Gamma}}^{\ell^{ k}}$ denote the Bergman kernel associated to $H^{0}_{L^{2}}(X_{\Gamma},\ell^{\otimes k})\subset H^{0}(X_{\Gamma},\ell^{\otimes k})$, vector subspace of $L^2$ global sections. For any $k\gg 1$, and $Z,W\in X_{\Gamma}$ , we derive estimates of the Bergman kernel $\|\mathcal{B}_{X_{\Gamma}}^{\ell^{ k}}(Z,W)\|_{\ell^{k}}$, when $\Gamma$ is a cocompact subgroup and when $\Gamma$ is an arithmetic subgroup.