On the block structure of regular unipotent elements from subsystem subgroups of type A × A in representations of the special linear group.

The behavior of regular unipotent elements from a subsystem subgroup of type A × A in p-restricted irreducible representations of a special linear group of rank greater than 5 over a field of characteristics p > 2 is investigated. For a certain class of such representations with locally small highest weights, it is shown that the images of these elements have Jordan blocks of all a priori possible sizes. In particular, the following is proved. Let K be an algebraically closed field of characteristics p, G = A( K), r ≥ 9, x ∈ G be a regular unipotent element from a subsystem subgroup of type A × A, and let φ be a p-restricted representation of G with highest weight $$ \sum\limits_{j = 1}^r {{a_j}{\omega_j}} $$. Set Assume that more than 6 coefficients a are note equal to p − 1 and that fo some i < r, a + a < p − 2 for p > 3 and a = a = 0 or 1 p = 3. Then the element φ( x) has Jordan blocks of all sizes from 1 to l. Bibliography: 30 titles. [ABSTRACT FROM AUTHOR]