Is there a ready reference for explicit computation of the heat kernel for Siegel upper half space $\mathbb{H}_n=\{Z=X+iY\in \mathrm{Mat}_n(\mathbb{C}) \vert Y>0\} $? I could find it for general symmetric spaces along with a few explicit computations here: http://www.univ-orleans.fr/mapmo/membres/anker/recherche/survey.pdf but haven't come across the Siegel case yet.