Special composition factors in restrictions of representations of special linear andsymplectic groups to subsystem subgroups with two simple components

Restrictions of $p$-restricted irreducible representations of special linear and symplectic groups in odd characteristic $p$ to subsystem subgroups of the maximal rank with two simple components are investigated. Our goal is to find in such restrictions composition factors which are big with respect to unipotent elements of one component and not very small for the other component. Here we call a composition factor $\psi$ of the restriction of a representation $\varphi$ to a subgroup big for a unipotent element $x$ of this subgroup if the minimal polynomials of the elements $\varphi(x)$ and $\psi(x)$ coincide. If the ranks of the simple components of the subgroups under consideration are not too small, we show that such factors exist for a wide class of representations and present some examples of representations whose restrictions have no such factors. For representations of symplectic groups with highest weights large enough with respect to $p$, we also find composition factors where for both components certain lower estimates for the values of their highest weights on the maximal roots hold. These results are applied to get lower estimates for the number of Jordan blocks of the maximal possible size in the images of certain unipotent elements in irreducible representations of special linear and symplectic groups. We emphasize that some of these elements do not lie in proper subsystem subgroups.