Small quadratic elements in representations of the special linear group with large highest weights

For almost all p-restricted irreducible representations of the group A_n(K) in characteristic p > 0 with highest weights large enough with respect to p, the Jordan block structure of the images of small quadratic unipotent elements in these representations is determined. It is proved that if φ is an irreducible p-restricted representation of A_n(K) with highest weight , not too few of the coefficients m_i are less than p − 1, and n is large enough with respect to the codimension of the fixed subspace of an element z under consideration, then φ(z) has blocks of all sizes from 1 to p. Bibliography: 15 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 343, 2007, pp. 84–120.