Orbits and Centralizers for Algebraic Groups in Small Characteristic and Lie Algebra Representations in Standard Levi Form

The purpose of this work is two-fold. First, we will explore what can be said about some particular conjectures concerning centralizers and orbits of algebraic groups when considering a ground field of small characteristic. Second, we attempt to understand non-restricted Lie algebra representations for standard Levi form by generalizing some existing machinery.;Specifically, in Chapter 2 we provide a proof of the existence of Levi decompositions of nilpotent centralizers in classical groups of bad characteristic. Then, in Chapter 3, we provide an initial approach to a conjecture of Steinberg in good characteristic related to understanding the orbits of an algebraic group by that of its faithful representations. This conjecture was previously known (due to Steinberg) in characteristic zero or "sufficiently large'', while our approach is valid for certain elements in almost good characteristic and provides a smaller restriction for the analogous case of certain elements in the Lie algebra. Finally, in Chapter 4 we generalize a construction of Jantzen in the special setting of standard Levi form. Here we study an important type of module called a baby Verma module and build its smaller parabolic analogue. It turns out that these both yield the same unique simple quotient.