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

Abstract

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 n with n>2 and p>2 are investigated. Let ω=∑ i=1 n m i ω i be the highest weight of a representation under consideration. If ω 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 n , the weight ω 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 n with n>3 and regular unipotent elements of naturally embedded A 2-subgroups, the weight ω 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.