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

AbstractImages 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 A2 in such representations of groups of type An with n>2 and p>2 are investigated. Let ω=∑i=1nmiωi 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 Bn and C2 and short roots where all such sizes congruent to mi+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 An and regular unipotent elements of naturally embedded subgroups of type A2 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 An, the weight ω is locally small if mi+mi+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 An with n>3 and regular unipotent elements of naturally embedded A2-subgroups, the weight ω is locally small if mi+mi+1+mi+2+mi+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