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 is an element of 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) greater than or equal to p-1 + c(x).