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Petersburg Department of Steklov Institute of Mathematics

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PREPRINT 6/2001

FRIEDRICH G\"OTZE and MIKHAIL GORDIN
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EXISTENCE OF LIMITING DISTRIBUTION
FOR THE THETA SERIES ON THE SIEGEL HALF SPACE

This preprint was accepted May 30, 2001

Contact:
` M.Gordin `

ABSTRACT:
Let $m \geq 1$ be an integer. For any $Z$ from Siegel
upper half space set $$\Theta (Z)=
\sum_{{\overline n} \in \Bbb Z^m} exp(\pi i \hskip 0.05cm
{}^t {\overline n} Z {\overline n}).$$ The function
$\Theta$ is unchanged under every substitution $X$
$\longmapsto X + P$ where $P$ is a symmetric matrix with
integral entries and even diagonal. Therefore, for any $Y > 0$ the
function $\Theta_Y ( \cdot) = (\det Y)^{1/4} \Th (\cdot+iY)$ may be
viewed as a complex-valued random variable on the torus $\Bbb T^{m(m+1)/2}$
with the Haar measure. It is asserted in the main theorem of this note
that there exists a weak limit of the distribution of
$ \Theta_{\tau Y}$ as $\tau \to \infty$ which does not depend on the
choice of $Y$.
This theorem is an extension to higher dimension of some known results for
$m=1$. 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.

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