Irina D. Suprunenko, 
Institute of Mathematics,  National Academy of Sciences of
Belarus,  Surganov str. 11, Minsk, 220072, Belarus

On an asymptotic behavior of elements of order $p$ in
irreducible representations of the classical algebraic groups with
large enough highest weights (to appear in Proceedings AMS).

\begin{abstract}
The behavior of the images of a fixed element of order $p$ in
irreducible representations of a classical algebraic group in
characteristic $p$ with highest weights large enough with respect
to $p$ and this element is investigated. More precisely, let $G$
be a classical algebraic group of rank $r$ over an algebraically
closed field $K$ of characteristic $p>2$. Assume that an element
$x\in G$ of order $p$ is conjugate to that of an algebraic group
of the same type and rank $m<r$ naturally embedded into $G$. Next,
an integer function $\sigma_x$ on the set of dominant weights of
$G$ and a constant $c_x$ that depend only upon   $x$, and a
polynomial $d$ of degree one are  defined.  It is proved that the
image of $x$  in the irreducible representation of $G$ with highest 
weight $\om$ contains more than $d(r-m)$ Jordan blocks of size $p$ 
if $m$ and $r-m$ are not too small and $\sigma_x(\om)\geq p-1+c_x$.
\end{abstract}