Orbits and Centralizers for Algebraic Groups in Small Characteristic and Lie Algebra Representations in Standard Levi Form.
Abstract: 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 Chapt... read moreer 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
Thesis (Ph.D.)--Tufts University, 2015.
Submitted to the Dept. of Mathematics.
Advisor: George McNinch.
Committee: Montserrat Teixidor i Bigas, Richard Weiss, and Eric Sommers.
Keyword: Mathematics.read less