On an asymptotic behavior of elements of order $p$ in irreducible representations of the classical algebraic groups with large enough highest weights

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 $\omega$contains more than $d(r-m)$ Jordan blocks of size $p$ if $m$ and $r-m$ are not too small and $\sigma_x(\omega)\geq p-1+c_x$.