Limiting distributions of theta series on Siegel half-spaces

Let $m>1$ be an integer. For any $Z$ from the Siegel upper half-space we consider the multivariate theta series
$$ \Theta(Z)=\sum_{\bar n\in\mathbb Z^m}\exp(\pi i^t\bar n Z\bar n). $$
The function $\Theta$ is invariant with respect to every substitution $Z\longmapsto Z+P$, where $P$ is a real symmetric matrix with integral entries and even diagonal. Therefore, for any real matrix $Y>0$ the function $\Theta_Y(\cdot)=(\det Y)^{1/4}\Theta(\cdot+iY)$ may be viewed as a complex-valued random variable on the torus $\mathbb T^{m(m+1)/2}$ with the probability Haar measure. We prove that there exists a weak limit of the distribution of $\Theta_{\tau Y}$ as $\tau\to0$, and this limit does not depend on the choice of $Y$. This theorem is an extension of known results for $m=1$ to higher dimension. We also establish the rotational invariance of the limiting distribution. The proof of the main theorem makes use of Dani–Margulis' and Ratner's results on dynamics of unipotent flows.