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

Abstract

For almost all p-restricted irreducible representations of the group An(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 An(K) with highest weight $$m_1 \omega _1 + ... + m_n \omega _n ,\sum\limits_{i = 1}^n {m_i \geqslant p - 1} $$ , not too few of the coefficients mi 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.