On the Jordan block structure of images of some unipotent elements in modular irreducible representations of the classical algebraic groups

Images of root elements in p-restricted irreducible representations of the classical algebraic groups over a field of characteristic p > 0 and images of regular unipotent elements of naturally embedded subgroups of type A(2) in such representations of groups of type A, with n > 2 and p > 2 are investigated. Let omega = Sigma(i-1)(n) m(i)omega(i) be the highest weight of a representation under consideration. If omega is locally small with respect to p in a certain sense, the sizes of all Jordan blocks (without multiplicities) in the images of root elements are found, except the case of the groups of type B-n and C-2 and short roots where all such sizes congruent to m(i) + 1 modulo 2 are determined with the ith simple root being short; for p > 2 and n > 3, all odd dimensions of such blocks for groups of type A(n) and regular unipotent elements of naturally embedded subgroups of type A 2 are found. Here the class of locally small weights with respect to p depends upon the type of a group and upon elements considered. For root elements in a group of type A, the weight w is locally small if m(i) + m(i+1) < p - 1 for some i. For root elements in other classical groups, the definitions of the relevant classes are more complicated and depend upon the root length; however, in all these cases locally small weights are determined in terms of certain linear functions of their values on two simple roots linked at the Dynkin diagram of a group. For groups of type A, with n > 3 and regular unipotent elements of naturally embedded A(2)-subgroups, the weight omega is locally small if m(i) + m(i+1) + m(i+2) + m(i+3) < p - 2 for some i with i < n - 2. For arbitrary p-restricted representations, the presence of blocks of certain sizes in the images of elements indicated above is established. (C) 2004 Elsevier Inc. All rights reserved.