Limiting distributions of theta series on Siegel half-spaces

Abstract:

Let $m \ge 1$ be an integer. For any $Z$ in the Siegel upper half-space we consider the multivariate theta series \begin{equation*}\Theta (Z)= \sum _{{\overline {n}} \in \mathbb {Z}^{m}} \exp (\pi i {}^{t} {\overline {n}} Z {\overline {n}}).\end{equation*} 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 $\Theta _{Y} ( \cdot ) = (\det Y)^{1/4} \Theta (\cdot +iY)$ can be viewed as a complex-valued random variable on the torus $\mathbb {T}^{m(m+1)/2}$ with the Haar probability measure. It is proved that the weak limit of the distribution of $\Theta _{\tau Y}$ as $\tau \to 0$ exists and does not depend on the choice of $Y$. This theorem is an extension of known results for $m=1$ to higher dimension. Also, the rotational invariance of the limiting distribution is established. The proof of the main theorem makes use of Dani–Margulis’ and Ratner’s results on dynamics of unipotent flows.